Application of the Adomian Decomposition Method (ADM) to Solving the Systems of Partial Differential Equations

Gabriel Bissanga,Joseph Bonazebi Yindoula,Justin Mouyedo Loufouilou

doi:10.11648/j.ijamtp.20210701.14

Abstract

Solving systems of partial differential equations (linear or nonlinear) with dirchelet boundary conditions has rarely made use of the Adomian decompositional method. The aim of this paper is to obtain the exact solution of some systems of linear and nonlinear partial differential equations using the adomian decomposition method.After having generated the basic principles of the general theory of this method, five systems of equations are solved, after calculation of the algorithm.Our results suggest that the use of the adomian method to solve systems of partial differential equations is efficient.However, further research should study other systems of linear or nonlinear partial differential equations to better understand the problem of uniqueness of solutions and boundary conditions.

Highlights

Over the past 25 years, the adomian decomposition method (ADM) [1], first introduced by American physicist George Adomian, has been used to efficiently and solve a large class of ordinary linear and nonlinear and partial differential equations
Adomian and Rach [16] have shown the efficiency of this method in solving nonlinear BVPs in several dimensions various ordinary and partial differential equations with Dirichlet conditions and Neumanntype boundary conditions
Dehghan [30] applied the ADM to solve a two-dimensional parabolic equation subject to non-standard limit specifications, little attention has been devoted to the application of ADM in solving systems of partial differential equations with Neumann boundary conditions, the mathematical difficulties encountered in solving these systems of partial equations this brought led the researchers to develop several techniques to obtain approximate or exact solutions capable of best describing the physical laws and the observed phenomena

Summary

Introduction

Over the past 25 years, the adomian decomposition method (ADM) [1], first introduced by American physicist George Adomian, has been used to efficiently and solve a large class of ordinary linear and nonlinear and partial differential equations. Dehghan [30] applied the ADM to solve a two-dimensional parabolic equation subject to non-standard limit specifications, little attention has been devoted to the application of ADM in solving systems of partial differential equations with Neumann boundary conditions, the mathematical difficulties encountered in solving these systems of partial equations this brought led the researchers to develop several techniques to obtain approximate or exact solutions capable of best describing the physical laws and the observed phenomena This is the case for systems of equations resulting from the Brusselator diffusion-reaction model, the resolution of which appealed among others to the Sumudu method [19] and variational iterations [24, 31, 32]. Taking into account the difficulties presented in the determination of the exact solution of this system, the decompositional method of Adomian seems to circumvent this one by the use of recursive relations developed in due course, the aim of this paper to determine the exact solutions of some systems of partial differential equations (linear and nonlinear) using the Adomian decompositional method

About the Adomian Decomposition Method

Remark

Example 1

Example 3

Example 4

Conclusion