IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 18, Issue 1 Ser. II (Jan. – Feb. 2022), PP 49-58 www.iosrjournals.org DOI: 10.9790/5728-1801024958 www.iosrjournals.org 49 | Page Adomian Decomposition Method for Solving Coupled Nonlinear System of Klein-Gordon Equations Bii Albert 1 , Oduor Michael 2 , Rotich Titus 3 1 Department of Mathematics and Computer Science, University of Eldoret, Kenya 2 Department of Pure and Applied Mathematics, Jaramongi Oginga Odinga University,Kenya. 3 Department of Mathematics, Moi University, Kenya. Abstract; In this paper, we apply Adomian Decomposition Method(ADM) for solving coupled nonlinear Klein-Gordon equations (CNLKGE) which arise in particle physics, wave theory and other physical phenomena of linear and nonlinear nature. The numerical solutions of CNLKGE have been compared to the exact solutions and presented graphically. The numerical results are in good agreement with exact solutions which shows the efficiency and reliability of the proposed algorithm. Keywords; Adomian Decomposition Method ;Adomian Polynomials; Coupled nonlinear Klein-Gordon Equations; Recursive algorithms. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 16-01-2022 Date of Acceptance: 31-01-2022 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction A wide variety of physically significant problems such as Coupled Nonlinear system of Klein-Gordon Equations (CNLKGE) , has been the focus of extensive studies for the last decades. In order to better understand these nonlinear behaviour, many researchers developed more accurate approaches to the solutions. The Nonlinear coupled Klein-Gordon equation was first studied by [1,2,3]. Recently, [6,7,8] have solved CNLKGE by using ADM and their modified and extended versions. In this paper, Our aim is to develop ADM for CNLKGE and compare the solutions with the exact solutions given by [1]. In 1994, the Adomian Decomposition Method (ADM) was introduced by George Adomian [5].The method does not require any transformation, discretization, perturbation or any restrictive assumptions but it utilises Adomian polynomials to handle nonlinear terms. The structure of the paper is as follows; Section II gives the basic concepts of ADM and how to generate the Adomian Polynomials. Section III develops the recursive algorithms for the CNLKGE. Section IV tabulates the results for various orders and compares them and their convergence. Section V concludes the study. The system to be solved is defined by; 3 3 1 2 ( , , , ) 2 2 ( 1) 2 2 0, [0, ], 0, [0, ], 0 (1) ( , , , ) 4 0, [0, ], 0, [0, ], 0 (2) xx tt xx tt f f t x t x t t f f F u vu u u u u u uv u u uv x L L t t t F uu v v v v uu x L L t t t                       Subject to the initial conditions     1 2 2 2 2 2 3 2 1 ( ,0) ( ) sec , [0, ], 0 (3) 1 1 ( ,0) ( ) sec tanh , [0, ], 0 (4) 1 1 1 2 ( ,0) ( ) sec , [0, ], 0 (5) 1 1 t c x u x x h x L L c c c x x u x x h h x L L c c c c x v x x h x L L c c                                                             Where 2 1 c 