Abstract
In this study we will investigate generalized regularized long wave (GRLW)equation numerically. The GRLW equation is a highly nonlinear partialdifferential equation. We use finite difference approach for timederivatives and linearize the nonlinear equation. Then for space discretizationwe use delta-shaped basis functions which are relatively few studiedbasis functions. By doing so we obtain a linear system of equationswhose solution is used for constructing numerical solution of theGRLW equation. To see efficiency of the proposed method four classictest problems namely the motion of a single solitary wave, interactionof two solitary waves, interaction of three solitary waves and Maxwellianinitial condition are solved. Further, invariants are calculated.The results of numerical simulations are compared with exact solutionsif available and with finite difference, finite element and some collocationmethods. The comparison indicates that the proposed method is favorableand gives accurate results.
Highlights
IntroductionIf α = 1, ǫ = 0, μ = 0, γ = 0, p > 2 Eq (1) corresponds generalized regularized long wave (GRLW) equation
Consider following generalized equation ut + αux + ǫx − μuxxt − γuxx = 0, (1)− ∞ < x < ∞, t > 0 if α = 1, ǫ = 0, μ = 0, γ = 0, p = 2 Eq (1) corresponds regularized long wave (RLW) equation, if α = 1, ǫ = 0, μ = 0, γ = 0, p > 2 Eq (1) corresponds generalized regularized long wave (GRLW) equation.in which t is time, x is spatial variable and u is the amplitude, and α ≥ 0, ǫ ≥ 0, μ ≥ 0, γ ≥ 0, p ≥ 2
The GRLW equation was first proposed by Peregrine [2, 3] for description of an undular bore and by Benjamin et al [4] GRLW equation suggested as a model for long waves with small amplitudes on the surface of water in a channel
Summary
If α = 1, ǫ = 0, μ = 0, γ = 0, p > 2 Eq (1) corresponds generalized regularized long wave (GRLW) equation. If α = 0, ǫ = 0, μ = 0, γ = 0, p = 2 Eq (1) corresponds to viscous Burgers’ equation, In this paper, we will study GRLW equation numerically. Numerical investigation of nonlinear generalized regularized long wave equation via delta-shaped . Due to highly nonlinear structure of the GRLW equation, developing efficient numerical methods for this equation is a challenging work. With Dirichlet boundary conditions u(a, t) = u(b, t) = 0 by employing finite difference and delta-shaped basis functions.
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