Using Adomian decomposition methods for solving systems of nonlinear partial differential equation

International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 10, Issue 2, pp: (14-22), Month: October 2022 - March 2023, Available at: www.researchpublish.com

Using Adomian decomposition methods for solving systems of nonlinear partial differential equations Abdallah Habila Ali1, Abrar Yousif Elriyah2 Sudan University of Science and Technology, College of Science, Department of Mathematics (Sudan) 1 Mashreq University, College of Science, Department of Mathematics (Sudan) 2 DOI: https://doi.org/10.5281/zenodo.7182781

Published Date: 10-October-2022

Abstract: In this paper, we apply the Adomian decomposition method (ADM) and Modified decomposition method (MDM) on two different types of nonlinear partial differential equations (PDEs), has been solved by using the homotopy perturbation method combined with new transform (NTHPM). But after solved by (MADM) we found (MADM) has less of computational work than (NTHPM), more effective, powerful and simple than (NTHPM). Keywords: Systems of nonlinear partial differential equations, Adomian decomposition method, Modified decomposition method, homotopy perturbation method combined with new transform.

I. INTRODUCTION The system of PDEs arises in many areas of mathematics, engineering and physical sciences. These systems are too complicated to be solved exactiy so it is still very difficult to get closed form solutions for most problems. A vast class of analytical and numerical methods has been proposed to solve such problems. Such as the Adomian decomposition method (ADM) [1,2], the variational iteration method [3,4], the homotopy perturbation method (HPM) [5-7], and the differential transform method [8,9]. But many systems such as system of high dimensional equations, the required calculations to obtain it is solution in some time may be too complicated. Adomian decomposition method has been receiving much attention in recent years in applied mathematics in general, and in the area of series solutions in particular[10-13]. The method proved to be powerful, effective, and can easily handle a wide class of linear or nonlinear, ordinary or partial differential equations, and linear and nonlinear integral equations [1317]. 1.1 The Adomian Decomposition Method In this section of nonlinear partial differential equations will be examined by using Adomian decomposition method. Systems of nonlinear partial differential equations arise in many scientific models such as the propagation of shallow water waves and the Brusselator model of chemical reaction-diffusion model. To achieve our goal in handling systems of nonlinear partial differential equations , we write a system in an operator form by 𝐿𝑡 𝑢 + 𝐿𝑥 𝑣 + 𝑁1 (𝑢, 𝑣) = 𝑔1

𝐿𝑡 𝑢 + 𝐿𝑥 𝑣 + 𝑁2 (𝑢, 𝑣) = 𝑔2

(1)

With initial data 𝑢(𝑥, 0) = 𝑓1 (𝑥), 𝑣(𝑥, 0) = 𝑓2 (𝑥),

(2)

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