A collocation method with cubic B-splines for solving the MRLW equation

Abstract

The modified regularized long wave (MRLW) equation is solved numerically by collocation method using cubic B-splines finite element. A linear stability analysis of the scheme is shown to be marginally stable. Three invariants of motion are evaluated to determine the conservation properties of the algorithm, also the numerical scheme leads to accurate and efficient results. Moreover, interaction of two and three solitary waves are studied through computer simulation and the development of the Maxwellian initial condition into solitary waves is also shown.

Solitary wave solutions of the MRLW equation using radial basis functions

“…[25] using a finite difference scheme, in Ref. [26] using a collocation method with cubic B-splines, in Ref. [27] using the collocation method with quadratic B-spline finite elements and used the central finite difference method for time derivative.…”

Variational Homotopy Perturbation Method for the Nonlinear Generalized Regularized Long Wave Equation

“…Khalifa [2,3] implemented the Adomian decomposition method and and the cubic B-spline collocation procedure to obtain the numerical solution of the GRLW equation. Wang Ju-Feng [11] used Meshless method for nonlinear GRLW.…”

A New Functional Iterative Algorithm for the Regularized Long-Wave Equation Using an Integral Equation Formalism

“…Next, we move on to the inverse Laplace transform L −1 on the second term of (2.13), 22) where the formula 1/(s − a) = L exp(at) for a ∈ C and Laplace convolution theorem are used [10,12]. And we take the inverse Fourier transform F −1 on the above result, giving a triple integral I 2 ,…”

“…For simulating a wave train, we first consider an initial value problem of the governing equation (1.1) and the initial conditions 17) or, alternatively, 18) for constants A i > 0 and B > 0, which are known as the Maxwellian conditions [22]. And then, the algorithm (3.24) and (3.25), which corresponds to α = 0, is iterated to solve the above initial value problem for three different A i > 0, i.e., A 1 = 0.01, A 2 = 0.20 and A 3 = 0.27, where the concerning computational domains are selected as 20m < x < 140 m (M = 481, x = 1/3 m) and 0sec< t < 35sec (N = 211, t = 1/6 sec); equivalently, -49< x * < 343( x * ≈ 0.82), 0< t * < T = 268( t * ≈ 1.28) in dimensionless form.…”