Spectral method for solving the equal width equation based on Chebyshev polynomials

Abstract

A spectral solution of the equal width (EW) equation based on the collocation method using Chebyshev polynomials as a basis for the approximate solution has been studied. Test problems, including the migration of a single solitary wave with different amplitudes are used to validate this algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The interaction of two solitary waves is seen to cause the creation of a source for solitary waves. Usually these are of small magnitude, but when the amplitudes of the two interacting waves are opposite, the source produces trains of solitary waves whose amplitudes are of the same order as those of the initial waves. The three invariants of the motion of the interaction of the three positive solitary waves are computed to determine the conservation properties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Comparisons are made with the most recent results both for the error norms and the invariant values.

Solitary waves for power-law regularized long-wave equation and R(m,n) equation

“…The dimensionless form of the RLW equation with power-law nonlinearity that is going to be studied in this paper is given by [1,8] …”

Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method

“…Therefore, the Hermite DAF has sufficient flexibility to handle complicated boundary conditions and geometries, like finite-difference and finite-element methods, but with an accuracy of the same order as spectral methods. Note that (15) assumes that the derivative of and approaches zero as → ±∞; this is convenient for approximating soliton solutions of the RLW equation, as they tend to zero as → ±∞. However, in general this is not the case and the complete numerical approximation must provide a way to handle boundary conditions.…”

“…In (15), if the kernel, ( ) , , is fixed to be symmetric (or antisymmetric) and invariant by translation, there must be cases where ( ) are located outside of the computational domain, [ , ], and their values are undefined there. In the present algorithm, such ( ) are to be obtained by boundary conditions.…”

“…In fact, there are two approximations involved in computing the matrix representation of the differential operator. First, the delta function is approximated by an even Hermite-series with leading Gaussian term (15), which is sufficiently smooth everywhere. Second, the integral in (13) is evaluated by Gauss quadrature approximation (14).…”

“…This corresponds to the modified regularized long wave (MRLW) equation. The MRLW was solved numerically by explicit finite-difference methods [12], Galerkin methods using cubic B-spline finite elements [13], Galerkin and Petrov-Galerkin methods using quadratic B-spline finite elements [14], spectral method based on Chebyshev polynomials [15], a least square technique using linear space-time finite elements and Petrov-Galerkin finite-element scheme with shape functions taken as quadratic B-spline functions 2 International Journal of Computational Mathematics [16], and the collocation method with quintic B-spline [17]. Recently, Mokhtari and Mohammadi [18] proposed a meshfree technique based on a global Sinc-collocation approximation to solve the GRLW equation.…”