Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations

Fadugba Sunday Emmanuel,Adebayo Kayode James,Ogunyebi Segun Nathaniel,Okunlola Joseph Temitayo

doi:10.11648/j.ijamtp.20200601.12

Fadugba Sunday Emmanuel, Adebayo Kayode James + Show 2 more

Open Access

https://doi.org/10.11648/j.ijamtp.20200601.12

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Abstract

Numerical analysis is a subject that is concerned with how to solve real life problems numerically. Numerical methods form an important part of solving differential equations emanated from real life situations, most especially in cases where there is no closed-form solution or difficult to obtain exact solutions. The main aim of this paper is to review some numerical methods for solving initial value problems of ordinary differential equations. The comparative study of the Third Order Convergence Numerical Method (FS), Adomian Decomposition Method (ADM) and Successive Approximation Method (SAM) in the context of the exact solution is presented. The methods will be compared in terms of convergence, accuracy and efficiency. Five illustrative examples/test problems were solved successfully. The results obtained show that the three methods are approximately the same in terms of accuracy and convergence in the case of first order linear ordinary differential equations. It is also observed that FS, ADM and SAM were found to be computationally efficient for the linear ordinary differential equations. In the case of the non-linear ordinary differential equations, SAM is found to be more accurate and converges faster to the exact solution than the FS and ADM. Hence, It is clearly seen that the ADM is found to be better than the FS and SAM in the case of non-linear differential equations in terms of computational efficiency.

Highlights

It is a known fact that several mathematical models emanating from the real and physical life situations cannot be solved explicitly, one has to compromise at numerical approximate solutions of the models achievable by various numerical techniques of different characteristics
The Adomian decomposition method (ADM) is an approximation method used for solving nonlinear differential equations, [12, 13]
The third order convergence numerical method devised via the transcendental function of exponential type for the solution of the initial value problem in ordinary differential equation is given by [20]

Summary

Introduction

It is a known fact that several mathematical models emanating from the real and physical life situations cannot be solved explicitly, one has to compromise at numerical approximate solutions of the models achievable by various numerical techniques of different characteristics. Development of numerical methods for the solution of initial value problems in ordinary differential equations has attracted the attention of many researchers in recent years. Problems for Ordinary Differential Equations discovered that with few iterations used, the ADM is simple, easy to use and produces reliable results [19]. Successive approximation method is a means of solving initial value problems in ordinary differential equations numerically. It is useful when the solution of the differential equations cannot be obtained analytically. The review of FS, ADM and SAM for the solution of linear and non-linear differential equations in the context of the exact solution is presented. Section Four presents the discussion of results and concluding remarks

Analysis of the Methods

Convergence Analysis of the Method

Analysis of the Adomian Decomposition Method

Analysis of the Successive Approximation Method

Illustrative Examples

Discussion of Results and Concluding Remarks