Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Abstract

Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.

Numerical solution of nonlinear sine–Gordon equation by using the modified cubic B-spline differential quadrature method

“…The L ∞ and RMS norms at the different time levels are shown in Table 2. From Table 3, it can be observed that the MCB-DQM results are better than the results obtained in [17,18,20]. Fig.…”

“…Dehghan & Shokri [14] 0.25 2.43 6 10 − × 5.46 6 10 − × 1.18 5 10 − × 2.32 5 10 − × 3.91 5 10 − × 5.89 6 10 − × 0.50 5.54 6 10 − × 7.39 6 10 − × 4.19 5 10 − × 4.11 5 10 − × [14], and Mittal & Bhatia [20]. Table 5 shows the 2 L and L ∞ error norms at the different time levels, quantatively.…”

“…Table 2 shows the 2 L and L ∞ error norms at the different time levels. It can be seen from [20]. The L ∞ and RMS norms at the different time levels are shown in Table 2.…”

“…Firstly we compute the numerical results in the domain [−1, 1] using the parameters: 0.0001 t ∆ = and the space step size 0.04 h = and compare our results with the results available in the literature [14,20]. Table 2 shows the 2 L and L ∞ error norms at the different time levels.…”

Numerical solution of some nonlinear wave equations using modified cubic B-spline differential quadrature method

“…In this work, we present the differential quadrature method based on modified cubic B-spline basis functions(as used in [7], [23]), to solve some one dimensional non-linear wave equations. The wave equation is converted into system of partial differential equations and then equations are discretized spatially by modified cubic B-spline functions based differential quadrature method.…”