Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

Abstract

In this article, the generalized Rosenau-KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie-Trotter and the second-order Strang time-splitting techniques combined with the quintic B-spline collocation by the help of the fourth order Runge-Kutta (RK-4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L 2 and L ∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.

A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

“…. , u n−1 and u n satisfy ( 18)- (20) for n ≤ N − 1. Next, we prove that there is a u n+1 that satisfies the discrete scheme ( 18)- (20).…”

“…We consider the Rosenau-KdV equation We discretize the problem (46)-(48) using the numerical scheme ( 18)- (20).…”

“…The authors of [19] proposed a Crank-Nicolson meshless spectral radial point interpolation (CN-MSRPI) method for the nonlinear Rosenau-KdV equation. The authors of [20] solved the equation by the first-order Lie-Trotter and second-order Strang time-splitting techniques combined with quintic B-spline collocation, while [21] studied numerical solutions by using the subdomain method based on sextic B-spline basis functions. Although various methods have been proposed, we wonder whether there might be a new method with higher accuracy and efficiency that can keep some properties of the original partial differential equation.…”

A Novel Generalized n-Dimensions Sixtic B-Spline Function to Solving n-Dimensions Mathematical Models

“…A modifed cubic B-spline diferential quadrature method was employed by Elsherbeny and colleagues to investigate the 2D-Poisson equation [7]. Kutluay et al utilized modifed bi-quintic B-splines to solve the twodimensional unsteady Burgers' equation, Poisson equation, and difusion equations [8][9][10]. Further advancements were made by Raslan et al, who explored the generalization of B-spline functions in n-dimensions for solving partial diferential equations.…”

Two efficient numerical methods for solving Rosenau-KdV-RLW equation

“…Uçar et al (2017) obtained numerical solutions of the equation using Galerkin cubic B-spline FEM. Kutluay et al (2019) obtained numerical solutions of the equation by time splitting techniques. If is taken in Equation (1), then it is called the Rosenau-RLW equation.…”