Abstract
In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L2 and L∞ error norms are computed to study the accuracy and the simplicity of the presented method.
Highlights
Discretization using finite differences in time and spectral methods in space has proved to be very useful in solving numerically non-linear Partial Differential Equations (PDEs) describing wave propagation
In [4] [5] the combination of spectral methods and finite differences is applied to well-known nonlinear PDE of the Boussinesq type which admits bidirectional wave propagation, has closed form solitary wave solutions and like the Korteweg de Vries (KdV) is completely integrable in one space dimension
The modified equal width wave equation based upon the Equal Width Wave (EW) equation [7] [8] which was suggested by Morrison et al [9] is used as a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes
Summary
Discretization using finite differences in time and spectral methods in space has proved to be very useful in solving numerically non-linear Partial Differential Equations (PDEs) describing wave propagation. A combination of spectral method and leap frog is applied to the modified equal width wave equation. Esen applied a lumped Galerkin method based on quadratic B-spline finite elements which have been used for solving the EW and MEW equations [14] [15]. Zaki considered the solitary wave interactions for the MEW equation by collocation method using quintic B-spline finite elements [17] and obtained the numerical solution of the EW equation by using least-squares method [18]. Evans and Raslan [22] studied the generalized EW equation by using collocation method based on quadratic B-splines to obtain the numerical solutions of a single solitary waves and the birth of solitons. The proposed method is validated by studying the motion of a single solitary wave, development of interaction of two positive solitary waves and development of three positive solitary waves interaction for the MEW Equation (1)
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