Solving a Class of Second Order Delay Differential Equation by Using Adams and Explicit Runge-Kutta Method

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Published on International Journal of Biology, Physics & Matematics
ISSN: 2721-3757, Volume 1, Issue 1, page 11 – 17
Publication Date: 9 September 2018

Zulqurnain Sabir, Muhammad Umar, & Canan Unlu

Zulqurnain Sabir & Muhammad Umar
Department of Mathematics, Unique Science College for Girls Dina
Jhelum, Pakistan

Canan Unlu
Department of Mathematics, Istanbul University
Istanbul, Turkey

Journal Full text PDF: Solving a Class of Second Order Delay Differential Equation by Using Adams and Explicit Runge-Kutta Method.

Abstract
In this study, an analysis has been carried out for solving a class of second order delay differential equation by exploiting the strength of the Adams and explicit Runge-Kutta method. The applicability and reliability of proposed approaches are validated by solving two different variants of delay differential equation. The obtained numerical results from both of the schemes are compared with the exact solution that shows the correctness and validity of the designed scheme. These obtained numerical results are also tabulated and shown graphically.

Keywords; Delay Equation, Adams Method, Explicit Runge-Kutta Method, Predictor, Corrector.

1. Introduction
In recent years, the studies of delay differential equations (DDEs) have developed very rapidly and intensively. Problems involving these equations in the modeling of non-Newtonian fluid mechanics, astrophysics and population dynamics [1-3]. Such problems also occur in the mathematical modeling of numerous physical and biological phenomena, for example, optically bi-stable devices [4], in evolutionary biology [5], narrative of the human pupil light reflex [6], physiological methods or diseases [7], activation of neuronal variability [8] and many more [9-10]. The solution of ordinary differential equations (ODEs) represents the only present state while DDEs comprise the past state.
There are several numerical techniques that have been suggested for solving the second order DDEs directly, like as the Adams Moulton technique [11], two-phase one-step block technique [12], spline collocation technique [13] and the Adomian decomposition method [14]. Many researchers such as San et al. [15], Suleiman [16], Ismail et al. [17] and Ishak et al. [18] modified the first order ODEs into DDEs. San et al. [15] solved directly the DDEs using direct one-point technique of multistep of second order. The one-point method workout the single solution at one step, Rasdi et al. [12] used to solve the DDEs directly using the two-phase and three-phase one-step block method. The techniques in Rasdi et al. [12] can compute simultaneously two/three solutions in a block. In the recent years, researchers’ community is interested to obtain the numerical solutions of the DDEs. A variety of numerical techniques have been accessible by Kadalbajoo and Sharma [19-20] for singular perturbed DDEs with only negative shift. Mirzaee and Hoseini [21] used collocation technique and matrices of Fibonacci polynomials to explain differential-difference equation with positive and negative shifts. Genga et al. [22] conversed numerical treatment of singularly perturbed DDEs by replicating kernel technique. Ramos [23] explained a nemerious exponential techniques for the numerical results of linear ordinary differential-difference equations with a slight delay, that is based on piece-wise analytic results of advection reaction diffusion operators. Moreover, Jugal Mohapatra and Srinivasan Natesan [24] assembled a numerical technique for singularly perturbed differential difference equation with minor delay.
The aim of the recent study is to investigate a class of second order DDEs using the Adams method and explicit Runge-Kutta (RK) method. The correctness of the proposed results is established to compare the exact results for each problem. The salient geographies of the given method are briefed as:
a. A novel application of numerical methods to determine the excellence solutions for nonlinear DDEs with higher accuracy and superior reliability;
b. The performance of calculated numerical results with the exact solution proves the correctness of the designed methodology …………

2. Mathematical Formulation
The second order DDEs are of the form of (Formulation on Table 1).
Where the initial function is (x) and the delay term is τ. Our purpose of studying the generic form of the DDEs and its solutions is to increase a wider understanding of the general form and to grow analytically tools to explore these equations to cover more applications.

3. Numerical Procedure
In this section, the procedures based on stochastic and deterministic numerical solvers has been implemented broadly in varied fields, for example nanotechnology [25], nonlinear algebraic equations [26], Thomas-Fermi model [27], doubly singular nonlinear systems [28] and multi-point boundary value problems [29]. In the present study, the strength of predictor-corrector Adams technique and explicit Runge-Kutta numerical technique is exploited to solve the second order DDEs………