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.

Automated summary

The reduced differential transform method (RDTM) is used to solve two-dimensional nonlinear sine-Gordon equations with appropriate initial conditions, generating analytical approximate or exact solutions in a convergent power series form. The method considers the use of appropriate initial conditions without discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the method are demonstrated by test problems, indicating promise for solving other types of nonlinear partial differential equations in science and engineering.