# A Tenth Order Second Derivative Numerical Integrator for the Solution of First-order Ordinary Differential Equations

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Published on International Journal of Biology, Physics & Mathematics
Publication Date: May 17, 2019

E. A. Areo, B. T. Olabode & A. L. Momoh
Department of Mathematical Sciences, Federal University of Technology
Akure, Nigeria

Abstract
This paper proposed a continuous second derivative method with -stability. The continuous method was derived by multistep collocation approach and was used to generate complementary methods for successful implementation of the derived method in block mode. The advantages of the second-derivative terms were taken in order to derive numerical methods with a high order of accuracy which is stable and suitable for the numerical solution of differential equations. In particular, second-derivative of the differential system gotten from equating the problem and assumed solution was obtained. This, however, gives freedom for more collocation at the second-derivative without loss of generality. The proposed algorithm was found to be convergent when analyzed based on the basic properties of Linear Multistep Method. The application of the method on some examples show that it is in good agreement with the exact solution.

Keywords: Continuous, Collocation & Multistep.

Introduction
This publication aimed at introducing a continuous second derivative method for numerical solution of system of first order initial value problems of ordinary differential equations of the
The need for the introduction of continuous second derivative method is as a result of its high order of accuracy, stability and suitability for numerical approximation of systems of first order ordinary differential equations of the type (1). In literature, a lot of numerical methods had been proposed for the numerical integration of equation (1) but due to their low-order of accuracies are not suitable for large stiff system of initial value problems of ordinary differential equations. This is unconnected to the fact we are familiar with the solution at grid points typically of discrete variable methods with Euler, Runge-Kutta, Adam’s, Stomer Cowell Methods etc. as examples. Several authors such as (-) have derived numerical methods where they include off-grid points in between the grid points and were generalized in both traditional numerical methods (Runge-Kutta and linear multistep methods) as a result of Dalhquist bearer theorem . In a similar vein, authors such as (-) had proposed methods where they included second derivative terms.
The continuous second derivative method has extra degree of freedom as a result of the extra stage which was exploited to increase its order of accuracy and modified the region of absolute stability. Block method generally retained the merits of one-step methods of being self-starting and of permitting easy change of step length during integration as recorded in . In this paper, we derived continuous-second-derivative block multistep method with higher order of accuracy, low error constants, large domain of stability and rapidly converge to the expected solution.

1 Analysis of the continuous second derivative block method
1.1 Order, Consistency, Zero-Stability and Convergence of the method
The associated linear difference operator of the multistep collocation is expressed as;
where is an arbitrary function which is continuously differentiable on Following  and , we can write the term in (12) as a Taylor series expansion about the point x to obtain the expression,
Definition 4 (consistency)
The continuous second derivative of high-order of accuracy methods (11) is said to be consistent if the order of method is greater than or equal to one, that is if . In addition to
Definition 5 (Zero-stability)
The second derivative block methods (11) is said to be zero-stable if the roots
Hence, the method is zero-stable.
Definition 6 (convergence) the necessary and sufficient condition for the continuous second derivative higher-order methods (11) to be convergent are that it must be consistent and zero-stable . Hence from Definitions 4 and 5 the second-derivative higher method is convergent.

1.2 Stability domain of the continuous second derivative method
The stability property of the continuous second derivative block methods as discussed in  is adopted to study the stability domain proposed method. In order to achieve this, the test equations and are applied to the block method (11) with and solving the characteristic equation
Problem I is moderately stiff ode. This was solved within the interval [0,1] with N = 500. The numerical results at some selected number of iterations are reported in table 1.
Problem II is a linear problem by Enright  which was solved by . We also solved the problem within interval [0,1] with N = 500.The numerical results are as reported on table 2. The absolute error was compared with those in .
Problem III is a linear stiff problem solved by Lambert , Yakubu  and others. The problem was solved with the interval with N = 500. The numerical results and absolute error are as reported in table 3

4. Conclusion
In this paper, we described the method of derivation of continuous second derivative method. The proposed method was analyzed and found that it has order p=10, consistent, and converge rapidly to the expected solution. Its efficiency is evident as shown in tables of results 1 − 3
The scheme is recommended for use where there is desire for higher accuracy.
Conflict of Interest: The authors declare that they have no conflict of interest.