Numerical Investigations of the Coupled Nonlinear Non-homogeneous Partial Differential Equations

Document Type : Research Paper

Authors
Z.M. Alaofi 1 , 2
T.A. El-Danaf 3
Faisal E.I. Abd Alaal 4
S.S. Dragomir 2
1 Department of Mathematics, College of Science and Arts, King Khalid University, Muhayil Asir, Saudi Arabia
2 Mathematics, College of Engineering & Science, Victoria University, Melbourne, Australia
3 Department of Mathematics, Faculty of Sciences and Arts, Taibah University, Medina, Saudi Arabia
4 Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt‎
10.22055/jacm.2022.39327.3388
Abstract
The mathematical description of various processes such as the nonlinear Klein-Gordon equation occurring in mathematical physics leads to a nonlinear partial differential equation. The mathematical model is only the first step, however, towards finding the solution of the problem under consideration. It has become possible to develop realistic mathematical models with the currently available computing power for complicated problems in science and engineering. To the best of our knowledge, systematically using the collocation method to acquire the numerical solution has not been previously used for the Klein-Gordon equation. The main aim of this paper is to systematically use the collocation method to acquire the numerical solution of the two coupled nonlinear non-homogeneous Klein-Gordon partial differential equations. We examine and analyze their stability, in detail. To this aim, we use the Von Neumann stability method to show that the proposed method is conditionally stable. A numerical example is introduced to demonstrate the performance and the efficiency of the proposed method for solving the coupled nonlinear non-homogeneous Klein-Gordon partial differential equations. The numerical results demonstrated that the proposed algorithm is efficient, accurate, and compares favorably with the analytical solutions.
Keywords
Nonlinear coupled hyperbolic Klein-Gordon equations
Nonlinear phenomena
Jacobi collocation method
Stability analysis‎
Subjects

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Journal of Applied and Computational Mechanics
Volume 8, Issue 3
Summer 2022
Pages 1054-1064