Abstract
In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.
Highlights
Wave propagation problems are studied in several areas of engineering, physics, and applied mathematics including relativistic quantum mechanics, acoustics, biomedical engineering, and field theory problems
The following theorem states the continuous dependence of weak solutions for (10), and this theorem will be used in the proof of uniqueness
We propose a unified numerical method to obtain more accurate results for the solution of an initial boundary value problem (IBVP) for one dimensional coupled sine-Gordon equations
Summary
Wave propagation problems are studied in several areas of engineering, physics, and applied mathematics including relativistic quantum mechanics, acoustics, biomedical engineering, and field theory problems (see, [1,2,3,4,6,9,11,14,22,25,27,31] and the references given therein). There have been extensive theoretical and numerical studies on nonlinear wave systems such as sine-Gordon, Klein–Gordon, and coupled sine-Gordon equations in the literature (see, [7, 10, 21, 24, 30] and the references given therein). Such type of problems attracted much attention in the last decades due to the presence of soliton solutions. Several types of prey–predator models are investigated in [35]
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