Crank-Nicolson-DQM based on cubic exponential B-splines for the approximation of nonlinear Sine-Gordon equation

Abstract

In this paper, Crank-Nicolson differential quadrature method based on cubic exponential B-spline (CExpB-spline) functions is presented to approximate the 1D nonlinear hyperbolic Sine-Gordon equation (SGE). The time derivative is discretized by the usual forward difference scheme while the differential quadrature method (DQM) is used to integrate the spatial derivatives. The discretization of the problem gives systems of linear equations. Three numerical examples are chosen to investigate the efficiency and accuracy of the method. It is observed that the proposed method provides excellent results than the existing methods. The rate of convergence (ROC) of the present method is obtained numerically showing that the method is second-order accurate in space.

Keywords

• Hyperbolic SGE
• DQM
• CExpB-spline functions
• Crank-Nicolson scheme
• ROC