Quintic B-Spline Collocation Method for Numerical Solution of the RLW Equation

Abstract

Quintic B-spline collocation schemes for numerical solution of the regularized long wave (RLW) equation have been proposed. The schemes are based on the CrankNicolson formulation for time integration and quintic B-spline functions for space integration. The quintic B-spline collocation method over finite intervals is also applied to the time-split RLW equation and space-split RLW equation. After stability analysis is applied to all the schemes, the results of the three algorithms are compared by studying the propagation of the solitary wave, interaction of two solitary waves and wave undulation.2000 Mathematics subject classification: 65D07, 65N30, 65N35, 76B25.

A numerical solution of the RLW equation by Galerkin method using quartic B‐splines

“…We solve four test problems to illustrate the efficiency of the proposed algorithm, which are the motion of a single solitary, interaction of solitary waves, development of an undular bore and wave generation. To compare results of our scheme with that of some previous studies, all computations are carried out with the parameters used in some earlier articles [7,8,12,13,17]. …”

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

“…The magnitude of the solitary wave was measured as 0.29998, whose crest is located at x = 22 at time t = 20 for both schemes. It can be observed from (Table V) 3.97781 0.80963 2.5762 0.515 0.181 [20] (Table I) 3.96467 0.80462 2.56972 0.015 1.501 [22] (Table IV) 3.97989 0.81046 2.57900 0.095 0.039 [24] (Table I) 3 The conservation of the invariants is achieved to remain almost the same with analytical ones for the solitary wave of amplitude 0.3, but when the motion of smaller solitary wave of magnitude 0.09 is studied, changes of C 1 and C 2 are observed in the third decimal digits.…”

“…The change in water level of magnitude U(x, 0) is centered on x = x c . To make comparisons with earlier studies [8,13,18,19,22,24], the parameters are taken as ε = 1.5, µ = 0.16666667, U 0 = 0.1, x c = 0, a = −36, b = 300, mesh size h = 0.24, and time step t = 0.1, d = 2, 5. Figures 20 and 21 show the developments of the undular bores at time t = 250.…”

“…The solution of the same time-split RLW equation was also obtained with the cubic B-spline Galerkin method and cubic B-spline collocation method, respectively [21,24]. To make comparison with results of these methods, the quartic B-spline collocation method is applied to the time-split RLW equation.…”

Computational Analysis of Shallow Water Waves with Korteweg-de Vries Equation

“…In this context, there are several models that govern this wave ow. A few of these commonly studied models are Boussinesq equation [1], Kawahara equation [2], Peregrine equation [3], Benjamin-Bona-Mahoney equation [4], and several others [5][6][7][8]. This paper studies the model with the aid of Korteweg-de Vries (KdV) equation.…”