Numerical treatment of the Sine-Gordon equations via a new DQM based on cubic unified and extended trigonometric B-spline functions

Abstract

The purpose of this work is to propose a new composite scheme based on differential quadrature method (DQM) and modified cubic unified and extended trigonometric B-spline (CUETB-spline) functions to numerically approximate one-dimensional (1D) and two-dimensional (2D) Sine-Gordon Eqs. (SGEs). These functions are modified and then applied in DQM to determine the weighting coefficients (WCs) of spatial derivatives. Using the WCs in SGEs, we obtain systems of ordinary differential equations (ODEs) which is resolved by the five-stage and order four strong stability-preserving time-stepping Runge–Kutta (SSP-RK_5,4) scheme. This method's precision and consistency are validated through numerical approximations of the one-and two-dimensional problems, showing that the projected method outcomes are more accurate than existing ones as well as an incomparable agreement with the exact solutions is found. Besides, the rate of convergence (ROC) is performed numerically, which shows that the method is second-order convergent with respect to the space variable. The proposed method is straightforward and can effectively handle diverse problems. Dev-C++ 6.3 version is used for all calculations while Figs. are drawn by MATLAB 2015b.