Quartic B‐spline collocation algorithms for numerical solution of the RLW equation

Abstract

The collocation method based on quartic B-spline interpolation is studied for numerical solution of the regularized long wave (RLW) equation. The time-split RLW equation is also solved with the quartic B-spline collocation method. Numerical accuracy is tested by obtaining the single solitary wave solution. Then, interaction, undulation and evolution of solitary waves are studied. Solutions are compared with available results. Conservation quantities are computed for all test experiments.

A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation

“…We substitute the single solitary wave solution (8) with (23) and (27) and plot the resulting functions in Figure 2 to determine the effects of the modified equations (23) and 27on the single solitary wave solution of the RLW equation 1. From the results indicated in Figure 2 we note that the impact of the modified equation 27on the single solitary wave solution of the RLW equation 1is smaller than that of the modified equation (23).…”

“…We can also conclude that the finite difference scheme (26) will produce better results than (22) when used to determine numerical solutions to the RLW equation. In both cases the functions obtained after substituting the single solitary wave solution of the RLW equation (1) with the modified equations (23) and (27) propagate with the wave velocity of the single solitary wave solution.…”

“…Bhardwaj and Shankar [25] developed a numerical technique based on quintic splines. Cubic [26], quartic [27], and quintic [28,29] B-spline collocation methods applied to a splitted RLW equation have been investigated to solve (1). Irk [30] extends the work of Bhardwaj and Shankar [25] by considering an Adams-Moulton time-integration scheme coupled with a quintic spline collocation method for the spatial variable.…”

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

“…The numerical solutions at the increasing time levels are similarly obtained as in the first time step. As it can be seen in the Lie-Trotter technique, the solution of the Rosenau-KdV-RLW equation (1) subject to initial and boundary conditions (2) and (3) is obtained by solving two initial boundary value problems (7) and (8) combined with the initial conditions. The approximate solutions of Equations (7) and (8) can be found by any numerical solution methods.…”

B-spline collocation algorithms for numerical solution of the RLW equation

“…As the use of splines [27] in numerical methods to obtain the numerical solutions of differential equations leads to low-cost manageable banded matrix system, numerical methods incorporated with spline functions are produced to have numerical solutions of the RLW equation. Thus, the implementations of both collocation [17,[20][21][22]25] and Galerkin [16,18,19,26] methods with quadratic, cubic, quartic, quintic, and septic B-splines have been done in finding the numerical solutions of the RLW equation. The investigation of the solution of differential problems with B-splines is obtained by the method of weighted residuals.…”