Abstract
A meshless method based on the singular boundary method is developed for the numerical solution of the time-dependent nonlinear sine-Gordon equation with Neumann boundary condition. In this method, by using a time discrete scheme to approximate the time derivatives, the time-dependent nonlinear problem is transformed into a sequence of time-independent linear boundary value problems. Then, the singular boundary method is used to establish the system of discrete algebraic equations. The present method is meshless, integration-free, and easy to implement. Numerical examples involving line and ring solitons are given to show the performance and efficiency of the proposed method. The numerical results are found to be in good agreement with the analytical solutions and the numerical results that exist in literature.
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