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]]>Publication on International Journal of Biology, Physics & Mathematics

Publication Date: March, 2020

**Abiodun O. Ajibade & Bolaji A. Shehu**

Department of Mathematic, Ahmadu Bello, University, Zaria

Department of Mathematics and Statistics, Kaduna Polytechnics, Kaduna

Nigeria

Journal Full Text PDF: The Approximate Solution of MHD Natural Convection Flow with Variable Properties Induced Magnetic Field, Viscous Dissipation and Ohmic Heating.

Abstract

This study is about the natural convection flow of a fluid that conducts electricity in a vertical channel with induced magnetic field, viscous and Ohmic heating together with variable thermal conductivity and variable viscosity. A transverse magnetic field is employed on the fluid through the plates. One of the plates is subjected to an isothermal condition. While the other plate remains at a temperature equal to zero. The dimensional governing equations are altered (using suitable transforms) into dimensionless coupled system of ordinary differential equations. These equations are nonlinear and hence solved numerically using the differential transform method (DTM). The consequences of varying the Eckert number, magnetic parameter, variable viscosity, variable thermal conductivity and the other relevant dimensionless parameters involved in the study on the velocity, temperature, magnetic field and the current density are observed. It is obtained that increase in the viscous dissipation of the fluid results in the increase in its velocity, temperature, current density and the heat transfer. While it decreases the magnetic field profile. In addition, the skin friction on both plates grows and the heat transfer increases as the magnetic parameter decreases.

Keywords: Viscous dissipation; Ohmic heating; induced magnetic field; variable fluid properties; differential transform method (DTM).

1. INTRODUCTION

Fluid flow through natural convection is caused entirely by buoyancy forces, which arise from density variations in a field of gravity. Unlike in forced convection where the velocity of flow of the fluid does not depend on temperature difference, in natural convection, the distribution of velocity and temperature are interconnected (coupled) and always considered together. If the fluid is incompressible then the density variation due to changes in pressure are negligible. However, density changes due to non-uniform heating of the fluid cannot be neglected. Since such changes are responsible for initiating free convection. An interesting feature of magneto-hydrodynamic heat transfer problems is that, the usual Reynolds analogy between skin friction and heat transfer, as in non-conducting fluids, does not, in general, hold good. This is because, in addition to viscous heating, there is a Joule heating due to the flowing electric current in the fluid.

Studies involving viscous dissipation and Ohmic heating on MHD natural convection flow of an electrically conducting fluid in a vertical channel with induced magnetic field is attracting a lot of interest because of its importance in the design of various industrial devices, which are subjected to large variation of gravitational force. Its application is found in heat exchangers designs, wire and glass fiber drawing and in nuclear engineering in connection with the cooling of reactors. Furthermore, Ohmic heating processes makes it possible to use high temperature at short time on suspended materials. This results in increase in the quality of products. A lot of researches considering different aspect of natural convection flow of fluids with viscous dissipation and Ohmic heating in different channels with the fluid being electrically conducting or not; and with or without magnetic field considerations were conducted recently.

Some of the works in this regard include the study by Jha and Ajibade (1) on the Effect of viscous dissipation on natural convection flow between vertical parallel plates with time-periodic boundary conditions. They solved the governing equations analytically and discovered that heat is being transferred from the fluid with viscous dissipation to the plate when the fluid has small Prandtl. Parveen and Alim (2) used implicit finite difference method of solution to the governing nonlinear differential equation as a tool to studied Joule Heating effect on Magneto-hydrodynamic Natural Convection Flow along a Vertical Wavy Surface. They observed that the effect of increasing Joule heating parameter results in the boundary layers becoming thicker; decrease in the local rate of heat transfer and increasing the local skin friction coefficient. The effects of hall current and viscous dissipation on MHD free convection fluid flow in a rotating system are studied by Abdul Quader and Alam (3). They deduced that the fluid temperature increases due to increase in the Eckert number. While it decreases due to increase in the prandtl number after solving the governing equation using the explicit finite difference scheme. Alam et al (4) used the finite difference method together with Newton’s linearization approximation to discuss the conjugate effects of viscous dissipation and variable viscosity on free convection flow over a sphere with joule heating and heat conduction. They argued that Increase in the values of viscous dissipation parameter enhances the velocity profile, the temperature profile and the local skin friction coefficient; while it decreases the heat transfer rate. That an enhanced Joule heating causes an increase in both the velocity and temperature profiles. Lakshmi et al (5), studied thermal energy increase and dissipation on MHD heat and mass concentration gradient flow over a plate. They employed the Crank-Nicolson’s implicit finite difference scheme to solve the governing differential equations and deduced that temperature of the fluid increases as the viscous dissipation parameter increases. Kabir et al (6) investigated the influence of viscous dissipation and thermal energy increase on free convection flow down a wavy surface. The equation of the problem was solved numerically by the keller-box method and they concluded that the velocity, temperature of the fluid and its local skin friction coefficient increases with increasing value of the viscous dissipation; whereas the rate of heat transfer decreases. Jaber (7) investigated the viscous dissipation and joule heating effects on the flow of a magneto-hydrodynamics fluid having variable properties past a vertical plate. He solved the governing equations numerically and deduced that the velocity increases, the temperature increases, the shear stress and the heat transfer increases at the wall as the viscous dissipation and the thermal conductivity parameters increase. Haque et al (8), worked on the effects of viscous dissipation on MHD natural convection flow over a sphere with temperature dependent thermal conductivity with heat generation. From this work, they were able to discover that the velocity and the temperature of the fluid improved with increasing thermal conductivity, increasing heat generation and viscous dissipation parameters. That skin friction increases with increasing viscous dissipation, heat generation and the thermal conductivity variation parameter. In another study, Nath and Parveen (9) used Keller-Box method to work on viscous dissipation and heat absorption effect on natural convection flow with uniform surface temperature down a wavy surface. In this work they concluded that increase in the viscous dissipation has increasing effect on the skin friction and decreasing effect on the rate of heat transfer. Emad et al (10) analysed the influence of dissipation and resistive heating on MHD free convection flow over a flat plate with the combined effect of Hall currents for the case of power-law variation of the wall temperature. They were able to do this by solving the transformed governing equation using a fifth-order Runge–Kutta–Fehlberg scheme with Newton–Raphson shooting method and concluded that the presence of dissipation and resistive heating decreases the local Nusselt number. Kishore et al (11) discussed the result of heat exchange and viscous dissipation on MHD temperature dependent natural convection flow over a vertical plate. They solved the governing equations numerically by the implicit finite difference method of Crank – Nicolson’s type, and discovered that velocity increases with increase in the thermal Grashoff number, acceleration parameter, Eckert number and time

Induced magnetic field plays a great role in the flow of fluids, which conducts electricity because it also produces additional magnetic field in the flowing fluid. This magnetic field generates a resistive force on the fluid flow and as a result, it modifies the original magnetic field. Therefore, in many concrete situations it is important to consider the influence of magnetic induction in the magneto-hydrodynamic equations. In the studies above, the influence of magnetic induction has been neglected so as to simplify the solution method of the problem. On account of this, Sarveshanand and Singh (12) discussed Magneto-hydrodynamic natural convection along vertical porous plates in which the induction of magnetic field is considered. They observed that the magnetic field induced in the fluid is significantly increased when the suction parameter, Prandtl number and the Hartmann number are increased. Prakash et al (13) in their study of MHD mixed convective flow over vertical plate where heat is generated due to viscosity and magnetization considered the influence of magnetic induction. And it is found that intensity of the induced magnetic field reduces when the quantity of the magnetic parameter, magnetic prandtl number and the viscous dissipation increases. Magnetic induction was also considered by Raju et al (14) in their work on the effect of heat transfer on a flowing fluid which dissipates heat due to viscous actions of its molecules past a vertical plate. They used a perturbative technique to solve the governing equations and deduced that the induced magnetic field decreases as the magnetic prandtl number is increased.

This study considers the influence of induced magnetic field and the heat generated due to the passage of the conducting fluid, through the magnetically induced vertical channels and that generated due to the interaction of the molecules of the viscous fluid on the MHD free convective flow with variable viscosity and thermal conductivity. The solution of the transformed governing boundary value equations involving the velocity, temperature and the induced magnetic fields are obtained by the differential transform method. Also, the expression for the electric current induced, the skin friction and the thermal exchange rate in terms of the Nusselt number are obtained. The corresponding results displayed in graphs and tables show that the Joule heating, viscous dissipation and the magnetic parameter have significant effects on the fluid flow considered in this study.

2. A GENERAL DESCRIPTION OF THE DIFFERENTIAL TRANSFORM METHOD (DTM)

Given a function y(x) its differential transform is defined by Zhou (15) as follows:

The inverse of which is defined as equation (2) below

In Equations (1) and (2), y(x) is the given function and Y (k) is the transformed function.

From the two Equations, we obtain equation (3) below.

From equation (3) it can be deduced that the notion of DTM is obtained from Taylor series expansion, but the differential transform method does not evaluate the derivatives symbolically as does in Taylor’s series. Rather, the relative derivatives are calculated by an iterative method. Following Zhou (15) and Ahmad et al (16) the basic operations of the DTM is defined as shown on table (1) below

Table 1. Basic operations of the one-dimensional differential transform method

In this study the given function is represented by the lower letter case and upper letter case represents the transformed function. The basic operations on table 1 are proved using equations (1) and (2). In practice, the function y(x) is written as a finite series using equation (2) in the form of equation 4 below

(4)

The value of n is decided by the convergence of the natural frequency in this work. Also, the sum of further terms written as the equation (5) below is taken to be negligibly small.

3. DISCRIPTION OF THE PROBLEM AND THE GOVERNING EQUATIONS

Consider a steady natural convection boundary layer flow of a viscous incompressible and electrically conducting fluid with induced magnetic field between two vertically parallel and infinitely long plates as illustrated in figure1 below.

Figure 1. Geometry of the problem

Taking and as the components of the velocities (along the plates) and (normal to the plates), respectively and h the distance between the two permeable vertical plates (figure 1). Then, keeping one of the plates at constant heat flux ( ) and the other maintained at constant temperature , so that for the steady incompressible boundary layer flow, the governing equations for the continuity, momentum and energy equations can be written as in Sarveshanand and Singh (12) adding the viscous dissipation and the Ohmic heating terms to the Temperature equation shown below.

With the boundary conditions

and k* represents the variable viscosity and variable thermal conductivity respectively. The fluid density is represented by ρ; Cp is the specific heat of the fluid at constant pressure. Because of the fluid’s electrical conductivity (σ) a magnetic field is induced as the fluid moves along the Equation (5) above is the expression for the velocity field with the temperature dependent viscosity taken as an inverse function of temperature following Singh and Agarwal (17) as

(10)

This is equivalent to equation (12) below

Equation (6) is the expression for the induced magnetic field and equation (7) is the expression for the Temperature field with the viscous dissipation and Joule heating terms. The variable thermal conductivity taken as an inverse function of temperature following Hazarika and Konch

Where a, b and T0 are constants and their values depend on the fluid’s reference state and its heat conductivity properties. That is the kinematic viscosity (ʋ) and thermal conductivity (k). Also, for liquids a > 0; b > 0 and for gases a < 0; b < 0.Where the dynamic viscosity and thermal conductivity of the baseline fluid (ambient fluid) are µ0 and k0 respectively.

From equation (11) let λ = aΔT which is the viscosity variation parameter. It is obvious that it increases with increasing temperature. So that equation (11) becomes equation (14) below

From equation (14), it is clear that the value of the dimensionless viscosity decreases with increasing temperature. That is when λ > 0. Since increasing temperature decreases the viscosity of liquids; while it increases that of gases, λ is represented with different signs for both. Likewise the same argument can be put forward for the dimensionless thermal conductivity presented in equation (15).

Where ε = bΔT is taking as the thermal conductivity variation parameter; which is taken to be negative in liquids and positive in gases.

The non-dimensional parameters given in equations (17a) and (17b) are employed to change the governing equations (5) to (9) to the non-dimensional form.

Hence, the nondimensional systems of nonlinear ordinary differential equations of the second order (18) to (22) are obtained.

Where Ha is the Hartmann number, Pm is the magnetic Prandtl number, Pr is the prandtl number, Ec is the Eckert number and M is the magnetic parameter.

4. SOLUTION BY DIFFERENTIAL TRANSFORM METHOD

In order to solve the non-linear governing continuity, momentum and energy equations, the equations (5) to (9) are reduced into the equations (18) to (22) using the set of non-dimensional parameters (17a) and (17b). The equations obtained constitute a system of nonlinear-coupled second order ordinary differential equations. These are solved numerically by using the Differential Transform Method (DTM). By which we obtain the iterative relations (23) to (26) below for the respective equations (18) to (22) in the system.

Also, applying DTM to the boundary conditions gives the expressions (26) below

U (0) = 0; U (1) = a; T (1) = -1; T (0) = c; B (0) = 0; B (1) = d. (26)

The expressions U (K), B (K) and T (K) are the differential transforms of the velocity equation U(y), the induced magnetic field equation B(y) and the temperature equation T(y). The constants ‘a, c and d’ are values from the boundary conditions and will be determined. Using the symbolic software Maple (21) we obtain the following solutions:

The solution of the induced current density equation is given by equation

The skin friction is given by the equation (31). The solution of which is given by equation (32); and those at y = 0 and y = 1 are given by the respective equations (33) and (34).

5. RESULTS AND DISCUSSIONS

A semi-analytic technique called the Differential Transform Method is used to solve the equations of free convection flow of an electrically conducting fluid in a vertical channel with induced magnetic field, viscous dissipation and Ohmic. For the purpose of discussing the results, the approximate solutions are obtained for the various flow parameters, which includes the Eckert number (Ec), the magnetic parameter (M), the magnetic Prandtl number (Pm), the thermal conductivity parameter(ε), the Prandtl number ( Pr ), the Hartmann number ( Ha ) and viscosity parameter( λ ). For these parameters, various computations have been carried out with values in the following range: , , , , . By so doing, relations for the velocity

profile, magnetic field profile, temperature profile, skin friction, the Nusselt number (which measures the heat flux) and the induced current density are obtained. As established in the work of Oke (19) the convergence of the iteration process is quite rapid. The effects of the flow quantities which affects the velocity field (U), the magnetic field (B), the temperature field (T) and the current density profile (J) are discussed with the help of the figures (2), (3), (4) and (5) respectively. Also, the effects of the relevant parameters on the skin friction (τ) is shown in table( 2) and on the Nuselt number (Nu) ( or the rate of heat transfer ) is shown in tables (3). Table (4), is used to compare the result of the present work with that f Sarveshnand and Singh (12).

Figure 2, Velocity profile varying the parameters (a) Ec, (b) M, (c) λ, (d) ε, (e) Pm, (f) Ha (g) Pr

In figure 2(a), it is seen that the velocity profile is increases as the Eckert number increase. Eckert number is the viscous dissipation parameter, so an increase in it means the dissipative force increases, which energize the fluid. Hence, the increase in its velocity profile. Figure 2(b) depicts the velocity variations for varying values of (M). It shows that as the magnetic parameter decreases the velocity increases. Magnetic parameter depicts the ratio of magnetic induction to the viscous force. Also, figure 2(c) shows an enhanced velocity profile as the viscosity parameter (λ) increase. This is expected since the viscosity parameter bears an inverse relationship with the viscosity of the fluid. The velocity is seen to be enhanced with decreasing thermal conductivity parameter as shown in figure 2(d). Increased velocity profile is also observed, as Pm is decreased shown in figure 2(e). When Pm is reduced, the strength of the magnetic field reduces which reduces the development of the flow resistive Lorentz force. This improves velocity of the flow. From figure 2(f) it is seen that decrease in Ha results in increase of the velocity of the flow. As the Hartmann number decreases, the viscous force becomes stronger than the magnetic force of the fluid thereby limiting the resistive influence of the Lorentz force. Hence, an enhanced motion of the fluid is achieved. Figure 2(g) shows that decreasing the prandtl number results in increase in the velocity of the fluid. This is because heat diffuses more readily than momentum.

Figure 3. Induced magnetic field profile varying the parameters (a) Ec (b) M (c) λ (d) ε (e) Pm (f) Ha (g) Pr.

In figure 3(a) it is seen that as the Ec increases, Pm is decrease. Increase in the Eckert number implies greater viscous dissipative heat, which is a consequence of work done against viscous stresses, which causes the reduction of the magnetic field. Figure 3(b) show that as the magnetic parameter increases the magnetic field also increases. This happens because increasing the magnetic parameter makes the magnetic induction in the fluid to be greater than the viscous force. Figure 3(c) shows that as the viscosity parameter decreases, the magnetic field increases. It is observed in figure 3(d) that increase in the thermal conductivity parameter causes the induction of magnetic field to increase. In figure 3(e) as Pm decreases, the magnetic induction is enhanced. Figure 3(f) depicts that as the Hartmann number increases, the magnetic induction decreases. From figure 3(g) it can be seen that as the prandtl number increases, the magnetic field decreases. All these observations agree with what is obtained in literature.

Figure 4, Temperature profile varying the parameters (a) Ec (b) ε (c) M (d) Pr.

The rise in temprature depicted in figure 4(a) is as a result of increasing the Eckert number (Ec). Increasing the Eckert number results in greater viscous dissipative heat which helps in raising the temperature profile. Figure 4(b) depicts that a rise in the magnetic parameter (M) results in the decrease of temperature. The Lorentz force developed due to the increase of M reduces the fluid velocity leading to the decrease of the temperature. Figure 4(c) shows that as the thermal conductivity parameter increases the temprature of the fluid decreases. This is borne from the fact that thermal conductivity affects the molecular diffusion of the fluid. Similarly, as Pr increases shown by figure 5, the temperature profile decreases. When the prandtl number is increased, the molecular movement of the fluid is reduced causing a drop of temperature in the flow field.

Figure 5, Induced current profile varying Magnetic Prandtl number (Pm)

Figure 20 shows that as the Eckert number increases, the induced current density also increases. When the dissipative force is increase as the Eckert number is being increased the fluid flows faster. This generates greater induced current due to the presence of the transverse magnetic field in the flowing fluid. In figure 21, we can see that as the magnetic parameter is increased, the induced current profile reduces. A growing magnetic parameter retards the velocity of the flow due to the magnetic pull caused by the Lorentz force which develops in the flow field. A consequence of the reduced velocity of a conducting fluid within transverse magnetic field is the reduced generation of induced current density. It is observed in figure 22 that as the viscosity decreases, the induced current (J) profile increases. This happens because the electrically conducting fluid is moving faster in the flow field, which contains magnetic fields. Figure 23 depicts that as the thermal conductivity parameter increases, J decreases. In figure 24 it is observed that as the prandtl number increases, the J also increases. As the Hartman number increases shown in figure 25, the J increases too. The induced current density increases when the Hartmann number (Ha) increases. Increase in Ha occurs when the electromagnetic force increases against the viscous force. This causes a faster motion of the fluid within the channel and resulting into greater induced current. Finally, in figure 26 it is seen that as the Pm decreases, the current also decreases.

Table 2 below shows the influence of the physical parameters namely Eckert number, magnetic parameter, viscosity variation parameter, thermal conductivity variation parameter, Hartmann number, magnetic prandtl number and the prandtl number on the coefficient of skin-friction which represent the shearing stress at the two plates. Table 3 represents the values of the heat transfer coefficient called the Nusselt number (Nu) for the various relevant parameters. Table 4 shows a comparison of the present work with that of Sarveshanand and Singh (12) with constant viscosity and thermal conductivity in the absence of viscous dissipation and Ohmic heating.

Table 2, The effect of (a) Ec and M (b) λ and ε (c)Ha and Pm (d) Pr, on heat transfer coefficient Nu

It is observed in table 2(a), that as the Eckert number (Ec) increases the skin friction (τ) also increases in the first plate; but in the second plate it decreases between the range 0.001 to 4.5 and increases between the ranges 4.5 to 8.0. In the same table as the magnetic parameter (M) increases, τ decreases on both plates. Table 2(b) shows how the viscosity parameter (λ) and the thermal conductivity parameter (ε) affect the skin friction around the two plates. The result is that decreasing (λ) decreases the skin friction on both plates and as ε increases, the fluid does not get heated easily which affect the velocity negatively. Hence, the result is a decrease in the skin friction as is shown in the table. Table 2(c) shows the effects of Hartmann number (Ha) and the Magnetic Prandtl number (Pm) on the skin friction. By increasing the (Ha) we observe the decreasing of the skin friction on both plates. Decrease in Pm number, results in increase in the skin friction on both plates. This is the case because increasing Ha and Pm reduces the velocity of the fluid due to Lorentz force pulling effect. Table 2(d) shows that increasing the prandtl number the skin friction on the first plate increases. Whereas it decreases from 0.7 to 4.0 and it increases from 4.0 to 6.0 on the second plate. This is because heat spreads away from the wall faster for higher values of Pr. Whereas, for smaller Pr the thermal boundary layer thickens increases hence; the rate of heat transfer is reduced. This is in agreement with the result obtained by Dada and Oladesusi (20).

Table 3, The effect of (a) Ec and M (b) λ and ε (c)Ha and Pm (d) Pr, on heat transfer coefficient Nu.

The effects of the Eckert number and the magnetic parameter on the heat transfer, taking place in the fluid are displayed in table 3(a). It is observed that as the Eckert number increases the Nuselt number increases but it behaves differently with the increase of the magnetic parameter. In table 3(b), as the viscous variation parameter (λ) increases negatively between – 0.005 to – 0.5 the Nuselt number increases but it decreases between the values – 0.5 to -1.0. Increase in the values of thermal conductivity parameter (ε) results in the decrease of the Nusselt number. The effect of Ha and the magnetic prandtl number on the Nusselt number shown on table 3(c). It shows that as Ha increase the Nu increases whereas the Nusselt number decreases as the magnetic prandtl number increases. Prandtl number affects the Nu in the same direction as shown on table 3(d).

Table 4, Comparison of the present work with Sarveshanand and Singh (12).

Table 4 shows the results of the comparison of the present work with that of Sarveshanand and Singh (12). We let Ec = 0, M = 0 which means that there are no effects of viscous dissipation and Joule heating. λ = ε = 0 means that the viscosity and thermal conductivity are constants. The table shows that the results agree to a very appreciable degree.

6. CONCLUSION

The presence of Joule heating, viscous dissipation as well as the magnetic parameter has significant effects on the free convection flow of a conducting fluid in a vertical channel with induced magnetic field just as it is presented in this study. The main findings together with the effects of various other paramount parameters on the velocity field, induced magnetic field, temperature field, induced current density, skin friction profiles and Nusselt number has been depicted in graphs and tables as seen above. It is found that:

(1) Increased dissipation in the fluid, results in the increase in its velocity, temperature, current density and the thermal exchange rate. While it decreases the magnetic induction. Furthermore, increasing the viscous dissipation of the fluid leads to increase of the skin friction in the first plate while it fluctuates in the second plate between some ranges.

(2) Decreasing the ratio of magnetic induction to the viscous force (magnetic parameter) in the fluid leads to increase in the velocity, the temperature of the fluid and the quantity of current induced in the fluid. Also, the drag caused by friction on both plates grows and the heat exchange rate increases as the magnetic parameter decreases. The magnetic induction of the fluid is directly related to the magnetic parameter.

(3) Enhanced velocity and electric current induction results from the reduction of the fluid viscosity. While increased induced magnetic field profile and reduced skin friction in both plates occurs as the viscosity of the fluid increases. In addition it is observed that the heat exchange rate increases at lower viscosity and decreases at higher viscosity.

(4) The temperature of the fluid and its induced current density decreases as the thermal conductivity parameter increases. So also the thermal exchange rate and the skin friction on both plates decrease as the thermal conductivity parameter increases. Whereas the velocity of the fluid increases as the thermal conductivity parameter decreases. While the magnetic induction increases with increasing thermal conductivity parameter.

(5) When the momentum diffusivity is lower than the magnetic diffusivity, the skin friction on both plates, the velocity of the fluid and the magnetic induction are enhanced; the induced current density is reduced. While the thermal exchange rate also decreases with decreasing magnetic prandtl number

(6) The velocity and the skin friction enhances with decreasing Hartmann number. The magnetic induction decreases as the Hartmann number increases. While the current produced by induction and the heat exchange rate increases as the Hartmann number increases.

(7) Also, the fluid velocity is enhanced when the prandtl number decreases. For increasing prandtl number the magnetic induction and the fluid temperature decreases. Induced current and the rate at which heat is transferred in the fluid have a direct variation with the prandtl number. Skin friction grows with increasing prandtl number on the first plate but it decreases for smaller values and increases for bigger values on the second plate.

The numerical results of this work, are compared with previously published ones, for some special cases and found to be in excellent agreement. It is hoped that the present results be used for understanding more complex problems dealing with natural convection flow of an electrically conducting fluid in a vertical channel with induced magnetic field, viscous dissipation and Ohmic heating.

7. Acknowledgements

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

8. Declaration of Conflicting Interests.

The Authors declare that there is no conflict of interest

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]]>The post Review on Use of Molecular Markers for Characterizing and Conserving of Plant Genetic Resources appeared first on Zambrut.

]]>Published on International Journal of Agriculture & Agribusiness

Publication Date: January, 2020

**Abdi Alemi Ketema**

Department of Plant Science, Faculty of Agriculture and Veterinary Science, Ambo University

P. O. Box 19, Ambo, Ethiopia

Journal Full Text PDF: Review on Use of Molecular Markers for Characterizing and Conserving of Plant Genetic Resources.

Abstract

Molecular markers have revolutionized and modernized our ability to characterize genetic variation and to rationalize genetic selection, being effective and reliable tools for the analysis of genome architectures and gene polymorphisms in crop plants. The area of plant genomics that has shown the greatest development with respect to the use of molecular marker technology is that of population genetics. All DNA polymorphism assays have proven to be powerful tools for characterizing and investigating Germplasm resources, genetic variation and differentiation of populations, on the basis of gene diversity and gene flow estimates. As a matter of fact, the number of loci for which DNA-based assays have been generated has increased dramatically, the majority using PCR as methodology platform. The information acquired is now being exploited to transfer different traits, including biotic stress resistances and improved quality traits, to important varieties by means of marker-assisted selection (MAS) programs. The most important challenges in the near future are certainly the molecular characterization of Germplasm collections for preserving them from genetic erosion and the identification of phenotypic variants potentially useful for breeding new varieties. Knowing the presence of useful traits, genes and alleles would help in making decisions on the multiplication of plant accessions and the maintenance of seed stocks. There are no doubts that the use of molecular markers for characterization and conservation of genetic resources should be implemented so that potentially useful genes and genotypes can be added to core collections to make them exploitable by breeders.

Keywords: Molecular markers, DNA markers.

1. INTRODUCTION

Molecular markers have proven to be powerful tools for analyzing germplasm resources and assessing genetic variation within as well as genetic differentiation among populations. In fact, the area of plant genomics that has shown the greatest development with respect to the use of DNA marker technology is that of population genetics. However, both RFLP and PCR-derived markers have also been extensively applied in plant genetics and breeding for mapping Mendelian genes and QTLs. The use of molecular markers for investigating and managing genetic resources should be implemented so that useful information on genes and traits can be added to core collections to make them exploitable by breeders (Barcaccia, 2009).

Conservation of genetic resources entails several activities, many of which may greatly benefit from knowledge generated through applying molecular marker technologies. This is the case for activities related to the acquisition of germplasm (locating and describing the diversity), its conservation (using effective procedures) and evaluation for useful traits. In all, the availability of sound genetic information ensures that decisions made on conservation will be better informed and will result in improved germplasm management. Of the activities related to genetic resources, those involving germplasm evaluation and the addition of value to genetic resources are particularly important as they help identify genes and traits, and thus provide the foundation on which to enhance use of collections. ‘Characterization’ is the description of a character or quality of an individual (Marriem 1991).

The word ‘characterize’ is also a synonym of ‘distinguish’, that is, to mark as separate or different, or to separate into kinds, classes or categories. Thus, characterization of genetic resources refers to the process by which accessions are identified or differentiated. This identification may, in broad terms, refer to any difference in the appearance or make-up of an accession. In the agreed terminology of gene banks and germplasm management, the term ‘characterization’ stands for the description of characters that are usually highly heritable, easily seen by the eye and equally expressed in all environments (IPGRI/CIP. 2003). In genetic terms, characterization refers to the detection of variation as a result of differences in either DNA sequences or specific genes or modifying factors. Standard characterization and evaluation of accessions may be routinely carried out by using different methods, including traditional practices such as the use of descriptor lists of morphological characters. They may also involve evaluation of agronomic performance under various environmental conditions. In contrast, genetic characterization refers to the description of attributes that follow a Mendelian inheritance or that involve specific DNA sequences.

In this context, the application of biochemical assays such as those that detect differences between isozymes or protein profiles, the application of molecular markers and the identification of particular sequences through diverse genomic approaches all qualify as genetic characterization methods. Because of its nature, genetic characterization clearly offers an enhanced power for detecting diversity (including genotypes and genes) that exceeds that of traditional methods. Likewise, genetic characterization with molecular technologies offers greater power of detection than do phenotypic methods (e.g. isozymes). This is because molecular methods reveal differences in genotypes, that is, in the ultimate level of variation embodied by the DNA sequences of an individual and uninfluenced by environment. In contrast, differences revealed by phenotypic approaches are at the level of gene expression (proteins).

2. LITERETURES REVIEW

2.1 Genetic markers in plant breeding:

Genetic markers are the biological features that are determined by allelic forms of genes or genetic loci and can be transmitted from one generation to another, and thus they can be used as experimental probes or tags to keep track of an individual, a tissue, a cell, a nucleus, a chromosome or a gene. Genetic markers used in genetics and plant breeding can be classified into two categories: classical markers and DNA markers (Xu, 2010). Classical markers include morphological markers, cytological markers and biochemical markers. DNA markers have developed into many systems based on different polymorphism-detecting techniques or methods (southern blotting – nuclear acid hybridization, PCR – polymerase chain reaction, and DNA sequencing) (Collard et al., 2005), such as RFLP, AFLP, RAPD, SSR, SNP, etc.

2.1.1. Classical markers

2.1.1.1 Morphological markers

Use of markers as an assisting tool to select the plants with desired traits had started in breeding long time ago. During the early history of plant breeding, the markers used mainly included visible traits, such as leaf shape, flower color, pubescence color, pod color, seed color, seed shape, hilum color, awn type and length, fruit shape, rind (exocarp) color and stripe, flesh color, stem length, etc. These morphological markers generally represent genetic polymorphisms which are easily identified and manipulated. Therefore, they are usually used in construction of linkage maps by classical two- and/or three-point tests. Some of these markers are linked with other agronomic traits and thus can be used as indirect selection criteria in practical breeding. In the green revolution, selection of semi-dwarfism in rice and wheat was one of the critical factors that contributed to the success of high-yielding cultivars. This could be considered as an example for successful use of morphological markers to modern breeding. In wheat breeding, the dwarfism governed by gene Rht10 was introgressed into Taigu nuclear male-sterile wheat by backcrossing, and a tight linkage was generated between Rht10 and the male-sterility gene Ta1. Then the dwarfism was used as the marker for identification and selection of the male-sterile plants in breeding populations (Liu, 1991). This is particularly helpful for implementation of recurrent selection in wheat. However, morphological markers available are limited, and many of these markers are not associated with important economic traits (e.g. yield and quality) and even have undesirable effects on the development and growth of plants.

2.1.1.2 Cytological markers

In cytology, the structural features of chromosomes can be shown by chromosome karyotype and bands. The banding patterns, displayed in color, width, order and position, reveal the difference in distributions of euchromatin and heterochromatin. For instance, Q bands are produced by quinacrine hydrochloride, G bands are produced by Giemsa stain, and R bands are the reversed G bands. These chromosome landmarks are used not only for characterization of normal chromosomes and detection of chromosome mutation, but also widely used in physical mapping and linkage group identification. The physical maps based on morphological and cytological markers lay a foundation for genetic linkage mapping with the aid of molecular techniques. However, direct use of cytological markers has been very limited in genetic mapping and plant breeding.

2.1.1.3 Biochemical/ protein markers

Protein markers may also be categorized into molecular markers though the latter are more referred to DNA markers. Isozymes are alternative forms or structural variants of an enzyme that have different molecular weights and electrophoretic mobility but have the same catalytic activity or function. Isozymes reflect the products of different alleles rather than different genes because the difference in electrophoretic mobility is caused by point mutation as a result of amino acid substitution (Xu, 2010).

2.1.2. DNA markers

DNA markers are defined as a fragment of DNA revealing mutations/variations, which can be used to detect polymorphism between different genotypes or alleles of a gene for a particular sequence of DNA in a population or gene pool. Such fragments are associated with a certain location within the genome and may be detected by means of certain molecular technology. Simply speaking, DNA marker is a small region of DNA sequence showing polymorphism (base deletion, insertion and substitution) between different individuals. There are two basic methods to detect the polymorphism: Southern blotting, a nuclear acid hybridization technique (Southern 1975), and PCR, a polymerase chain reaction technique (Mullis, 1990). Using PCR and/or molecular hybridization followed by electrophoresis (e.g. PAGE – polyacrylamide gel electrophoresis, AGE – agarose gel electrophoresis, CE – capillary electrophoresis), the variation in DNA samples or polymorphism for a specific region of DNA sequence can be identified based on the product features, such as band size and mobility. In addition to Sothern blotting and PCR, more detection systems have been also developed. For instance, several new array chip techniques use DNA hybridization combined with labeled nucleotides, and new sequencing techniques detect polymorphism by sequencing. DNA markers are also called molecular markers in many cases and play a major role in molecular breeding.

Since Botstein et al. (1980) first used DNA restriction fragment length polymorphism (RFLP) in human linkage mapping, substantial progress has been made in development and improvement of molecular techniques that help to easily find markers of interest on a largescale, resulting in extensive and successful uses of DNA markers in human genetics, animal genetics and breeding, plant genetics and breeding, and germplasm characterization and management. Among the techniques that have been extensively used and are particularly promising for application to plant breeding, are the restriction fragment length polymorphism (RFLP), amplified fragment length polymorphism (AFLP), random amplified polymorphic DNA (RAPD), microsatellites or simple sequence repeat (SSR), and single nucleotide polymorphism (SNP). According to a causal similarity of SNPs with some of these marker systems and fundamental difference with several other marker systems, the molecular markers can also be classified into SNPs (due to sequence variation, e.g. RFLP) and non-SNPs (due to length variation, e.g. SSR) (Gupta et al., 2001).

Table 1. Comparison of most widely used DNA marker systems in plants; Adapted from Collard et al. (2005), Semagn et al. (2006a), Xu (2010), and others.

2.2 Genetic Characterization and Its Use in Decision-Making for the Conservation of Crop Germplasm

Characterization, at present is carried out either based on morphological traits or on molecular markers (biochemical and DNA markers). Morphology-based characterization has some limitations in the accurate identification of the accessions, such as limited number of traits to characterize (Rao 2004). The characterization, conservation and exploitation of crop plant germplasm maintained in gene banks propound a number of challenges to the researchers dedicated to the investigation of plant genetic resources. Common problems include the development of strategies for sampling representative individuals in natural and experimental populations, the improvement of tools and technologies for long-term conservation and for high-throughput characterization of large numbers of stored accessions. The knowledge of the genetic diversity present in a gene bank is crucial for developing sustainable conservation strategies and it is also essential for the profitable exploitation of a gene bank by specific breeding programs. As a matter of fact, germplasm characterization of plant accessions deposited in gene banks has been limited and this likely represents a major cause for the limited adoption of conserved accessions in crop breeding programs (Ferreira 2006). Consequently, the genetic characterization of accessions belonging to a given collection and the examination of genetic relationships among them should be strengthened and perpetrated not only for maintaining but also for exploiting crop genetic resources.

Conservation of the genetic resources in the agro-ecosystem in which they have evolved (in situ conservation) is now being more widely considered, as complementary to strategies based on gene banks (ex situ conservation), for limiting genetic erosion and so preserving genetic diversity. If it is true that in situ conservation has been proposed essentially for wild relatives of cultivated plants, it is also true that when considered for major crops this alternative can very often be unfeasible from a socio-economic perspective (Negri et al. 2000; Lucchin et al. 2003). Genomic DNA-based marker assays have revolutionized and modernized our ability to characterize genetic variation and to rationalize genetic selection (Lanteri and Barcaccia 2006). Molecular markers are known as particularly effective and reliable tools for the characterization of genome architectures and the investigation of gene polymorphisms in crop plants.

Besides linkage mapping, gene targeting and assisted breeding, the plant DNA polymorphism assays are powerful tools for characterizing and investigating germplasm resources and genetic relatedness. These techniques include restriction fragment length polymorphism (RFLP) markers and PCR-based molecular markers, such as simple sequence repeat (SSR) or microsatellite markers (Morgante and Olivieri 1993), amplified fragment length polymorphism (AFLP) markers (Vos et al. 1995).

2.3 Genetic Diversity and Similarity Statistics for Characterizing Plant Germplasm at the Population Level

Genetic diversity and similarity measurements are very useful for describing the genetic structure of populations. The genetic structure of natural populations of a crop plant species is strongly influenced by the reproductive system of their individuals and the union types occurring within populations. Breeding schemes that can be adopted as well as variety types that can be constituted depend on the reproductive barriers and mating systems of plants (Barcaccia 2009). Natural populations of species that reproduce by apomixis or that propagate vegetatively are polyclonal, being composed by several genetically distinct clones and usually dominated by a few well-adapted genotypes. Therefore, genetic variation within populations is distributed among clones and most populations are characterized by different levels of differentiation among genotypes.

Landraces of self-pollinated species (e.g., bean, lentil, wheat and barley) are composed of a mixture of pure lines, genetically related but reproductively independent each other. Thus, genetic as well as phenotypic variation is mainly detectable among lines due to the presence within natural populations of fixed genotypes mainly homozygous for different alleles. Spontaneous hybridization is however possible to some extent depending on the species, environmental factors and germplasm stocks. Cultivated varieties of selfing species are usually represented by pure lines obtained by repeated self-pollination of a number of hybrid individuals stemmed from two parental lines chosen for complementary morphological and commercial traits. Maize is one of the most commercially important cross-pollinated species. In many countries, existing landraces are selected by farmers for their own use and eventually sale to neighbors. Traditionally, landraces are developed by mass selection in order to obtain relatively uniform populations characterized by valuable production locally. Synthetics are also produced by intercrossing a number of phenotypically superior plants, selected on the basis of morpho-phenological and commercial traits. More rarely, plants are also evaluated genotypically by means of progeny tests. Compared to landraces, synthetics have a narrower genetic base but are equivalently represented by a heterogeneous mixture of highly heterozygous genotypes sharing a common gene pool. However, newly released varieties are exclusively represented by F1 hybrids developed by private breeders and seed companies using inbred lines belonging to distinct heterotic groups.

Genetic characterization is providing new information to guide and prioritize conservation decisions for crop plants. The most urgently required action is the effective protection of all remaining wild ancestral populations and closely related species of crop plants, most of them now endangered. They are the only remaining sources of putative alleles of economic values that might have been lost during domestication events. It is equally important to ensure that the plant genetic resources selected for conservation include populations from the geographic areas representing the different domestication centres where high estimates of genetic diversity within and differentiation among populations are expected(Barcaccia 2009).

2.4 Using molecular characterization to make informed decisions on the conservation of crop genetic resources

Information about the genetic make-up of accessions helps decision making for conservation activities, which range from collecting and managing through identifying genes to adding value to genetic resources. Well-informed sampling strategies for germplasm material destined for ex situ conservation and designation of priority sites (i.e. identifying specific areas with desirable genetic diversity) for in situ conservation are both crucial for successful conservation efforts. In turn, defining strategies is dependent on knowledge of location, distribution and extent of genetic diversity.

Molecular characterization, by itself or in conjunction with other data (phenotypic traits or geo-referenced data), provides reliable information for assessing, among other factors, the amount of genetic diversity (Perera et al.,2000), the structure of diversity in samples and populations (shim et al.,2000, Figliuolo et al 2004), rates of genetic divergence among populations (Maestri et al.,2002) and the distribution of diversity in populations found in different locations (Maestri et al ., 2004, Perera et al.,2000).

A recent study on the genetic diversity of cultivated Capsicum species in Guatemalan home gardens compared the diversity present in an array of home gardens in the Department of Alta Verapaz with a countrywide representative sample of 40 accessions conserved ex situ in the national collection (Guzmán et al., 2005). The results showed that home gardens of Alta Verapaz (H = 0.251) contained as much diversity as the entire national ex situ collection (H = 0.281). These results thus suggest that, (1) home gardens are indeed an extremely important resource for in situ conservation of Capsicum germplasm in Guatemala, and as such they should not be neglected; (2) if further collecting activities were to be undertaken, special emphasis should be given to collecting in Alta Verapaz; and (3) additional collecting in Alta Verapaz alone could disclose novel genetic diversity that is absent from the national collection. Conservation of clonally propagated crops demands more complex and expensive procedures. If these crops are maintained on-farm, their existence is endangered by several factors, one of which being the introduction of alternative improved varieties. Conservation efforts need then to be based on solid knowledge of clonal diversity. This was the case for Abyssinian banana or ensete (Ensete ventricosum (Welw.) Cheesman) from Ethiopia, which was analysed with AFLP markers (Negash, A. et al 2002) Of the 146 clones from five different regions, only 4.8% of the total genetic variation was found between regions, whereas 95.2% was found within regions. The results led to a reduced number of clones for conservation and indicated the existence of a common practice of exchange of local types between regions, which, in its turn, emphasized the need to collect further in different farming systems.

A study on taro (Colocasia esculenta (L.) Schott) genetic diversity in the Pacific, using SSR markers, showed that many of the accessions from countries of the Pacific region were identical to those of Papua New Guinea. This indicates that originally the cultivars may have been introduced throughout the region from Papua New Guinea (Mace, E.S. et al 2005) and that collection of taro genetic diversity could focus on Papua New Guinea alone. Molecular characterization also helps determine the breeding behaviour of species, individual reproductive success and the existence of gene flow, that is, the movement of alleles within and between populations of the same or related species, and its consequences (Papa, R. & Gepts P. 2003 Papa, R. & Gepts P. 2003). Molecular data improve or even allow the elucidation of phylogeny, and provide the basic knowledge for understanding taxonomy, domestication and evolution (Nwakanma, D. C., et al 2003).

As a result, information from molecular markers or DNA sequences offers a good basis for better conservation approaches. Management of germplasm established in a collection (usually a field, seed or in vitro gene bank) comprises several activities. Usually, such activities seek to ensure the identity of the individually stored and maintained samples, to ensure the safeguarding of genetic integrity and genetic diversity and to have the material available for distribution to users. These tasks are primarily a responsibility of gene bank managers and curators, and involve the control of accessions on arrival at the facilities, as well as their continuous safeguarding for the future through regeneration and multiplication. For all these routine activities, information about the genetic constitution of samples or accessions is critical and provides possibly the most important means of measuring the quality of the work being performed. Börner et al. (2000) analysed bulk seed of wheat accessions to test their genetic integrity after 24 cycles of regeneration and after more than 50 years of storage at room temperature in a gene bank. They found neither contamination nor incorrect manipulation effects such as mechanical mixtures, but did identify one case of genetic drift in one accession.

However, in the same gene bank, a study examined the genetic constitution of rye accessions that underwent frequent regeneration. Results showed that (1) a significant number of alleles present in the original sample was lacking in the newly regenerated material, and (2) new alleles in the new material were not present in the first regeneration sample (Chebotar, S., et al 2003). Thus, the use of molecular markers can quickly help check whether changes in alleles or allele frequencies are taking place. Molecular information has been used to weigh the need for decreasing the size of germplasm collections, which otherwise would add costs to the long-term conservation of germplasm. For instance, Dean et al. (1999) used microsatellite markers to analyse the genetic diversity and structure of 19 sorghum accessions known as ‘Orange’ in the USDA’s national sorghum collection. They found two redundant groups (involving five entries) among the 19 accessions evaluated. They also found that much of the total genetic variation was partitioned among accessions. As a result, the authors concluded that the number of accessions held by the US National Plant Germplasm System (NPGS) could be significantly reduced without risking the overall amount of genetic variation contained in these holdings.

Markers were also helpful in examining genetic identities and relationships of Malus accessions (Hokanson, S.C., et al 1998.). Eight primer pairs unambiguously differentiated 52 of 66 genotypes in a study that calculated the probability of any two genotypes being similar at all loci analysed as being about 1 in 1,000 million. The results not only discriminated among the genotypes, but were also shown to be useful for designing strategies for the collection and in situ conservation of wild Malus species. Selected molecular technologies render cost-effective and comprehensive genotypic profiles of accessions (‘fingerprints’) that may be used to establish the identity of the material under study. Simultaneously, these technologies can detect contaminants (and, in the case of material mixtures, contamination with introgressed genes from other accessions or commercial varieties), as well as the presence of redundant materials (or ‘duplicates’) (McGregor, et al 2002).

Moreover, molecular data provide the baseline for monitoring natural changes in the genetic structure of the accession (Chwedorzewska,et al., 2002)or those occurring as a result of human intervention (e.g. seed regeneration or sampling for replanting in the field). Whatever the case, analysis of molecular information allows the design of strategies for either purging the consequences of inappropriate procedures or amending them to prevent future inconveniences (de Vicente, M.C. 2002). A small number of potential duplicates were identified in a core collection of cassava (Manihot esculenta Crantz) when isozyme and AFLP profiles were compared (Chavarriaga-Aguirre, P., et al 1999). The core collection had been assembled with information from traditional markers, which proved to be highly effective for selecting unique genotypes. Molecular data were used for efficiently verifying the previous work on the collection and ensure minimum repetition. Thus, gene bank managers can easily realize the potential value of using molecular methods to support and possibly modify or improve a gene bank’s operations.

A special and increasingly important role of genetic characterization is that of identifying useful genes in germplasm, that is, of maximizing conservation efforts. Because the major justification for the existence of germplasm collections is use of the conserved accessions, it is important to identify those valuable genes that can help develop varieties that will be able to meet the challenges of current and future agriculture.

Characterization has benefited from several approaches resulting from advances in molecular genetics such as genetic and QTL mapping, and gene tagging (Yamada, T.,et al 2004 , Kelly, J.D.,et al 2003). Research in this field has led to the acknowledgement of the value of wild relatives, in which modern techniques have discovered useful variation that could contribute to varietal improvement (Xiao, J., et al 1996, de Vicente, M.C. et al 1993). Knowledge of molecular information in major crops and species and of the synteny of genomes, especially conservation of gene order, has also opened up prospects for identifying important genes or variants in other crop types, particularly those that receive little attention from formal research.

Until now in India, identification and classification of Hibiscus have mainly been based on morphology and according to (Wachira F, Tanaka J and Takeda Y, 2001) even if these descriptors are useful, they show limited levels of inter and intra-varietal polymorphism and hence, may not account for all the diversity in the species. Since it is difficult to identify cultivar based entirely on these morphological features, several kinds of methods which can be used to measure levels and patterns of it is important to find an effective method to accurately identify the varieties to meet research needs. The novelty of this project lies in the use of different molecular markers with increasing order of specificity to study genetic diversity which will help in development of new cultivars of Hibiscus varieties with superior properties to meet changing agronomic requirements. polymorphism and hence, may not account for all the diversity in the species. Since it is difficult to identify cultivars based entirely on these morphological features, several kinds of methods which can be used to measure levels and patterns of it is important to find an effective method to accurately identify the varieties to meet research needs.

Modern molecular techniques have been developed in order to meet the demands of the horticulture industry genetic variation, which range from morphological characterization to various DNA-based markers such as restriction fragment length polymorphism (RFLP), randomly amplified polymorphic DNA (RAPD), amplified fragment length polymorphism (AFLP) and simple sequence repeats (SSR) (Crawford, D. J.2000, Newton, A. C., et al 2002, Martinez, L.,et al 2003, Fontaine, C., et al 2004, Murtaza, N.2006, Ferdousi Begum.,2013). Identification and characterization of germplasm is essential for the conservation and utilization of plant genetic resources (Suvakanta-Barik., et al 2006). Characterization of plant with the use of molecular markers is an ideal way to conserve plant genetic resources. Molecular characterization helps to determine the breeding behaviour of species, individual reproductive success and the existence of gene flow, the movement of alleles within and between populations of the same or related species, and its consequences (Papa R and Gepts P.2003).

Molecular data improves the elucidation of phylogeny, and provide the basic knowledge for understanding taxonomy, domestication and evolution of plants (Nwakanma D C., et al 2003). Random amplified polymorphic DNA (RAPD) technique has been widely used in many plant species for varieties analysis, population studies and genetic linkage mapping (Williams J. G. K., et al 1990, Yu K., et al 1993, Rout G. R., et al 2003). Optimization of the RAPD analysis depends on selection of primers. Although, the RAPD method uses arbitrary primer sequences, many of these primers must be screened in order to select primers that provide useful amplification products. By contrast, single-locus markers are usually characterized by co-dominance (i.e. both alleles identified in heterozygous individuals) and thus are more flexible and supply more robust and comparable data (Karp, 2002).

An appropriate use of molecular markers techniques requires to clearly define the issues addressed, what type of information will be needed (on genetic diversity) and to know what the different techniques can offer not only in terms of genetic information but also resource requirements, reproducibility, adaptability for automation. Furthermore, it is of pivotal importance to consider how the information will be gathered and the way in which the data will be scored and analysed. For accurate and unbiased estimates of genetic diversity adequate attention has to be devoted to:

a. sampling strategies,

b. utilization of various data sets on the basis of the understanding of their strengths and constraints,

c. choice of genetic similarity estimates or distance measures, clustering procedures and other multivariate methods in analyses of data and

d. objective determination of genetic relationships (Mohammadi and Prasanna, 2003).

For all these reasons, choosing the most appropriate technique may be difficult and often a combination of techniques is needed to gather the information one is interested in. Up to now most conservation efforts have focused on agriculturally important crops and about one third of all ex situ accessions in gene bank represents just five species: i.e. wheat (Triticum sp.), barley, rice, maize and beans (Phaseolus spp). The relative over-representation of five species does not necessarily mean that their genetic diversity has been fully covered (Graner et al. 2003) but, on the other hand, there is significant lack of knowledge about the diversity and geographic distribution of less utilized crops as well as their wild relatives (Hammer et al. 2003). Genetic studies in selected crops have demonstrated that widespread and localised alleles occurring in the entire collection are usually contained in the core subset, with only rare localized alleles excluded (van Hintum et al. 2000). Findings suggest that, although a high variability can be found among plants, most of their genotypes belong to the same landrace locally called ‘Nostrano di Storo’ (Barcaccia et al., 2003).

3. CONCLUSION

In conclusion, the most important challenges in the near future are certainly collections for preserving crops from genetic erosion, the molecular characterization of germplasm, and the identification of useful variation in germplasm, potentially useful for breeding new varieties. Knowing the presence of useful traits, genes and alleles would help in making decisions on the multiplication of accessions and the maintenance of seed stocks for responding to an expected higher demand of materials. Such information may also help in making decisions on heterogeneous accessions, where only some genotypes may possess useful alleles. Thus, the gene bank curator may have to decide to maintain the original material as it is and separate a subpopulation carrying the desirable alleles and give it new accession numbers and management protocols. This will facilitate germplasm use and add value to the collections.

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Publication Date: January, 2020

**Gilder Cieza Altamirano & Rafaél Artidoro Sandoval Núñez**

Department of General Studies, National Autonomous University of Chota

Chota, Peru

Journal Full Text PDF: Numerical Solution of Nonlinear Third Order Van Der Pol Oscillator.

Abstract

In the present study, numerical investigations have been carried out for solving a class of third order Van der Pol oscillator for differential equation related to nonlinearity for both parameters based on stiff and non-stiff conditions. The variants of the proposed scheme have been numerically solved and the comparison of results are presented based on the two schemes named as Adams numerical scheme and implicit Runge-Kutta scheme which is used to solve the third order nonlinear Van der Pol oscillator. The details of the achieved numerical results in the form of tables have been numerically discussed.

Keywords: Third order Van der Pol oscillator, Adams method, implicit Runge-Kutta method, nonlinear, stiff & non-stiff.

1. Introduction

The angular frequency based functions having the derivatives of first order has achieved diverse attention of the researchers community using the models of nonlinear oscillators [1-4]. Recently, these schemes represent generally the nonlinear differential equations of second-order. A superior class of an oscillatory system is called oscillator of Van der Pol that is used to govern nonlinear damping and represent the second-order nonlinear ordinary differential model. In this study we solve the nonlinear third order Van der Pol oscillator which is basically a third order differential equation. The generic form of nonlinear third order Van der Pol equation is given as:

(1)

The above model is presented on the basis of second order Vander-pol oscillators which is extensively used in science, engineering and has the huge amplitude planner flexural free sensations of an inextensible slender cantilever beam component [5]. Jockon and Mickens [6] accessible the comprehensive fundamental study for the second order nonlinear oscillators and familiarized the generic properties of the oscillator. Additionally, Van der pol oscillator rises in many submissions of engineering and applied science. Numerous methods have been applied to solve these types of models. By keeping a view on it, a good variety and source of assessment article [7] has been obtainable in which variety of numerical techniques for nonlinear Van der Pol oscillators are nominated in detail.

The goal of the present investigation is to explore and exploit a class of nonlinear third order Van der Pol oscillators using Adams method and Implicit Runge-Kutta method. In this regard, two numerical problems based on third order nonlinear Van der Pol oscillators along with two cases of each have been present and results are checked from two schemes that are found to be accurate.

The remaining parts of the article is organized as follows: Section 2 narrates the numerical technique, the numerical outcomes are provided in Section 3 and conclusions along with future research directions are listed in the final Section.

2. Numerical methods

In this study section, the communications based on deterministic and stochastic numerical solvers have been effected generally in extensive fields, for example linear/ nonlinear algebraic models [8], nanotechnology based equations [9], system of nonlinear doubly singular equations [10], model based on Thomas-Fermi equations [11], an extensive class of delay differential second order equation [12] and boundary value problems based on multi-point equations [13]. In this study, predictor-corrector numerical Adams scheme and numerical Implicit Runge-Kutta method are applied to solve the third order nonlinear Van der Pol oscillators.

2.1 Predictor-corrector Adams numerical technique

To find the numerical solution of nonlinear third order Van der Pol oscillator, the corrector- predictor numerical scheme is applied that takes two stages to complete further. The approximate outcomes of prediction is accomplished in the first step, while to find the numerical outcomes of correction is proficient with the same contributions of prediction.

(2)

Two steps for the generalized Adams-Bashforth numerical technique by using the predictor-corrector scheme:

(3)

Adams-Moulton two-stages corrector is provided as:

(4)

The 4-steps predictor corrector is as follows:

(5)

The 4-step Adams-Bashforth Moulton is given as:

(6)

2.2 Implicit Runge-Kutta numerical technique

To solve the nonlinear Van der Pol oscillator, the Implicit Runge-Kutta technique is designed. The generic form of Implicit Runge- Kutta technique is considered as:

(7)

In the first step, to consider the obtained initial outcomes and slopes are documented for all variables. Taking these achieved numerical outcomes for slopes (the ) at the central point of the interval to form the designs of the dependent variable. In the second step, the slopes of the middle-point (the ) are achieved by taking these attained mid-point values. The designed numerical outcomes for slopes are twisted back to the first point to variety the other set of middle point outcomes that are originate to the new slope of predictions at middle-point (the ). These numerical values are supplementary applied to create predictions to development slopes at the final point of the interval (the ). Likewise, all the outcomes for are achieved composed to make another set of increase functions. Finally, take at the start point to make the final prediction.

3. Results and Discussions

In this section, numerical experimentations are presented for nonlinear third order Van der Pol Oscillators with stiff and non-stiff parameters. Two problem along with two cases have been solved and their numerical results are tabulated in Tables 1 and 2.

Problem 1

Case 1: Consider the Van der Pol Oscillator with non-stiff conditions for and

(8)

Case 2: Consider the Van der Pol Oscillator with non-stiff conditions for and

(9)

To find the numerical result of the problems, we have applied the Adams and Implicit Runge-Kutta by using the Mathematica built in functions presented in Table 1 and Table 2. The results calculated for the both methods of the two problems are given in Table 1 and Table 2 for inputs with a step size of . It is cleared that the proposed solutions show same results as of Adams and Implicit Runge-Kutta results and proved very good agreements.

Table 1: Comparison of Adams and Implicit Runge-Kutta for case 1 and case 2

Case 1 Case 2

x Adams Implicit RK Adams Implicit RK

0 2 2 3 3

0.2 1.99772 1.99772 3.120105 3.120105

0.4 1.984585 1.984585 3.143269 3.143269

0.6 1.956347 1.956347 3.145955 3.145955

0.8 1.913439 1.913439 3.144768 3.144768

1 1.858406 1.858406 3.142849 3.142849

1.2 1.794106 1.794106 3.140789 3.140789

1.4 1.722751 1.722751 3.138701 3.138701

1.6 1.645572 1.645572 3.136605 3.136605

1.8 1.562811 1.562811 3.134505 3.134505

2 1.473817 1.473817 3.132402 3.132402

2.2 1.377129 1.377129 3.130296 3.130296

2.4 1.270461 1.270461 3.128186 3.128186

2.6 1.150547 1.150547 3.126074 3.126074

2.8 1.012786 1.012786 3.123959 3.123959

3 0.850614 0.850614 3.121841 3.121841

3.2 0.654462 0.654463 3.11972 3.11972

3.4 0.410171 0.410171 3.117596 3.117596

3.6 0.096847 0.096847 3.115468 3.115468

3.8 -0.31514 -0.31514 3.113338 3.113338

4 -0.86383 -0.86383 3.111205 3.111205

4.2 -1.59221 -1.59221 3.109068 3.109068

4.4 -2.56449 -2.56449 3.106929 3.106929

4.6 -3.99054 -3.99054 3.104786 3.104786

4.8 -6.73794 -6.73794 3.102641 3.102641

5 -17.3345 -17.3345 3.100492 3.100492

Problem 2

Case1: Consider the Van der Pol Oscillator with stiff conditions for and

(10)

Case2: Consider the Van der Pol Oscillator with stiff conditions for and

(11)

Table 2: Comparison of Adams and Implicit Runge-Kutta for case 1 and case 2

Case 1 Case 2

x Adams Implicit RK Adams Implicit RK

0 5 5 3 3

0.2 5.199783 5.199783 3.199927 3.199927

0.4 5.399122 5.399122 3.399702 3.399702

0.6 5.598003 5.598003 3.599319 3.599319

0.8 5.796412 5.796412 3.798773 3.798773

1 5.994337 5.994337 3.998055 3.998055

1.2 6.191762 6.191762 4.197159 4.197159

1.4 6.388675 6.388675 4.396078 4.396078

1.6 6.585061 6.585061 4.594806 4.594806

1.8 6.780908 6.780908 4.793334 4.793334

2 6.976203 6.976203 4.991657 4.991657

2.2 7.170932 7.170932 5.189767 5.189767

2.4 7.365083 7.365083 5.387658 5.387658

2.6 7.558642 7.558642 5.585321 5.585321

2.8 7.751597 7.751597 5.78275 5.78275

3 7.943935 7.943935 5.979938 5.979938

3.2 8.135644 8.135644 6.176878 6.176878

3.4 8.326712 8.326712 6.373562 6.373562

3.6 8.517127 8.517127 6.569984 6.569984

3.8 8.706877 8.706877 6.766137 6.766137

4 8.89595 8.89595 6.962014 6.962014

4.2 9.084335 9.084335 7.157607 7.157607

4.4 9.27202 9.27202 7.35291 7.35291

4.6 9.458994 9.458994 7.547915 7.547915

4.8 9.645247 9.645247 7.742617 7.742617

5 9.830767 9.830767 7.937008 7.937008

4. Conclusion

In the present study, the numerical treatment based on third order nonlinear stiff/ non-stiff Van der Pol oscillator presented by manipulating the strength of the numerical Adams scheme and Implicit Runge-Kutta scheme. The numerical results are found to be accurate and consistent from both of the schemes. The proposed scheme is valuable and appropriate for solving linear/ nonlinear second order Van der Pol oscillator for two problems each has two cases. The software used for solving the nonlinear second order Van der Pol oscillator is Mathematica 10.4. In future, this scheme is applied to solve nonlinear system of second order and third order Van der Pol oscillators.

5. References

[1] Kovacic, I., 2011. Forced vibrations of oscillators with a purely nonlinear power-form restoring force. Journal of Sound and Vibration, 330(17), pp.4313-4327.

[2] Mickens, R.E. and Oyedeji, K., 2011. Comments on the general dynamics of the nonlinear oscillator x¨+(1+ x˙ 2) x= 0. Journal of Sound and Vibration, 330(17), pp.4196-4200

[4] Feng, J., Zhu, W.Q. and Liu, Z.H., 2011. Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems. Communications in Nonlinear Science and Numerical Simulation, 16(8), pp.2978-2984.

[3] Eigoli, A.K. and Khodabakhsh, M., 2011. A homotopy analysis method for limit cycle of the van der Pol oscillator with delayed amplitude limiting. Applied Mathematics and Computation, 217(22), pp.9404-9411.

[5] Hamdan, M.N. and Shabaneh, N.H., 1997. On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass. Journal of Sound and Vibration, 199(5), pp.711-736.

[6] Mickens, R.E. and Oyedeji, K., 2011. Comments on the general dynamics of the nonlinear oscillator x¨+(1+ x˙ 2) x= 0. Journal of Sound and Vibration, 330(17), pp.4196-4200.

[7] Kerschen, G., Worden, K., Vakakis, A.F. and Golinval, J.C., 2006. Past, present and future of nonlinear system identification in structural dynamics. Mechanical systems and signal processing, 20(3), pp.505-592.

[8] Raja, M. A. Z., Sabir, Z., Mehmood, N., Al-Aidarous, E. S., & Khan, J. A. (2015). Design of stochastic solvers based on genetic algorithms for solving nonlinear equations. Neural Computing and Applications, 26(1), 1-23.

[9] Mehmood, A., Zameer, A., & Raja, M. A. Z. (2018). Intelligent computing to analyze the dynamics of Magnetohydrodynamic flow over stretchable rotating disk model. Applied Soft Computing, 67, 8-28.

[10] Raja, M. A. Z., Mehmood, J., Sabir, Z., Nasab, A. K., & Manzar, M. A. (2017). Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Computing and Applications, 1-20.

[11] Sabir, Z., Manzar, M. A., Raja, M. A. Z., Sheraz, M., & Wazwaz, A. M. (2018). Neuro-heuristics for nonlinear singular Thomas-Fermi systems. Applied Soft Computing, 65, 152-169.

[12] Sabir, Z., Umar, M. and Unlu, C., Solving a Class of Second Order Delay Differential Equation by Using Adams and Implicit Runge-Kutta Method.

[13] Sabir, Z., & Raja, M. (2014). Numeric treatment of nonlinear second order multi-point boundary value problems using ANN, GAs and sequential quadratic programming technique. International Journal of Industrial Engineering Computations, 5(3), 431-442.

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]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: August, 2019

**Hnin Hnin Khaing & Yee Yee Htun**

Lecturer, Department of Engineering Mathematics, TU (Maubin), Ayarwaddy Division

Lecturer, Department of Engineering Mathematics, TU (Hmawbi), Yangon Division

Myanmar

Journal Full Text PDF: Implementation of Numerical Solutions for Nonlinear Equations using MATLAB.

**Abstract**

Problems that most frequently encountered are nonlinear equations in sciences and engineering problems. In this paper, we will focus on MATLAB solutions to nonlinear equations by studying various methods. In this paper, the numerical methods for solving nonlinear equations using MATLAB can be carried out. This present the most widely used iterative methods for nonlinear equations and MATLAB features for finding numerical solutions. The numerical examples are considered and implemented in this paper.

**Keywords:** nonlinear equations, MATLAB, numerical methods, iteratives methods.

**1. INTRODUCTION**

Compare to linear algebraic equations, most frequently encountered are nonlinear equations in sciences and engineering problems. Solving nonlinear equations could be computationally expensive; therefore, the solving of approximate linear equations was indispensable especially in the early times when the computers were not powerful enough. Today, with the rapid development of computing technology, solving directly nonlinear equations is becoming increasingly important. In this paper, various methods of solving nonlinear equations problems are studied and we will focus on MATLAB solutions to nonlinear equations. The three methods of solutions to nonlinear algebraic equations will be presented in this technical approach paper. The graphical method for nonlinear equations with one and two unknown variables can be analysis with polynomial equations. Numerical solutions to nonlinear equations and nonlinear matrix equations can also be implemented in this paper. (Dingyü Xue, 2009).

Graphical and numerical methods will be presented and the simplest solution is implemented using MATLAB. In this paper, the linear programming, quadratic programming and general nonlinear programming will be studied and MATLAB-based solutions will be carried out. As a procedure, the theory background of Bisection Method, Secant Method and Newton Raphson Method of solving nonlinear equations problems are studied and carried out. Then, to be approached to simplest and fast way, MATLAB instructions relative to nonlinear equations solving process are also studied and tested. Actually, various possible problems are applied to three methods and tested. But, only the main points or examples are expressed in this paper as a portion of Mathematical approach to Engineering Problems.

**2. NUMERICAL METHODS FOR NONLINEAR EQUATIONS**

2.1 Bisection method

The idea behind the Intermediate Value Theorem can be stated: when we have two points connected by a continuous curve, as shown in Picture 1:

• one point below the line

• the other point above the line

• then there will be at least one place where the curve crosses the line.

Picture 1(a). Intermediate values Picture 1(b). Bisection Algorithm

(www mathsisfun com) (www codewithc com)

The first step in iteration is to calculate the mid-point of the interval [ a, b ]. If c be the mid-point of the interval, it can be defined as: c = ( a+b)/2. The function is evaluated at ‘c’, which means f(c) is calculated.

• f(c) = 0 : c is the required root of the equation.

• f(b) * f(c) > 0 : if the product of f(b) and f(c) is positive, the root lies in the interval [a, c].

• f(b) * f(c) < 0 : if the product of f(b) and f(c) is negative, the root lies in the interval [ b, c].

In the second iteration, the intermediate value theorem is applied either in [a, c] or [ b, c], depending on the location of roots. And then, the iteration process is repeated by updating new values of a and b. Picture 2 shows how iteration done in bisection method.

Picture 2. Iteration Process in Graph

2.2 Secant Method

This method uses two initial guesses and finds the root of a function through interpolation approach. For each successive iteration, two of the most recent guesses can be used as two most recent fresh values to find out the next approximation. Features of Secant Method can be shortly expressed as:

• No. of initial guesses – 2

• Type – open bracket

• Rate of convergence – faster

• Convergence – super linear

• Accuracy – good

• Approach – interpolation

• Programming effort – tedious

The Procedure of Secant Method can be shown as flowchart as in Picture 3 compare with Newton Raphson Method and this will give the algorithm to implement in MATLAB.

(a) Secant Method Algorithm (b) Newton Raphson Method Algorithm

Picture 3. Comparison of Secant Method and Newton Raphson (www codewithc com)

2.3 Newton Raphson Method

The theoretical and mathematical background behind Newton-Raphson method and its MATLAB program (or program in any programming language) is approximation of the given function by tangent line with the help of derivative, after choosing a guess value of root which is reasonably close to the actual root. The x- intercept of the tangent is calculated by using elementary algebra, and this calculated x-intercept is typically better approximation to the root of the function. This procedure is repeated till the root of desired accuracy is found. Lets now go through a short mathematical background of Newton’s method. For this, consider a real value function f(x) as shown in the Picture 3(c).

Picture 3(c). Newton’s Method (Dingyü Xue, 2009)

Let’s try the example problem using Newton-Raphson method, solving it numerically. The function is to be corrected to 9 decimal places. For a given function: x3−x−1 = 0, which is differentiable,

**3. MATLAB FUNCTIONS FOR SOLVING NONLINEAR EQUATIONS**

3.1 Function “roots”

The Syntax : r = roots(p) returns the roots of the polynomial represented by p as a column vector. Input p is a vector containing n+1 polynomial coefficients, starting with the coefficient of xn. A coefficient of 0 indicates an intermediate power that is not present in the equation. This function can study as shown in Picture 4(a). For example: p = [3 2 -2] represents the polyno-mial 3×2+2x−2.The roots function solves polynomial equations of the form p1xn + …+ pnx + pn+1 =0. Polynomial equations contain a single variable with nonnegative exponents.

(a) roots() (b) fzero()

Picture 4. Study on MATLAB Functions

3.2 Function: “fzero”

Root of nonlinear function (fzero) can be write in syntax as:

3.3 Function: “inline”

The constructing of inline object can be scriptable in MATLAB as the fucncitons: inline (expr), inline (expr,arg1,arg2,…) or inline (expr,n). “inline (expr)” constructs an inline function object from the MATLAB expression contained in “expr”. The input argument to the inline function is automatically determined by searching expr for an isolated lower case alphabetic character, other than i or j, that is not part of a word formed from several alphabetic characters. If no such character exists, x is used. If the character is not unique, the one closest to x is used. If two characters are found, the one later in the alphabet is chosen. “inline (expr,arg1,arg2,…)” constructs an inline function whose input arguments are specified by arg1, arg2,…. Multicharacter symbol names may be used. “inline (expr,n)” where n is a scalar, constructs an inline function whose input arguments are x, P1, P2, … .

Three commands related to inline allow you to examine an inline function object and determine how it was created. “char(fun)”converts the inline function into a character array. This is identical to “formula(fun)”. “argnames(fun)” returns the names of the input arguments of the inline object fun as a cell array of character vectors. “formula(fun)” returns the formula for the inline object fun.A fourth command “vectorize(fun)” inserts a . before any ^, * or /’ in the formula for fun. The result is a vectorized version of the “inline” function. (www mathworks com)

3.4 Function “num2str”, “abs”, “plot”

The simple functions of “num2str” which convert number to string, “abs” wich convert the valuse to be absolute or real only and “plot” which make data to graph or curve are also studied and apply in solution of nonlinear equations of MATLAB. Picture 5 shows example expression of these functions which can be studied by help of MATLAB.

(a) num2str() (b)abs()

(c) Plot()

Picture 5. Study on MATLAB Functions

**4. IMPLEMENTATION RESULTS**

4.1 M-Scripts for Three Methods

Input data are made as variable for any nonlinear equations, not only by getting answer, plotting graph process was followed at the end of program. The m-script for each method was carried out as in the following Table 1:

Table 1. Implemeting Codes of MATLAB

4.2 Simulation Results

The simulation results are shown in Picture 6 to Picture 8. We can study, analyze, compare and solve fastly and simply with our implemented MATLAB program for any nonlinear equations.

(a) m-file Running Results (b) Error Plot

Picture 6. Bisection Method Results

(a) m-file Running Results (b) Error Plot

Picture 7. Secant Method Results

(a) m-file Running Results (b) Error Plot

Picture 8. Newton Raphson Method Results

Picture 9. Results Comparison for ( )

**5. DISCUSSION, CONCLUSION AND RECOMMENDATION**

The Implementatin of Numerial Solutions for Non Linear Equations are sucessfully done in this research paper using MATLAB. By this practicle approach, we can easily compare and anlyse any nonlinear equations which are mostly represented in real- world Engineering process. How we can compare and used each method of Bisection, Secant and Newton Raphson’s can be express as shown in Picture 9 for a given same nonlinear equation of . We can proof that Bisection Method was simplest form. For Secant Method, it is faster than other numerical methods, except the Newton Raphson method and there is no need to find the derivative of the function as in Newton-Raphson method. If the function is not differentiable, Newton’s method cannot be applied. By this research paper, we can clearly analyze how nonlinear equations are easily be solved using various method by the help of MATLAB.

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]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: August, 2019

**Saddam Husain Dhobi, Ram Chandra Pangeni, Kishori Yadav & M. D. Jahangeer Rangrej**

Science and Techonology, Tribhuvan University (Physics), Kathmandu, Province-3, Nepal

Journal Full Text PDF: Experimental Calculation of Voltage Generated by Numbers of Atom When Photon Incident and Cross-section Area of Single Photon.

**Abstract**

The main work of this research work is to calculate voltage generated by incidence of red photon on the surface atoms of poly crystalline silicon base solar panel. This is calculated by passing red leaser photon through aluminum pinhole of different diameters and then incidence on the solar panel pinhole whose surfaces closed by highly absorption materials i.e. whole panel is cover except the pinhole of desire diameter. On taking the different data i.e. voltage vs incidence photon through pinhole we find that on decrease the diameters of pinhole of aluminum plate for fixed pinhole of solar panel the voltage also decrease and vice-versa. On the other hand, we also calculate voltage generated by the number of atoms on which photon incidence in certain cross-section area. These data are listed on observation table1 and with the help of these data we are able to calculated the average cross-section area of red photon.

**Keywords:** Aluminium Pinhole, Poly Crysatlline Silicion, Photon, Cross-section area, Red Photon.

**1. Introduction**

The phenomena which is used to find the voltage across the atoms or voltage generate by atom when photon is incidence it. When photon incidence on the surface of atoms it ejected electron from the atom after distribution of photon energy. In this experiment, photon energy are able to ejected the electron from the atom which is bounded or correlated with other atoms. This phenomena can also called photoelectric effect.

In this work we are trying to introduce the the energy lose or distribution by a single photon energy which are unable to eject the electron from the atom is divided into numbers of packet energy. Due to the division of these numbers of packet energy of single photon, the work function is greater then theses packet energy generated by single incidence photon. This phenomena is only seen when photon is incidence on an atom expect normally i.e. the division of single photon is only experience when photon is incidence with certain angle expect perpendicular.

This is observed when we focus for the perpendicular incidence the photon on the pinhole of solar panel. Because during the focusing for perpendicular incidence of photon on the solar panel considerable surface the voltage obtained from multi meter is less than voltage obtained at perpendicular incidence of photon on considerable surface.

Figure 1: Sketch of different component of solar panel and circuit connection to obtained Voltage.

Some part of our experiment contain this whole system of figure 1. and more extra are design to obtained the data which is shown in experimental set up. The short description of different component of solar panel are given below:

1.2 Electrons & Hole: An intrinsic or pure silicon crystal at room temperature has sufficient heat or thermal energy for some valence electrons to jump the gap from the valence band into the conduction band and hence becoming free electrons. These free electrons are called conduction electrons. When an electron jumps to conduction band, a vacancy is left in valence band within silicon crystal. These vacancy is called a hole. For every electron raised to the conduction band by external energy, there is one hole left in the valence band, creating what is called an electron-hole pair. Recombination occurs when a conduction-band electron loses energy and falls back into a hole in the valence band

.3 Depletion Layer: The free electrons in the n-region are randomly drifting in all directions. At the instant of the pn junction formation, the free electrons near the junction in the n region begin to diffuse across the junction into the p region where they combine with holes near the junction. When pn junction is formed, the n region loses free electrons as they diffuse across the junction. This creates a layer of positive charges near the junction. As the electrons move across the junction, the p region loses holes as the electrons and holes combine, which creates a layer of negative charges near the junction. These two layers of positive and negative charges form the depletion region. As electrons continue to diffuse across the junction, more and more positive and negative charges are created near the junction as the depletion region is formed. A point is reached where the total negative charge in the depletion region repels any further diffusion of electrons into the p region and the diffusion stops. In other words, the depletion region acts as a barrier to the further movement of electrons across the junction.

1.4 N-type Silicon: To increase the number of conduction-band electrons in intrinsic silicon, pentavalent impurity atoms are added. These are atoms with five valence electrons such as arsenic, phosphorus (P), bismuth (Bi), and antimony (Sb). Each pentavalent atom forms covalent bonds with four adjacent silicon atoms. Four of the antimony atom’s valence electrons are used to form the covalent bonds with silicon atoms, leaving one extra electron. A conduction electron created by this doping process does not leave a hole in the valence band because it is in excess of the number required to fill the valence band. The electrons are called the majority carriers in n-type material. Holes in an n-type material are called minority carriers.

1.5 P-type Silicon: To increase the number of holes in intrinsic silicon, trivalent impurity atoms are added. These are atoms with three valence electrons such as boron, indium, and gallium. Each trivalent atom forms covalent bonds with four adjacent silicon atoms. All three of the boron atom’s valence electrons are used in the covalent bonds. Because the trivalent atom can take an electron, it is often referred to as an acceptor atom. The number of holes can be carefully controlled by the number of trivalent impurity atoms added to the silicon. A hole created by this doping process is not accompanied by a conduction electron. The holes are the majority carriers in p-type material. Conduction-band electrons in p-type material are the minority carriers.

1.6 Anti Reflection Coating: To reduces the reflection of light from the surface of the solar cell, to further reduce the reflection of incoming radiation from sun in order to maximize the absorption of light, a silicon nitride film (SiNx) or other such properties material, which acts as Anti reflection coating, is deposited by plasma enhanced chemical vapor deposition or any other method on the front surface of the solar cell. This film serves as a passivating layer for the front surface of the solar cell in addition to serving as an anti-reflection coating. This film must be optimized to absorb the majority of incoming radiation as well as passivate the surface satisfactorily. The target thickness of anti refelcting coating for baseline cells is set at 780nm1.

1.7 Metallic conducting Strips: Continuous efforts to develop new materials and modeling techniques for solar cells are being made in order to produce new photovoltaic devices with improved electrical performances. In addition to the new semi conducting materials, solar cells consist of a top metallic grid or other electrical contact to collect electrons from the semiconductor and transfer them to the external load. In a solar cell operating under the normal conditions, even a small deviation from the optimum power condition can cause a loss of conversion efficiency2.

**2. Review**

The rays of light neither mutually color each other, nor mutually illuminate each other, nor mutually impede each other in any way. This is just like one physical motion’s not impeding another as study by Kepler and first observe the scattering of photons by photons in an experiment seems to have been undertaken in 1928 in the Soviet Union by S. I. Vavilov. In the experiment, no experimental sign of photon-photon collisions was found and Hughes and Jauncey give as bound for the cross section.

When light and sound simultaneously pass through a medium, the acoustic phonons of the sound wave scatter the photons of the light beam. This scattering of light from acoustic modes is called Brillouin scattering. A particularly interesting effect of Brillouin scattering has to do with the frequency of the scattered light. An incident photon can be converted into a scattered photon of slightly lower energy, normally’ propagating in the backward direction, and a phonon. For a Stokes process, where a phonon is generated, the frequency of the scattered light is decreased; for an anti-Stokes process, where a phonon is annihilated, the frequency of the scatted light is increased. Increased frequency, by the equation E = hv, means increased photon energy. The difference between the energy of the scattered photon and the incident photon is called the Brillouin shift4.

QED is normally used to describe the scattering of electrons. It is recognized in QED, the quantum mechanical formalism describing the scattering of photons would be different than the scattering of electrons as the cross sections for scattering or their interaction coefficients with the lattice would be different. In describing diffraction consistent with the conventions of QED, we adopt a momentum representation for the Coulomb potential associated with the lattice. The most probable values for the magnitude of the y-momentum of the virtual photons associated with the scattering potential are integral multiples of h/2d. On recognize these energies are the eigenvalues of the photon standing wave eigenfunctions for the particle-in-a-box problem in quantum mechanics where the length of the box is d. The scattering probability distribution observed with photon diffraction is derived from a function of the y-momentum exchanged from the scattering by virtual photons of the lattice summed over the probabilities or densities of the virtual photons with the different momentum values5.

Comparing spectra, the Brillouin shift is much smaller than the Raman shift because the velocity of acoustic waves is much less than the velocity of light. This was already known from the spectrum of spontaneous scattering, where the Raman process gives a much larger shift than Brillouin scattering. Parametric processes require conservation of both energy and momentum. Stimulated Brillouin scattering occurs when a beam of laser light generates a parametric process that simultaneously produces an exactly retrorefected Stokes beam and an acoustic wave traveling in the forward direction. Energy conservation requires that the Stokes beam frequency is reduced from the laser frequency by the frequency of the acoustic wave. Historically the term “Stokes was named for Sir George Gabriel Stokes, who in 1852 described the change in wavelength of fuorescence, which is always at lower photon energy than the incident light. When Raman scattering was discovered, similar shift to lower photon energy was called Stokes light. When Raman scattering was discovered to have weak signals at shorter wavelength than the incident light, this was called “anti-Stokes” light6.

One or several scattering processed Rayleigh scattering, Brillouin scattering, and Raman scattering, can occur due to the interaction of an incident wave with a medium. When the intensity of light is low the resulting scattering process will be spontaneous. However, when the incident intensity reaches a certain threshold, stimulated scattering will be observed with a strong interaction between light fields and matter. In Brillouin scattering is both stimulated and spontaneous, a pump photon at a frequency up produces an acoustic phonon and a red-shifted. The energy and momentum conservation requirements on Stokes and anti-Stokes Brillouin scattering are as follows. Slow light, the propagation of an optical pulse at a very low group velocity, is of interest for enhancing the interaction of light and material and to provide higher controllability of the gain spectrum bandwidth. SBS has been studied in a variety of gas, liquid, and solid media for different applications, and SBS capabilities vary greatly depending on the choice of gain medium. Thus overall potential for SBS applications is broad, selection of the right gain material for a particular application is critical. In practice, many more parameters need to be taken into consideration when selecting a medium, including the transmittance at the wavelength of interest, gain coefficient, generation and damage threshold, Brillouin frequency shift and line width, environmental sensitivity, toxicity, as well as available size7.

The molecular theory of matter starts with quantum mechanics and statistical mechanics. According to the quantum mechanical Heisenberg Uncertainty Principle, the position and momentum of an object cannot be determined simultaneously and precisely. The Heisenberg Uncertainty Principle helps determine the size of electron clouds, and hence the size of atoms. Heisenberg’s Uncertainty Principle applies only to the subatomic particles like electron, positron, photon, etc. It does not forbid the possibility of nanotechnology, which has to do with the position and momentum of such large particles like atoms and molecules. This is because the mass of the atoms and molecules is quite large, and the quantum mechanical calculation by the Heisenberg Uncertainty Principle places no limit on how well atoms and molecules can be held in place8. The atomic radius is taken as half of the inter atomic distance in a crystalline state. The atomic radius of elements and the relationship with their position in the periodic table, together with the numerical values of atomic radius9.

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]]>The post Case Study of Shortest Path Algorithms and Implementation using MATLAB appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: July 25, 2019

**Yee Yee Htun**

Ph.D (Applied Maths), Department of Engineering Mathematics

TU (Hmawbi), Hmawbi Township, Yangon Division, Myanmar

Journal Full Text PDF: Case Study of Shortest Path Algorithms and Implementation using MATLAB.

**Abstract**

Shortest path problems are among the most studied network flow optimization problems withinteresting application across a range of fields. In this paper, three shortest path algorithms arediscussed via Dijkstra’s Algorithm (one to all pairs of nodes), Floyd Warshall’s Algorithm (all to allpairs of nodes) and Linear Programming Problems (LPP). These algorithms are also solved usingMatlab software, which gives quick results for larger nodes. By this research, we can sucessfully study how many ways to find shortest path. Graph technique in Matlab can also be applied to be simply solved the shortest path problems. The application of Direct Graph and Undirect Graph of shortest path was implemented for the route of ferry bus, North Dagon township to TU (Hmawbi).

**Keywords:** Shortest path, Dijkstra’s Algorithm,Floyd Warshall’s Algorithm, Linear Programming Problems, Matlab software & Direct Graph.

**1. INTRODUCTION**

Finding the shortest path is an important task in network and transportation related analysis. Shortestdistance problems are inevitable in road network applications, such as city emergency handling anddriving system, where optimal routing has to be found. Therefore, network optimization has alwaysbeen the heart of operational research. Also, as traffic conditions of a city change from time to time,there could be a huge amount of request occurring at any moment, for which an optimal path solutionhas to be found quickly. Hence, efficiency of an algorithm is very important to determine the shortestroutes are between nodes in a network.

There are many algorithms that can be used to determine the shortest route between two nodes in anetwork. In this paper, two standard algorithms Dijkstra’s algorithm and Floyd Warshall’salgorithm are discussed and also solved using Matlab software. The linear programming formulation of shortest route problem solved using (0-1) binary integer programming technique is also discussed.

The dual of formulated linear programming and shortest route problem solved by algebraic method is demonstrated for small number of nodes, as it is difficult to solve for large number of nodes. In such cases, Matlab software can be the best choice. Further, the shortest distance and shortest routedetermined using Complementary Slackness Theorem.(Dr. Roopa, K.M., 2013).

**2.GRAPH IN MATLAB**

A directed graphwith four nodesand three edges is as shown in Picture 1 in Matlab. Graph theory functions in the toolbox apply basic graph theory algorithms to sparse matrices. A sparse matrix represents a graph, any nonzero entries in the matrix represent the edges of the graph, and the values of these entries represent the associated weight (cost, distance, length, or capacity) of the edge. Graph algorithms that use the weight information will cancel the edge if a NaN or an Inf is found. Graph algorithms that do not use the weight information will consider the edge if a NaN or anInf is found, because these algorithms look only at the connectivity described by the sparse matrix and not at the values stored in the sparse matrix.(matlabexpo.co.kr, 2016).

Picture 1. Direct Graph example

Sparse matrices can represent four types of graphs:

1) Directed Graph — Sparse matrix, either double real or logical. Row (column) index indicates the source (target) of the edge. Self-loops (values in the diagonal) are allowed, although most of the algorithms ignore these values.

2) Undirected Graph — Lower triangle of a sparse matrix, either double real or logical. An algorithm expecting an undirected graph ignores values stored in the upper triangle of the sparse matrix and values in the diagonal.

3) Direct Acyclic Graph (DAG) — Sparse matrix, double real or logical, with zero values in the diagonal. While a zero-valued diagonal is a requirement of a DAG, it does not guarantee a DAG. An algorithm expecting a DAG will not test for cycles because this will add unwanted complexity.

4) Spanning Tree — Undirected graph with no cycles and with one connected component.There are no attributes attached to the graphs; sparse matrices representing all four types of graphs can be passed to any graph algorithm. All functions will return an error on nonsquare sparse matrices.

2.1 Finding the Shortest Path in a Directed Graph

1) Create and view a directed graph with 6 nodes and 11 edges.

2) Biograph object with 6 nodes and 11 edges.

3) Find the shortest path in the graph from node 1 to node 6.

4) Mark the nodes and edges of the shortest path by coloring them red and increasing the line width.

This example Graph results can be shown as in Picture 2 (a).

2.2 Finding the Shortest Path in an Undirected Graph

1) Create and view an undirected graph with 6 nodes and 11 edges.

2) Biograph object with 6 nodes and 11 edges.

3) Find the shortest path in the graph from node 1 to node 6.

4) Mark the nodes and edges of the shortest path by coloring them red and increasing the line width.

This example Graph results can be shown as in Picture 2(b).

(a) (b)

Picture 2. Direct Graph and Undirect Graph in Matlab for 6 nodes(www.mathwork.com)

**3. SHORTEST PATH ALGORITHMS**

3.1 Dijkstra’s Algorithm

Dijkstra’s algorithm considers two sets:

i) set P, which at any specific point consists of all the nodesthat were encountered by the algorithm;

ii) set S, a precedence set, which at any specific pointconsists of the precedent node for each node in the network. Apart from these sets, the algorithmutilizes the following distance information.

qij, for i, j=1, 2, 3, 4…n and i≠j, denote the weight of the directed edge (arc) from vertex i to vertex j.

If there is no arc from i to j, then qij, is set to be infinity.tj, for j=1, 2, 3, 4…n and j≠s where s is the start index. Also,

tj = q1j for j=2,3,4…n (1)

In each iteration, the sets P and S as well as the set of all tj, for j=1, 2, 3, 4…n and j≠s, that are outputfrom the previous iteration are taken as inputs. Initially P = {s}. S is a set of size n populated with

i) 0 if tj = infinity, ii) s if tj = finite value

The steps involved in each iteration for finding the shortest distance are summarized below:

Step 1: Identify minimum among the computed tj values. Let tk be the minimum.Add k to the set P.

Step 2: Now P = {1, k}. For each of the nodes not in P and with finite qkj, for j=1, 2, 3, 4…n and j ∉

P, recalculate tj using the below expression:

tj = min{ tj, tk + qkj } (2)

Only if (tk + qkj) < tj, then update the jth entry in S to k. Continue the iterations until the end node, e, is added to the set P.Similarly, the steps to trace the shortest path between nodes s and e, using Dijkstra’s algorithm aregiven below:

Step 1: Take node e as the last node in the shortest path

Step 2: Find the eth entry in the set S, let this be x. Add this prefix node x to the partially constructed

shortest path.

Step 3: Check whether x is equal to s. If so, go to Step 4; else go to Step 3.

Step 4: The required shortest path from node s to node e is thus constructed.

Picture3 shows an example of using Dijkstra’s Algorithm.

Picture 3. Example of Dijkstra’s Algorithm

3.2Floyd Warshall’s Algorithm:

Floyd Warshall’s Algorithm is a graph analysis algorithm to find the shortest route between any twonodes in a network with positive or negative edge weights with no negative cycle. This algorithm usesthe dynamic programming technique to solve the shortest path problem between all pairs of nodes (allto all) in a directed network. It represents the network as a square matrix with n-rows and n- columnsand at the end of the algorithm each (i,j) of the matrix gives the shortest distance from node i to node j.If there is a direct link between node i to node j, then the value at (i,j) is finite, otherwise it is infinite,i.e, d (i,j)= ∞.

The steps to trace the shortest path between two nodes, say i and j using Floyd-Warshall’s

algorithm are given below:

Step 1: Take node j as the last node in the shortest path.

Step 2: Find the value S [i, j] from the precedence matrix Sn, let it be x. Add this Prefix node x t

partially constructed shortest path.

Step 3: Check whether x is equal to i. if so, go to step (4); else, set j = x and go to the step 3.

Step 4: The required shortest path from node i to node j is constructed.

Picture 4. Shortest Route Network

To determine the shortest distance and shortest paths between all pairs of nodes in atransportation network as shown in Picture 4, using Iteration (3): Set k=3. Consider third column and third row of D3 as pivot column and pivot rowrespectively. Except d (1, 3), all the entries in the pivot column are infinity and also except d (3, 5) andd (3, 7), all the entries in the pivot row are infinity. Further, apply transitivity property to obtain thefollowing results:

(i) Since, d(1,4)=17 , d(1,5)=14 and d(1,7)=32 . So, d (1,7) = 32 cannot be improved .

(ii) Set precedence matrix S2 as S (1, 4) = 3, S (1, 5) = 3. The changes are as shown in the

matrix D3and S3.

Continuing in this way, the final matrix in the last iteration where none of the entries in the d (i,j) canbe improved by transitivity property, because all the elements in the last row are infinity.Finally, the shortest distance between any two nodes is determined from the matrix D7 as shown in Picture 5.

Picture 5. Iteration 7 Results ( Dr. Roopa, K.M., 2013).

3.3 Algebraic Method for solving the dual Linear Programming Problem

The dual linear programming problem can also be solved using algebraic method for only small

number of variables. However, solving the above dual Linear Programming Problem throughalgebraic method, by introducing slack variables which gives better result compared to any othersoftware packages. By using the first and final tableaus of algebraic method , the dual problem can also be solved using Matlab software. For example, if the solutions obtained from Matlab software aregiven below:

y1 = –10.4979, y2 = 3.6415, y3 = –0.4979, y4 = 6.5021, y5 = 3.5021, y6 = 5.5021, y7 = 11.5021

The value of Z = 22 gives the shortest distance from node 1 to node 7. By considering, the solutions thatsatisfy the above constraints the following routes: 1-3, 3-4, 3-5, 4-7, 5-7, 5-6 and 6 -7 are obtained. Fromthese sequence of routes 1-3, 3-4, 4-7, the shortest route 1—3—4—7, which is of distance 22 units fromnode 1 to node 7 is traced. Similarly, other alternate shortest routes that can be obtained are: 1—3—5—7and 1—3—5—6—7 respectively.

The shortest route can also be determined using Complementary Slackness Theorem . As the

sequence of routes 1-2, 3-2, 2-7, 2-4, 4-6 and 4-5, do not satisfy the constraints in the dual problem, fromthe Complementary Slackness Theorem it follows that x12=x32 = x24 =x27=x45 =x46 = 0.

Substituting these variable values in the primal problem, the following systems of equations are obtained:

x13 = 1; x13- x34 –x35 =0;x34-x47=0;x35-x56 – x57 =0;x56–x67 =0; x47+x57 +x67 =1 (3)

By solving above system of linear equations using Gauss Elimination Method, the system in echelon formbecomes:

x34+ x35 = 1; x35+ x47 =1; x47+x56 + x57 =1; x47+x57 +x67 =1 (4)

In the above system of equations, there are 4 equations (r=4) with 6 unknowns (n=6) and two freevariables (x35, x56,). Hence, the possible choices are: (0,0),(0,1),(1,0),(1,1). Each of these possible choicesmay or may not be the solution points because the dependent variables have the restriction, xij = 0 or 1.( Dr. Roopa, K.M., 2013).

**4.IMPLEMENTATION OF SHORTEST PATH**

(BETWEEN NORTH DAGON-MAWATA BUS STOP AND TU (HMAWBI))

The case was assigned as to find shortest path between Mawata Bus Stop of North Dagon township and TU(Hmawbi). By solving this case, the author can apply low cost and minimum time to arrive TU(Hmawbi) from her home. There are 9 main Bus Stations as focal points. But there are a lot of Bus Stops Between each focal Bus Stations. Unit are per miles to be considered. The m-script of this case is shown in Picture 6.

Table-1 Nodes Assignment

No. BUS STOPS NODE

1 Ma wa ta 1

2 Bay lie 2

3 Aung migalar highway station 3

4 8 miles 4

5 Saw bwar gyi kyone 5

6 Htaut kyant 6

7 Toll gate 7

8 Hmawbi market 8

9 TU Hmawbi 9

Picture 6. M-script of Implementation Case

**5. RESULTS**

As a result of implementing shortest path in Matlab, author can make effort of cost and time to go to office from home everyday. The Resut is to use Bus No (99) between Mawata Bus stop to Sawbwargyi Goan Bus stop, and again to TU(Hmawbi) with Bus No (37). The minimun cost for a route is 700 MMK. The results from Matlab program are shown in Picture 7 (a) and (b).

(a) Direct Graph (b) Undirect Graph

Picture 7. Implementation Results

**6. CONCLUSION AND RECOMMENDATION**

This paper has presented the results of implementing or application of shortest path algorithms in real case. Effectiness of Matlab software can also be proved in this paper. All three algorithms of shortest path finding methods are studied and compared. It is evident that Dijkstra’s algorithm takes a relatively lesser time than Floydsand Binary integer programming in finding shortest route. However, Dijkstra’s algorithm is the betteroption for identifying the shortest path in larger networks such as railway, water, power distribution andgas pipeline networks. For simplicity, the author mainly applied Direct Graph methods that used Dijkstra’s algorithm in background of Matlab. This research paper carried out mainly case study of three shortest path finding concepts and analysis with real world application.

**7. REFERENCES**

1. Dr. Roopa, K.M., Apoorva, H.R1., Srinivasu,V.K.2 and Viswanatah, M.C 3,“A Study on Different Algorithms for Shortest Route Problem”, International Journal of Engineering Research & Technology (IJERT) ,Vol. 2 Issue 9, September,2013.

2. Algorithms in Java, Chapter 21,http://www.cs.princeton.edu/introalgsds/55dijkstra.

3. Floyd, Robert W. (June 1962). “Algorithm 97: Shortest Path”. Communications of the ACM 5 (6): 345. doi:10.1145/367766.368168 (http:/ / dx. doi. org/ 10. 1145/ 367766. 368168).

4. http://www.matlabexpo.co.kr, 2016.

5. http://research.microsoft.com/users/goldberg.

6. http://www.mathwork.com

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]]>The post Method Development Model Consensus on Analytic Hierarchy Process appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: July 23, 2019

**Maman Budiman, Dwi Gunarto & Randy Asmuni**

Pancasakti University, Tegal

Widya Dharma University, Klaten

Indonesia

Journal Full Text PDF: Method Development Model Consensus on Analytic Hierarchy Process.

**Abstract**

This paper propose a model of consensus on the method of AHP (Analytic Hierarchy Process) to decision-making group. The consensus method can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, the structure of the cluster, the object identification on decision “problematic” and a marginal opinion. Formation of the consensus made by Delphi method approach, discussion (exchange of information) is performed to obtain a homogeneous opinion. Rules do when the termination discussion found more homogeneous the opinion of 66.6 percent, or based on time constraints.

**Keyword:** Analytic Hierarchy Process, consensus & homogeneous.

**I. INTRODUCTION**

Group decision-making process in general produces a complex decision. Each party involved in the decision making process, can have the values and principles are different. The difference in value or principle of life, can make a difference / conflict of goals / interests, which can cause differences in preferences between the parties involved, the situation is called a conflict. In conflict situations, ideally one party does not impose desires (strategy) to achieve his own wishes, but must be willing and able to work together (win-win) through negotiation.

The process of cooperation is based on the intention sincere and open, will achieve mutually beneficial results, even can give a synergy effect, so that the results will be obtained by each party will be better than the process that is competitive (win-loose), Putro and Tjakraatmadja [1].

The purpose of group decision making occurs when the owner of the decision (decision) subjectively included as participants in decision-making, which in some cases the participants did not know all the activities as a whole, Brugha [2]. In fact, to be able to describe the problem well in the evaluation, a decision, a decision maker needs to consider the opinions of others who understand the problem.

This paper propose a model of consensus on the method of AHP (Analytic Hierarchy Process) to decision-making group. The model developed is the development of Techniques for Analyzing Consensus Relevant Data / ACRD developed by, Ngwenyama [3].

**II. LITERATURE REVIEW**

Forman and Peniwati [4], said the process of synthesis by the AHP method can be done in one of two ways:

• Aggregation of individual assessment method (the aggregation of individual judgment / AIJ), or

• Method of aggregation of individual priorities (the aggregation of individual priorities / AIP).

For AIJ method it is assumed that each member of the group acts as a single entity and can no longer act on the individual’s identity. The assessment results of each individual in the group are aggregated with the geometric mean method. For this case, the Pareto principle is not relevant so it does not need to be considered.

In the AIP method, it is assumed that the group members act as individuals who are independent of the other group members. The priority sequence produced by each individual in the group are aggregated with the average method geometric or arithmetic average, and both of them do not violate the Pareto principle.

In producing the preferences of individual preference groups, to consider the fulfillment of the Pareto principle. According to the Pareto principle, if the two alternatives, the alternatives alternatives a1 and a2, compared and each individual group members preferred alternative compared to alternative a1 a2, then the group should prefer an alternative rather than an alternative a1 a2.

Zahir [5], stating that in a large group there are various possible patterns of thinking, namely:

a. All members of the group have the same thought

b. The views of members vary but they are a member of a coherent homogeneous group

c. There are several clusters or homogeneous sub-groups within a large group.

Homogeneous group does not demand identical preferences of each individual group members. In a homogeneous group, each individual opinion remains varied, but has a “similarity”.

The similarity of this argument can be seen by comparing the value of the cosine angle formed by a pair of individual preferences of group members with a limit of homogeneity (). If the value of the cosine angle formed by a pair of individual preference greater than or equal to the limit value homogeneity, Then a couple of individuals can be said to have a “similarity”.

2.1 Techniques for Analyzing Consensus Relevant Data/ ACRD (Ngwenyama et. Al., 2006)

The decision making process in computer-supported group allows anonymous process takes place with the help of a facilitator as a steering discussions. To generate information for the facilitator, [3] proposed a number of techniques and approaches for analyzing data group preferences in the decision process.

Data analysis techniques relevant to the consensus (the techniques for analyzing the consensus of relevant data / ACRD) proposed by [3], is utilizing the preference data individually produced by each decision maker (value and sequence) in a group against a set of alternative decisions using AHP.

The analysis was performed based on the similarity of individual partner preferences of group members, which is indicated by the cosine of the angle is greater than or equal to the value limits of agreement.

The disagreement between a pair of individuals reached if the cosine angle value less than or equal to the limit value of disagreement.

Value, which probably is 0985 (cosine angle 10o) and value which probably is 0.966 (cosine angle 15o). Rationalization of making these angles is that the largest possible angle between two vectors weight is 90o. So 0o 10o-90o on a scale equivalent to 1 on a scale of 1-9, and 15o equivalent to 1.5 on a scale of 1-9.

Conceptually, this approach to support this ACRD techniques can be divided into three stages, namely:

a. The pre-evaluation stage

b. Stage generate preferences

c. Phase data analysis and reporting

Pre-evaluation stage includes the selection of alternatives for evaluation and determination of the evaluation criteria, as well as the delimitation agreement to define the rules of the termination of the decision-making process, such as the achievement of an allocated time interval or a certain level of agreement on the issue of (partial or full).

While on stage do the sorting preference produce alternative and presentation of the data comparison using AHP.

At the stage of data analysis and reporting analyzing the preference data performed by the decision makers to identify their position in group decision-making process. In this phase also the identification of possible coalitions, identify problematic decision alternatives, and identification of key individuals who have a preference position that enables the negotiation of consensus.

This approach allows the facilitator in the group to assess the level of group consensus at every stage of the group, so that the resulting information can help the facilitator to negotiate the formation of a consensus within the group. Consensus can be achieved when the consensus map has been identified that describe the preferences that can be accepted by the group.

In an ideal situation, should be reached complete agreement within the group. But in general, taking into account the differences of opinion, this is not possible, so it takes the rules of termination.

Individual indicators expressed by the Consensus Individual Vector (ICVt), which is used to identify individuals who have a good level of agreement with other group members and do not have barriers that make it difficult. The key individuals have the greatest ICVt value (ICVtmaks). Individual keys are used to facilitate the formation of a group consensus.

2.2 Cluster algorithm Zahir (2009)

For medium-sized groups (intermidiate-sized group) or a group of large-sized (large group), group homogeneity can not be guaranteed or achieved, so [5] proposed a clustering algorithm based on the method VAHP (the Vector Space Formulation of the Analytic Hierarchy Process).

By using this algorithm, in a group consisting of N members can be formed each cluster homogeneous, where 1 N,

Clusters are naturally determined by the value of cluster membership boundary (). The value of cluster membership boundary () Varies depending on the type of problem. This cluster membership limit values set by agreement of members of the group.

To determine the membership of a decision-maker to a cluster, cosine of the angle between the weight vector of the decision makers and the resultant weight vector of all decision makers in the cluster than the cluster membership boundary value ().

If the cosine of the angle between the weight vector of the decision makers and the resultant weight vector is greater than or equal to the limit value of cluster membership (), Then the decision maker was elected to be a member of the cluster.

Vice versa, if the cosine of the angle formed between the decision makers of the weight vector and the resultant weight vector is smaller than the limit value of cluster membership, the decision was delayed to have become a member of the cluster.

This delayed decision makers should wait for a re-elected into the cluster until there is another group of decision makers who are elected cluster. If not selected, the pending decision makers have to wait to be placed in another cluster.

The formation of clusters in the algorithm, [5] using a Monte Carlo simulation.

**III. METHOD DEVELOPMENT MODEL**

Systematics design modeling cycle as a model to follow;

As an initial step in the development of the model is done the problem definition. Furthermore, based on the definition of the problem formulated a conceptual model that shows the relationship between the variables that determine the behavior of the model. This model includes verbal model which only outlines the relationship issue, a system, and the purpose of the study.

Objective studies provide indications of performance to be achieved and provides a framework of conceptual models that form the expected performance. To operationalize the conceptual model of symbolization and determination of quantitative rules. Idealization and simplification linkage model variables referred to as the characterization phase models. Model formulation conducted as early development of formal models that show the size of the model performance as a function of the variables of the model.

In the formulation of the model used teleologik principle (review the modeling purposes) for memfungsionalkan attributes by looking at the destination (Teleos) of the system. Through a systems approach, the existence of the system and its environment can be understood by knowing the elements of the system, the relationship between elements and attributes of each element.

Environment system is a collection of objects outside the limits (boundaries) system that affects (affected) systems.

After the initial formulation of the model is complete, then the model’s ability to reproduce the properties and behavior of the real system testing. In this case the testing is based on three criteria to evaluate the model, namely:

a. tested the suitability of the model behavior with the behavior of the real system represents

b. testing the structure between the model variables.

c. estimates for variables, testing the availability of estimated values for key variables.

**IV. DEVELOPMENT MODEL CONSENSUS ON AHP METHOD**

For group decision making, decision-making is done by a group of individuals who are considered worthy to determine a decision. The decision group is considered better than individual opinions. Assessment conducted by many participants will be possible to produce a different opinion from one another, Anonymous [6].

Preferences group with AHP method is generated by synthesizing the preference of each individual group members. However, to carry out the synthesis of individual preferences, there are prerequisites that its decision-making should be ensured homogeneous [5].

Analysis of the achievement of group consensus can be done with attention to the “sameness” of individual partner’s opinion partisan group members / respondents, the opinions of those individuals can be incorporated into a homogeneous clusters,

Formation of the cluster structure can be made by utilizing the information on individual agreements, the level of individual disagreements and disagreements range of individuals who will be included in a cluster. The resulting cluster structure can provide information to identify the possible presence of a marginal opinion of individual members of the group, which can be used as a reference for the group members to obtain the consensus of the group.

The structure of the cluster can be used to identify objects either problematic decision criteria and alternatives, taking advantage of the aggregate value of the cosine on that object. Based on this information, for objects that have a low weight, do a study on the possibility of the removal of that element from the process of decision analysis.

4.2 Formulation Model

4.2.1 Analysis of Consensus in Group Decision Making

Development of consensus methods made to the development of group decision based on concepts that have been previously known. Development of the consensus method is divided in four main stages, namely:

a. Pre-evaluation

At this stage an agreement to establish the boundaries that form clusters. Objects decisions made for each criteria and alternatives. While the agreement stipulated limit is the value of the deal retang individuals in the group, The value of disagreement, And the limit value of cluster membership, And the rules of termination, either a time limit or level of consensus reached.

b. Generate preferences

At this stage, the individual preferences of each member of the group of objects decisions made based on the method of AHP, including testing for consistency.

c. Data analysis

At this stage, the determination of the value of the aggregation of individual preferences into preference groups, the analysis of the level of agreement and disagreement individuals, as well as the formation of cluster structures.

d. Formation of consensus

Formation of the consensus made Delphi method.

4.2.2 Preferences group

Preferences group is the value of “average” individual preferences. preferences group AIP is calculated based method [4], it is assumed that the group members act as individuals who are independent of the other group members. The priority sequence produced by each individual in the group are aggregated with the average method arithmetic average.

… (1)

Where:

= Weighting preferences to-element group i

= Weight of individual preferences to-element i

n = Number of individuals in the group

To see the relationship between the value of individual preferences with the preferences of the group is determined based on the amount of the angle, To facilitate the analysis is then performed the conversion value the value cos, Under the condition:

– cos approaches a value of 1, indicating strong agreement between the preferences of individuals with a preference group

– cos approaching a value of 0, indicating a weak agreement between individual preferences with the preferences of the group

Figure 1. Individual preferences and the preferences of the group in Vector Spaces

For the purposes of data analysis, carried out the determination of the level of agreement and disagreement individuals, as well as the formation of cluster structures.

4.2.3 Range of agreement

The range of agreements taken in this paper is the value ranges of the agreement Of 1.0 on a scale of AHP, or the difference 10oin a vector space. The basis used to determine the range of the deal is the rationalization of making these angles is that the largest possible angle between two vectors weight is 90o. So 0o 10o-90o on a scale equivalent to 1 on a scale of 1-9, and 15o equivalent to 1.5 on a scale of 1-9, [3].

4.2.4 Value limits of agreement and disagreement

Agreements limit value set at (For a strong agreement) are set to the value of 0985 (cosine angle 10o) and (For strong disagreement) is determined by the value of 0.966 (cosine angle 15o), the members of the group said to have a strong agreement if the cosine smaller than the value and a strong disagreement if cosine greater than,

4.2.5 Formation of the cluster structure

The structure of the cluster using cluster membership delimitation. Ordinance on the development of consensus analysis in this paper uses membership limits as big as 10o. Cluster structure was formulated on the development of the model are presented in Table 1.

Table 1 Cluster structure

clusters restriction limitation cosine

Cluster-1 0 ° – 5 ° 1000-0996

Cluster-2 5o – 15o 0996-0966

Cluster 2 ‘

Cluster 3rd 15o – 25o 0966-0906

Cluster to-3 ‘

Cluster 4th 25o – 35o 0906-0819

4th cluster ‘

Cluster 5th 35o – 45o 0819-0707

Cluster to-5 ‘

Cluster 6th 45o – 55o 0707-0574

Cluster 6th ‘

Cluster 7th 55o – 65o 0574-0423

Cluster 7th ‘

Cluster 8th 65o – 75o 0423-0259

Cluster 8th ‘

Cluster 9th 75o – 85o 0259-0087

Cluster 9th ‘

Cluster 10th 85o – 90o 0087-0000

Cluster 10th ‘

Information:

(A) Cluster to-n indicates the direction of the horizontal axis

(B) Cluster to-n ‘indicates the direction of the vertical axis

As an illustration, if the preference groups showed a weight of 0.5 for the ith element and 0.5 for element j, then the cluster structure for this problem is shown in Figure 2.

Figure 2 Individual preferences in space Cluster Group

4.2.6 Coherence individual preferences

To identify individual coherence, the analysis is done by calculating the coherence of individual preferences ( ) based on the average value of cos for each element based on individual preferences to-i

… (2)

Where:

= to-individual coherence i

= Angle formed between the preferences of individuals and groups

n = Number of comparisons related preferences to-element i

Elements that have values 0.966 () Indicate that the individual is “problematic”.

4.2.7 Coherence elements

To identify objects problematic decisions, the analysis done by calculating coherence elements ( ) based on the average value of cos to each individual’s preferences with regard to the i-th element.

Where:

= coherence to-element i

= Angle formed between the preferences of individuals and groups related to the i-th element

n = Number of comparisons related preferences to-element i

m = The number of individual members of the group

Elements that have values 0.966. Indicate that the element (object decision) are “problematic”. Objects problematic decision is a decision if the object is removed from the preference vector will increase the aggregate value of coherence elements.

4.2.8 Termination Rule

Rules set before the termination discussion proper assessment of the decision object. This termination rule can be in the form of a certain level of agreement or a certain time limit which if achieved by the group, then the group should do the synthesis of individual decisions even though the level of agreement has not been reached.

The level of agreement of 0.66 or more indicates that there is a majority in the group (more than two-thirds of the members of the group have the same level of agreement). This means that members of the group had an order of preference are very close. However, if the level of agreement (individuals who are in cluster 1) is less than 0.66, then the group repeats the stages of generating preference.

4.2.9 Formation of Consensus

Formation of the consensus on this model using the rules of the Delphi method. Discussion (exchange of information) was first performed by distributing a marginal opinion on group preferences. The second iteration is done by distributing the opinion that had the most powerful closeness value to the preference group formed in the previous stage (after iteration 1).

If consensus is not formed after two iterations, the facilitator identifies the object decision “problematic”, then held discussions with the members of the group about the possibility of the removal of the element. The next iteration is done and stop until a homogeneous common preference or based on time constraints.

The method developed consensus on this method can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, the structure of vector preference clusters and each cluster, objects mengidentifikansi decision “problematic” and a marginal opinion.

**V. CONCLUSION**

Consensus method developed in this paper can provide information about the level of agreement and disagreement, and disagreement range of individuals and groups, cluster structure, coherence individual preferences, and the coherence of the object elements of the decision. This analysis provides decision object information “problematic” and that the marginal individual opinion. Formation of the consensus made by Delphi method approach.

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]]>The post Increasing Problem Solving Ability and Motivation Learning Through Grand of Math Teacher Comments on Results Homework (PR) appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: June 14, 2019

**Agustinus Setiawan, Natha Fransiska & Nadine Feronica**

Indraprasta PGRI University, South Jakarta

Atma Jaya Catholic University, South Jakarta

Indonesia

Journal Full Text PDF: Increasing Problem Solving Ability and Motivation Learning Through Grand of Math Teacher Comments on Results Homework (PR).

**Abstract**

Mathematical problem solving ability and student motivation Dipawangi Cianjur SDN is considered still low, and the low level of participation of students with homework, the reason for the emergence of this study. This study used mixed methods Embedded Design. The population is all students of SDN Dipawangi with grade samples VA and VB. The instrument used was a test and non-test instrument. The test instrument in the form of question pretest and posttest, while the non-test questionnaire, observation sheets, and interviews. This research resulted in several conclusions:1) The mathematical problem solving ability of students treated teachers commenting on the results of PR is not better than students who did not receive such treatment; 2) there is no difference in mathematical problem solving abilities increase significantly, between students who received treatment commenting on the results of PR teachers and students who did not receive such treatment; 3) The students’ motivation treated teachers commenting on the results of PR better than students who did not receive such treatment; 4) there is a positive correlation between mathematical problem solving ability of students with student motivation.

**Keywords:** Comments Teacher, Homework (PR), Troubleshooting, Motivation.

**1. Preliminary**

Education has an important role in the life of the nation in an effort to create quality human resources. Basic education is the beginning for further education, and is an integral part-kan of the overall national education system. To improve the quality of education, the government has run to-wins the 9-year basic education, 6 years at the elementary level and three years at junior high school level. Primary education provides basic supplies to the students, to be able to develop lif-THEIR and ready to follow further education.

Mathematics, which is the basis for every discipline, needs to be given to all students from primary schools. Students are provided with the ability to think logically, analytical, systematic, critical, problem solving, creativity, and ability to cooperate. These competencies necessary for students to have the ability to acquire, manage, and use information in order to survive in a state that is always changing, uncertain and competitive.

Some problems are quite disturbing to teachers and parents, among them about homework (PR). Does having a PR or no PR? How long the student is expected to learn at home? At what age, class and where to start penuga ladder-san PR? Can students be successful in achieving a good achievement without PR? Is it worth the time to check homework and teachers write comments on students’ homework has been completed? It would be very desirable to answer all these questions for all subjects and all levels of schooling.

Students tend to be menyele-saikan tasks (including PR) and improve the quality of their learning when they get consistent feedback and constructive (Paulu, 1995:18). Arends (2012:232) states that the feedback provided can be teacher comments on the results of PR students. The provision of these comments can be given in writing or oral (verbal). According Ghandoura (1982:80), the students were treated writing comments on the results of PR, was obtained a score higher than students who are not given such treatment

Beutlich (2008:11) states that homework (PR) has varying degrees to their effectiveness. It is important for teachers to know what elements are doing homework (PR) to be more effective for students. Two factors contribute to the effectiveness of homework that student motivation and parental involvement.

Based on the reviews and the above phenomenon, the author felt the need to do research with the title “Upgrading Pemeca-han and Motivation Mathematical Problems Through a Master’s Comments on Results Homework (PR)”. It is expected that the problem solving and student motivation can be increased with treated teachers commenting on the results of PR.

**2. Homework**

Homework (PR) is a specific task or job either written or oral that must be done outside school hours (especially at home). PR deals with subjects that have been submitted by teachers to improve the mastery of concepts or skills and provide development. PR done by the students and checked by the teacher (Cooper, 1989:1)

Work assignments (homework) is very important in the defense-jarkan students at home and there is no direct communication between teachers, students, and parents. Therefore, using the strategy of estab-belajaran homework (homework) given by teachers in schools, as a support to maximize student learning outcomes, as well as the attention of the parents also become supporters (Paulu, 2006:1).

Homework is not just about academic values that would be obtained at the school. This is consistent with the opinion Arends (2012:45), that PR can be a means for social communication among students, and the source of the interaction between the students and their parents.

Arends (2012:312) suggest guidelines for assigning homework is as follows:

a. is an interesting, potentially fun and make sure students understand their duties.

b. give students a challenging homework and convince them to complete successfully.

c. provided with a frequency that often and little, rather than rare but significant amounts.

d. inform parents about the level of involvement is expected of them.

e. make clear rules on deadlines and other details necessary things.

The Apostle Paul (1998:16) also established guidelines on how long students should spend the time to do homework per day. The guidelines are as follows:

a. grades 1-3 :< 20 minutes

b. grade 4-6 :20-40 minutes

c. grade 7-9 :< 2 hours

d. 10-12 class :1 ½ – 2 ½ hours

**3. Feedback**

According to Slavin (1997:80), feedback or feedback is information about the results of the efforts that have been made student learning. Another definition of feedback is also conveyed by the Arends (1997:62), that the information given to students about their performance; for example on the knowledge they gained from learning

Arends (2012:232) argues that teachers can provide feedback to students in various ways, such as verbal, video or sound recording, testing, or through written comments. The guidelines are quite important about the feedback is as follows:

1) provide feedback as soon as possible after exercise.

2) strive for specific dsan clear feedback.

3) feedback is aimed directly at behavior.

4) maintain proper feedback to the developmental level of students.

5) give praise and feedback on the correct performance.

6) if it gives negative feedback, should be shown how to do it right.

7) help students concentrate on process and not results.

**4. Mathematical Problem Solving Ability**

According Duncker (Adams, 2007:16), a problem arises when the living creatures have a purpose, but do not know how to achieve that goal. Based on the structure, Reitman (Adams, 2007:17) states that the problem can be divided into two types, namely:(1) The problem defined perfectly (well-defined) or closed matters and (2) the problem is defined as a weak (ill-defined) or problems. While based on the context Carpenter and Gorg (Prabawanto, 2013:19) mengiden-tifikasi problem becomes:(1) a mathematical problem related to the real world (outside of mathematics) and (2) the problem mathematically pure (pure mathematical problems) are attached as a whole in mathematics.

Mathematical problem solving ability is very dependent on the problems that exist in mathematics. Therefore, the need for a discussion of mathematical problems (Prabawanto, 2009:54).

Arifin (Kesumawati, 2010:38) reveals troubleshooting indicators, namely (1) the ability to understand the problem; (2) the ability to plan problem solving; (3) ability to perform the work or calculations; and (4) the ability to perform inspection or pengece-kan back.

Students can succeed in solving the problem, if teachers are confident in completing various types of mathematical problems and was able to teach a variety of skills needed (Yee, 2009:54).

According to Caballero (2011:282), when students solve problems, they are often an adventure with feelings and emotions that cause tension during the search for solutions / strategies, to find a solution to these problems. This could have led to an interest, or even vice versa, hampered by negative emotions that trigger anxiety.

According to Rachmat (2001:80) there are four factors that influence the problem solving process:(1) motivation; (2) beliefs and attitudes are wrong; (3) a habit; (4) emotion.

**5. Motivation**

Motivation is generally defined as a state of the self that can generate, directing and maintaining behavior (Woolfolk, 2009:186). Motivation is not observed directly, but rather inferred from some clues as verbalization, choices tasks and activities directed at a specific destination. Motivation is a clear concept that helps us to understand why people behave the way they do (Schunk, 2012:346).

Waege (2009:85) states that the motivation of students can be shown in the consciousness (cognition), emotions (emotion) or habits (behavior). For instance, the motivation of students to get a good performance in mathematics can be shown in the excitement (emotion), if you get a high score on a test. Motivation of students can be shown by the study (behavior) to face a test, as well as in new learning concept (cognition) when studying for a test.

According to Woolfolk (2009:188), teaching can create intrinsic motivation by linking student interest and competencies that support growth. If the teacher has always stressed the intrinsic motivation to energize all of their students, he will be disappointed. Teachers should encourage and foster intrinsic motivation, extrinsic motivation while ensuring that support learning.

The following points indicate that the feedback (feedback) mem-possess strong relationships with student motivation:

a. To increase motivation to learn, the important thing to remember when giving feedback teachers, especially the negative ones is a sense of security (comfortable) students. Teachers should blow-Give negative feedback with warmth, hospitality, and far from being mocking or condescending. So students still comfortable despite getting a correction or negative feedback (Arends, 1997:160).

b. Moreover, according to Kulik (Slavin, 1997:32), so that feedbackcan provide motivation to the students, then the feedback should be given-right with a clear and specific. It is important for all levels of student development, especially for lower grade students.

c. Kulhavy and Stock (1989:280) states that feedback specific informational and motivational (moti-vasi improve student learning).

d. Clifford (1990:23) states that once a negative feedback-pun can enhance children’s learning motivation, origin focuses on the desired performance of teachers (not to the inability of students in general).

**6. Method**

The method used is the method mix (mixed method) model of Embedded Design. This method combines qualitative and quantitative methods together. In this model, there are methods of primary and secondary methods. Researchers chose quantitative methods as the primary method. And as a secondary method, researchers used qualitative data obtained from instruments of observation and interviews, in order to describe the learning process of students’ motivation to learn.

The study involved two samples of equivalent grade categories, namely, the experimental class and control class. The sample classes are formed using an existing class. Both the experimental class and the control class were not chosen at random (Sugiyono, 2015:118). In the experimental group was given treatment teacher comments on the results of homework (PR) students, while the control class is not given treatment teacher comments on the results of homework (PR) students.

The design of this study using nonequivalent control group design (Ruseffendi, 2005:52) the following:

by:

X = Giving teacher comments on the results of homework (PR) students

O = Pretest / posttest

**7. Population and Sample**

The population in this study were all students of SDN Dipawangi District of Cianjur of Cianjur Regency. Based on the understanding of researchers, students at this school have problems in problem-solving ability and motivation to learn. The samples in this research were two samples taken at random from the population. One class of samples taken serve as the experimental group, while the other class as the control class. Randomized class is a class V.

The test instrument used in this study is a test instrument and nontes. The test instrument consisted of five questions that have been tested explanation that hasthe validity, reliability, difficulty index, distinguishing as follows:

**8. Results and Discussion**

Based on analysis of mathematical problem solving ability test, the value of significance (2-tailed) was 0,311. Because Sig. (2-tailed)> α, then H0 is accepted. So that means an increase in the abi-pared with students’ mathematical problem solving treatment given teacher commenting on the results of homework (PR) is no better than students who were not given the treatment, in terms of the whole student.

Based on the results of questionnaire data calculation motivation to learn through SMI method (Method of Successive Interval) values obtained significance (2-tailed) was 0,287. Because Sig. (2-tailed) > α, then H0 is accepted. This means that the average final grade students’ learning motivation experimental and control group did not differ significantly. It can be concluded, given the treatment of student motivation teacher commenting on the results of homework (PR) is no better than students who are not given such treatment.

Through observation and interviews, researchers discovered facts on the ground that that provision of teacher comments on the results of PR can increase students’ motivation. This is in accordance with what was presented by Orsmond (Muir, 2006:26), that motivation can be generated from the feedback of written comments provided by the teacher.

But there are obstacles when mela-kukan treatment to provide written comments on the results of PR student, that teacher must provide additional time to write comments in the form of a correction, a word of praise or encouragement at every PR students. Moreover, if the teacher is assigned to the class of 40 students as the number of students in primary schools in general.

Based on the analysis on the correlation of test data that has been done, it is stated that there is no significant correlation between the abi-pared with mathematical problem solving and motivation of students. It bertenta-ngan with research results obtained Callard (2009:1), that if the student has the ability pemeca-han that will either lead to a high learning motivation.

**9. Conclusion**

Based on the results of data processing and the findings obtained in this study, obtained some conclusions as follows:

a. Mathematical problem solving ability of students treated the provision of teacher comments on homework better results than students who did not receive such treatment.

b. No difference-tan peningka mathematical problem solving ability significant, between students who received treatment commenting on the results of PR teachers and students who did not receive such treatment.

c. Student motivation treated teachers commenting on the results of PR better than students who did not receive such treatment.

d. There is no positive correlation between mathematical problem solving ability of students with student motivation.

The post Increasing Problem Solving Ability and Motivation Learning Through Grand of Math Teacher Comments on Results Homework (PR) appeared first on Zambrut.

]]>The post Impact of Two Contrasting Vermicomposts on the Fertility Status of a Sandy Soil appeared first on Zambrut.

]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: June 5, 2019

**Nweke, I. A., Ejinkonye, C. & Ogugua, U.V.**

Department of Soil Science Chukwuemeka Odumegwu Ojukwu University, Nigeria

College of Agriculture and Environmental Science, South Africa University, South Africa

Journal Full Text PDF: Impact of Two Contrasting Vermicomposts on the Fertility Status of a Sandy Soil.

**Abstract**

Vermicompost contains chemical nutrients of which has a positive effect on soil and crop life cycle. The study investigated the effect of fruit and vegetable vermicomposts respectively on the chemical characteristics of a sandy soil. The results of the study reveals that the vermicomposts studied increased the macro-nutrient (P, N, Ca Mg K and Na) contents of the sandy soil. The pH of the soil which was very acidic was increased to alkaline (9.09) by vegetable vermicompost (VGC) and neutral level (7.07) by fruit vermicompost (FVC). The OC and MC content of the soil showed elevated levels which were almost double the value recorded in the initial soil before vermicompost treatment. The two vermicomposts mediated over two fold increase in electrical conductivity, 251% for FVC and 100% for VGC respectively. When the two vermicomposts are compared the effect of VGC were more effective on pH, N, Ca, Mg, K, Na and MC of the soil compared with FVC results that was more effective on P, OC and EC contents of the sandy soil. The findings of the present study reveal that the vermicompost from biodegradable wastes has a great future for poor resource farmers and food production generally in the study zone. Hence soil prone to excessive leaching and erodible like sandy soil will no longer be a barrier to crop production when amended with vermicompost as its fertility status will be increased and erodibility greatly reduced.

**Keywords:** Electrical conductivity, exchangeable bases, organic matter, available phosphorous, sandy soil.

**1. Introduction**

Sandy soils pose many challenges to agricultural productivity in humid tropical climate and areas where there are seasonal hot dry climate. They are noted to have low water and nutrient holding capacity due to their low organic matter content and cation exchange capacity (CEC). Their plant available water according to the report of Allen (2007) is ≤ 50 -110mm per meter of soil couple with the fact that due to high soil temperatures in the tropics, soil organic carbon is rapidly lost (Jabbagy and Jackson 2000). The storage capacity for carbon of sandy soil is typically less than 1% because of the low potential to protect carbon from microbial activity (Six et al., 2006). The actual soil carbon content is however much lower than this due to low plant productivity, thus low carbon input rates. Farmers reliant on sandy soils need carefully designed and well integrated water and nutrient management system to increase their productivity and reduce adverse effects on ground water and soil acidity. The fertility of sandy soil can be upgraded through the help of alternative measure such as the use of vermicompost.

Vermicompost is finely divided mature organic matter with high level of plant nutrient availability increased surface area, aeration and drainage, microbial activity and water hold capacity, high porosity etc stabilized by interactions between earthworms and microorganisms (Nweke 2013; Aracon et al., 2008). Soils amended with vermicompost have the capacity according to the works of the following authors, Atiyeh et al. (2002), Arancon et al. (2003), Posstma et al. (2003), Perner et al. (2006), Mba and Nweke (2009) to improve soil moisture, soil aggregation, CEC, higher level of plant growth hormones and humic acids, higher microbial population and activity and less root pathogen or soil borne diseases as well as overall improvement in crop growth and yield (Arancon, et al., 2004; Nweke 2016). This study therefore, attempts to examine the influence of two contrasting vermicomposts on the fertility status of a sandy soil.

**2. Materials and Methods**

**2.1 Study Environment**

The experiment was set up at Faculty of Agriculture Chukwuemeka Odumegwu Ojukwu University Igbariam Campus Anambra State. The area is located between the Latitude 5’40 and 6’45N and longitude 6’40 to 7’20E.

**2.2 Collection of Materials**

The soil sample that was used for the experiment was collected from the Faculty of Agriculture Chukwuemeka Odumegwu Ojukwu University Igbariam campus. The soil was collected using shovel 15-20cm deep after scraping off 0-5cm from the Faculty of Agriculture premises in a plastic container. After the collection of the soil sample, dirty particles, stones and hard clods where carefully removed, aim is to ensure of fine silt before using it for the experiment. 300g of soil was weighed using weighing balance, 10 polythene bags was brought the 10 polythene bags was divided into 2, each 5 polythene bag contained 300g of sandy soil mixed with 50g fruit vermicompost (FVC) and the other 5 bags containing same measurement 300g of soil sample was also mixed with 50g vegetable vermicompost (VGC). The experiment was incubated for 2 months. The chemical properties of soil before incubation are recorded in Table 1. At the end of the study an aliquot of the sample was used to analyze for the chemical properties of soil based on the principles of Black (1965)

Table 1. Chemical properties of sandy soil before incubation with vermicompost

**2.3 Data analysis**

Data generated were subjected to T-Test analysis and mean values were compared using LSD at 5% alpha level.

**3. Result**

The nutrient content of the soil sample before incubation with the vermicompost showed that the chemical properties tested were at their lowest level (Table 1). The pH of the soil was 4.48; the available phosphorous (P) content of the soil was 10.80mgkg-1. The values of total nitrogen 0.03% (TN), organic carbon 0.49% (OC), and exchangeable bases (Ca2+, Mg2+, K+, and Na+) were generally low. The exchangeable acidity (EA), electrical conductivity (EC) and moisture content (MC) were 0.72 cmolkg-1, 110.00 µscm-1 and 16.78 % respectively. The soil contains low level of major nutrient elements. Hence the studied soil is considered poor in these essential plant nutrients.

The result presented in Table 2 showed that the mean value of soil pH measured in water (H2O) for VGC was 9.09 of which is alkaline, while FVC value obtained recorded neutral pH. The application of fruit vermicompost and vegetable vermicompost that was used to amend sandy soil respectively showed that there was great increase in the mean pH value of VGC and FVC. The available P value recorded from Table 2 in FVC was 0.82mgkg-1 higher than the value obtained from VGC. The value obtained from percentage nitrogen (N) from FVC showed that fruit vermicompost that was added to the soil sample (sandy soil) recorded low value compared to the VGC vermicompost. Organic carbon (OC) content showed that there was slight increase in FVC with the value of 0.042% while in the VGC the value obtained were low.

The Ca value from the result in Table 3 showed that both vegetable vermicompost and fruit vermicompost had great impact on the sandy soil. Mg of value 9.44cmolkg-1 recorded in VGC showed that the application of the vermicompost added to the soil sample contributed positively in Mg content of VGC than in FVC. The available K value in both FVC and VGC was low but FVC showed little change in value. For the Na; the value showed that there was slight increase in VGC than FVC. From Table 3 the study showed that the mean value of soil EA in FVC is greater than the value in VGC. The highest EC value was obtained from VGC (Table 3). Percentage moisture content (MC) result in the study showed higher value in VGC compared to FVC in the soil sample (sandy soil).

Table 2 Effect of two contracting vermicompost on sandy soil

Table 3 Effect of two contracting vermicompost on the parameters of sandy soil

**4. Discussion**

The studied soil is acidic in reaction according to the ratings of USDA-SCS (1974) and Chude et al. (2012) who considered soils of pH 4.8-5.1 to be strongly acidic in reaction. The low levels of exchangeable bases (Ca, Mg, K and Na) of the studied soil which are below their critical levels of 2.0-5.0 cmol+kg-1 moderate (Ca), 3.0-8.8 cmol+kg-1 very low to high (Mg), 2.0 cmol+kg-1 (K) (USDA, 1986) indicated that the studied soil is of low base status. This suggests appropriate amendment to provide the deficit between the inherent basic nutrients status, the amount removed by the crops and leaching losses for good crop performance. Since most of the parameters are at their lowest levels it is expected that the studied soil will benefit from vermicompost treatment because vermicompost is known to influence soil parameters positively. The implication of this is that appropriate soil amendment should be practised to realise optimum production capacity of the soil.

The findings from the study showed that there was great effect of FVC and VGC on the soil studied. This was in agreement with the report of (Lazcano et al., 2010). An increased pH range between7.07-9.09 seems to encourage mineralization of plant available nutrient observed in the study through the assistance of microbial decomposition of the vermicomposts. This invariable lead to increased EC status of the soil which could be due to reduced permeability and leaching of the soluble salts. The high OC recorded in the incubated soil visa vie the initial soil may be due to decomposition of OC compound from vermicompost. Soils with low OC have been reported to have low ability to hold cations in the exchangeable forms (Krasilinikoff et al., 2002). Increased moisture content (MC) in vermicompost amended sandy soil probably may be attributed to aggregation of the soil particles by the actions of microorganisms in the vermicompost which provide cementing action between the soil particles. Parthasarathi et al. (2008) observed that composted and worm-worked sludge increased the available soil moisture of a sandy loam soil from 10.5% to 54.4% and 31.6% respectively. In comparison of the two vermicomposts the impact of VGC on the properties of sandy soil was more effective compared to the FVC. Vermicompost emerges as one of the most feasible alternative techniques compared to conventional aerobic composting. This process is not only rapid, easily controllable, cost effective, energy saving, and zero waste, but also accomplishes the most efficient recycling of organic waste and nutrient.

**5. Conclusion**

From the findings of the study VGC performed better than the FVC in nutrient release content. The use of organic material has long been recognized in agriculture as beneficial for plant growth, yield, maintenance of soil fertility and soil amendment. Since worm worked vegetable waste and fruit waste are all use as organic manure which is chemically free and environmental safe, cheap, effortless and affordable, it is advised that farmers should make use of vermicompost in the production of crops and soil amendment.

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]]>Published on International Journal of Biology, Physics & Mathematics

Publication Date: June 3, 2019

**Forteck Aloysius Betangah**

University of Buea

South West Region, Cameroon, Central Africa

Journal Full Text PDF: An Appraisal of Mathematics Content Knowledge Learnt and Implications in the Teaching of Mathematics in English Speaking Primary Schools (Studied in South West of Cameroon, Central Africa).

**Abstract**

The study appraised mathematics content knowledge learnt and implications in the teaching of mathematics in English speaking primary schools in South west region of Cameroon. The survey research design was used on a target population of 31 mathematics teacher trainers and 6482 primary school teachers. 31 mathematics teacher trainers and 45 primary school teachers formed the accessible population. The instruments used to collect the data were a questionnaire made up of 40 close ended items. Validity and reliability of the instruments were insured with the aid of a primary school head teacher, a mathematics teacher trainer, and the supervisors. The researcher used the direct delivery technique to collect data from 31 mathematics teacher trainers on the questionnaire, 5 mathematics teacher trainers on one interview, and 45 primary school teachers on another interview. All the mathematics teacher trainers were from primary school teacher training colleges located in South West Region, while all the primary school teachers were from primary schools located in the South West Region. Data were analysed descriptively using frequencies and percentages, and also inferentially using Spearman Rank correlation. The results showed that: The mathematics content knowledge learnt in PSTTC has no significant impact on the teaching of mathematics in English Speaking Primary Schools in South West Region. Based on the findings, it is recommended that: More time should be allocated for the teaching/learning of mathematics in PSTTC, the mathematics syllabus for PSTTC should align with that of English Speaking Primary Schools, the weighting of mathematics in PSTTC should be increased so as to motivate student teachers to learn mathematics and the teaching of mathematics. Further research could be carried out on the strategies that could be employed in PSTTCs and during in-service training of primary school teachers, that would enable student teachers and primary school teachers develop positive attitudes and enthusiasm towards mathematics and the teaching of mathematics.

**Keywords:** Mathematics content knowledge, teaching of mathematics, learnt and implications.

**1. INTRODUCTION**

Mathematics is a compulsory subject in primary and secondary schools, as well as in primary school teacher training colleges (PSTTCs) in Cameroon. It is ‘the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them’ (McIntosh, 2013 p. 883). Ali (2013) opines that mathematics is an international language, a way of thinking and organizing a logical proof and it is the subject that is recognized as the mother of all learning with other subjects deriving their concepts from it, in both arts and sciences.

According to Ali (2013), mathematics is regarded as the queen of all sciences such as chemistry, physics, biology and economics, reason why any individual who is competent in mathematical sciences, can equally have the ability to do any other course. Close (2006) says that mathematics facilitates the study of academic subjects especially in the physical and social sciences, problem solving in our personal, educational, and occupational lives, and for studying and making sense of the world around us. Ali (2013) posits that mathematics can be used to determine whether an idea is true or not, or at least, whether it is probably true as a way of thinking, since it gives insight into the power of human mind and becomes a challenge to intellectual curiosity. Ali (2013) adds that mathematics is used in handling money, measurement in fashion and carpentry, as well as in technical economics. MINEDUB (2018) states that mathematics is a creative and highly inter-connected subject that is essential to everyday life, critical to science, technology, and agriculture and engineering, and also necessary for financial literacy and most forms of employment. According to MINEDUB (2018), mathematics develops logical and inferential thinking, as well as the ability to deduce and visualize in time and space. Mathematics according to Maliki, Ngban, and Ibu (2009) is described as a subject that affects all aspects of human life at different degrees (p. 131). According to The National Mathematics Advisory Panel (2008), mathematics is used throughout our daily lives. In Cameroon, mathematics is a prerequisite for admission into some professional programmes such as medicine, engineering, accountancy, agriculture and banking. It also forms part of the study of single subjects like physics, chemistry, economics, biology and geography. Generally, people in all works of life make use of some knowledge of numeracy either consciously or unconsciously. Therefore, the importance of mathematics cannot be over emphasized.

According to Close (2006, p. 53), “mathematics is a key subject in school curricula.” Therefore acquiring quality mathematics education requires quality primary education. There have been calls for primary education to be of desired quality in Cameroon and perhaps other parts of the world. Reason why one of the major concerns of the Growth and Employment Strategy Paper in Cameroon (GESP) (2010, p. 50) is to “Encourage quality primary education for all and nationwide.” In addition, one of the reforms envisaged with regard to Vision 2035 is “quality basic education” (GESP, 2010, p.74).

Apart from quality primary education, section 9 of Law No 98/004 of 14 April 1998 which laid down guidelines for education in Cameroon stipulates that primary education shall be compulsory for children of school going age to acquire basic literacy, numeracy and survival skills. Other educational programs such as the Millennium Development Goals (MDG) (2000) and the World Declaration on Education for All (1990) also emphasize the need for all children of school-going age to acquire primary education. The Draft Document of the Sector Wide Approach on Education which reflects a common and coherent vision of education in Cameroon (2005, p. 27) looks at primary school as “the major system of training, to which the state has the objective of providing a solid base for continuous training for the Cameroonian chi

**1.1 STATEMENT OF THE PROBLEM**

The mathematics curriculum for Primary School Teacher Training Colleges (PSTTCs) prescripts mathematics content and pedagogic knowledge. It therefore requires of student-teachers adequate competency in the teaching and learning of mathematics. Upon graduation, student-teachers are expected to use the mathematics content and pedagogic knowledge as well as the skills and attitudes acquired during their training, to teach mathematics in primary schools.

However, literature, experience and statistics show that percentage pass in mathematics in English Speaking Primary Schools in South West Region has remained low irrespective of the class, level and the type of school (public, confessional or lay-private). Poor performance in mathematics may result in pupils developing a negative attitude towards mathematics which they may carry along to higher classes and post-primary institutions. Pupils’ poor performance in mathematics could hinder them from studying subjects that have links with mathematics like economics, chemistry, and physics, as well as hinder them from studying professions that make use of mathematics like medicine, engineering and accounting. Generally, their use of numeracy would likely be hindered by their poor performance in mathematics. Pupils’ inability to study professions and subjects that have links with mathematics implies that by 2035 that Cameroon would be expected to emerge, there would be relatively fewer English Speaking Cameroonians as engineers, medical doctors, accountants and architects. This would likely slow down the rate at which Cameroon would emerge by 2035.

Primary school teachers are the main implementers of the mathematics curriculum for primary schools. Primary school teachers are also the guarantors of the quality of mathematics education in primary schools. Pupils’ poor performance in mathematics would suggest amongst other reasons that the teaching of mathematics in English speaking Primary Schools in South West Region still lags behind despite reforms in the mathematics syllabuses for PSTTCs in Cameroon.

Teachers’ behaviour in mathematics classrooms probably affects pupils’ performance in mathematics. The improvement in the quality of mathematics teaching is likely to succeed only if there is an adequate supply of suitably qualified mathematics teachers. It is against this backdrop that the researcher observes that there is a problem and wants to find out the impact that mathematics curriculum for Primary School Teacher Training Colleges has in the teaching of mathematics in primary schools, with the hope that findings from this study would help in improving the performance in mathematics in English speaking Primary Schools in the South West Region.

Objective, This study aims at investigating the extent to which Mathematics content knowledge learnt in PSTTC has an impact on the teaching of mathematics in primary schools.

Research Question, To what extent does the mathematics content knowledge learnt in PSTTC have an impact in the teaching of mathematics in primary schools?

**1.2 BACKGROUND**

In Cameroon indigenous education, those who played the role of teachers were parents, elders and members within age groups. According to Atayo (2000), they inculcated survival values which centred on man’s basic needs such as food, drink, health and sex. They also inculcated trans-survival values which touched directly on the quality of life. Ojong (2008) opines that teaching was done following the principle of Cameroonian indigenous education such as Functionalism. The survival values made use of some knowledge of numeracy in one way or the other. For example, counting was done in the vernacular. According to him, this form of indigenous education was functional because the curriculum was learnt and immediately applied in society. The curriculum was not written but was organised in sequence to fit the expectations of the different developmental stages recognized by the culture (Nsamenang, 2005). For example, after a child had learnt how to count in the vernacular, he/she applied it in number of grains planted in the farm, the quantity and time to take medication, as well as the number of wives and the spacing of children. The children developed interest in cooperative study, independent study and group work especially as the teaching methods were observation, imitation, and participation.

During the colonial period, primary teacher education in Cameroon was provided by missionary bodies and the government at different times, with each having a particular role, and addressing a particular population of student-teachers (Tchombe and Agbor, 2007). The earliest kind of teacher education in Cameroon appeared in the training of men to teach Christian doctrines of various religions by the late 19th century (Tchombe, 2000). Their efforts were geared towards evangelisation and training individuals to fulfil their roles as catechists, interpreters and teachers. That is why earlier missionaries trained teachers and leaders only in the basic 3Rs namely, writing, reading and arithmetic. There was no defined curriculum. However, arithmetic enabled teachers and leaders to trade, while writing and reading enabled them to read the bible. For example, by 1885 and 1907, Alfred Saker (a Baptist missionary) and the Roman Catholic Mission respectively provided such training in Douala (Gwei, 1975).

Gwei (1975) says that formal teacher training started in 1925, with the opening of a normal class at Government Primary School Victoria. The school had a secondary department that opened in 1924. Graduates from the normal class were awarded a third-class teacher certificate. Contract pupil teachers and those who passed out of class two of the secondary department were trained as teachers for two years. Ojong (2008) opines that the first formal teacher training institution in the country was opened in 1932. This was the normal class that eventually metamorphosed into the Government Teacher Training College (GTTC) in Kake, in the outskirt of Kumba. The students who were being trained at the Kake Teacher Training Centre were awarded the Grade III Teacher Certificate and prepared for teaching in the lower primary school. The curriculum included subjects such as principles and practice of education, general methods, school organisation and management, physical education, and child study as well as in the areas of pedagogy, didactics, school administration, educational psychology, environmental education, agriculture, and arts and craft. These subjects equipped the teachers with the necessary knowledge, skills and attitudes which enabled them communicate and write letters.

According to Tchombe (2000), the second phase of teacher training was categorized into two main stages. The first stage colleges were Elementary or Grade III colleges. Pupil-teachers, “C” teachers and able standard six and latter class seven pupils were admitted and underwent a three-year course which qualified them to teach in infant and junior primary schools. Students who succeeded at the completion of the course were awarded the Teachers’ Grade III Certificate while the unsuccessful ones were designated as Grade III CTR, that is, Grade III trained but uncertificated. The second stage colleges were Higher Elementary or Grade II Colleges. These colleges admitted Grade III Certificated Teachers and Secondary School Leavers and offered a two-year course leading to the Teacher’ Grade II Certificate. Those who failed at the end of the course were regarded as Grade II CRT, that is, Grade II trained but uncertificated. Tchombe in Ndongko and Tambo (2000) opines that in some cases, a four-year course was undertaken, leading to the award of a Grade II teacher’s certificate. The curriculum included subjects such as principles and practice of education, general methods, school organization and management, physical education, child study and other primary school subjects like arithmetic. The curriculum for arithmetic prepared the teachers to teach arithmetic either in the junior or senior primary school, depending on the grade of the teacher. For example the arithmetic curriculum for pupil-teachers, “C” teachers and able class seven pupils enabled them to teach arithmetic in the junior primary classes.

In 1944, the Teacher Training College in Kake was moved to its permanent site in Kumba. Since that period, more teacher training colleges have been opened by government and private agencies for both Grades III and II courses and in latter cases, for Grade I. For example, the confessional teacher training institutions started in 1944 with the Catholics opening a teacher training college in Bambui. However, mission teacher training colleges were more concerned with character formation than with intellectual development (Tchombe in Tambo, 2000). Ojong (2008) argues that between 1944 and 1957, the curriculum of the teacher training colleges included the following subjects: Arithmetic, English Language, Principles and Methods of Education and Practical Teaching, Oral English, Practical and Theoretical Rural Science, Hand Work, Physical Education, Teaching Aid and Classroom Exposition. The curriculum for females added the following: Domestic Science – Needle Work, Cookery, Child Care and Hygiene. The curriculum for arithmetic enabled teachers to teach primary school mathematics in order to prepare Cameroonians to succeed the colonial masters, especially in jobs that made use of some knowledge of numeracy.

Tchombe (2010) opines that by the early 1960s, twelve teacher training colleges were opened in English-Speaking Cameroon; one by the government and the rest by confessional agencies notably Catholic, Presbyterian, and Baptist. Of the twelve teacher training colleges, five were located in the North West Region, and seven in the South West Region. worthy of note is the fact that the lone government owned teacher training college was located in the South west Region. Seemingly, the increased number of teacher training colleges did not only increase the number of trained teachers but also increased the possibility for the curriculum of arithmetic to be learnt by many more teachers.

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