Solutions Of Two-dimensional sine-Gordon Equation Research Articles

Analytical Solution of Two-Dimensional Sine-Gordon Equation

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.

Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements

This article studies the boundary element solution of two-dimensional sine-Gordon (SG) equation using continuous linear elements approximation. Non-linear and in-homogenous terms are converted to the boundary by the dual reciprocity method and a predictor–corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons where considered in references [Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional sine-Gordon solitons. Comput Methods Appl Mech Eng 1991;86:1–26; Bratsos AG. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2005;2(2):189–211, Bratsos AG. A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer Algorithms 2006;43:295–308; Bratsos AG. The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 2007;206:251–77; Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simul 2007;76:271–8; Christiansen PL, Lomdahl PS. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D: Nonlinear Phenom 1981;2(3):482–94; Djidjeli K, Price WG, Twizell EH. Numerical solutions of a damped sine-Gordon equation in two space variables. J Eng Math 1995;29:347–69; Dehghan M, Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 2008;197:476–86]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated.