A Numerical Solution for Hirota-Satsuma Coupled KdV Equation

Abstract

A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubicB-splines as test functions and a linearB-spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newton’s method is used to solve the block nonlinear pentadiagonal system we have obtained. The resulting schemes are of second order accuracy in both directions, space and time. The von Neumann stability analysis of the schemes shows that the two schemes are unconditionally stable. The single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitons, three solitons, and birth of solitons is also discussed.

A numerical treatment based on Haar wavelets for coupled KdV equation

“…Table 4, as it can be seen from the table our results are agree with Ref. [18]. The positions and amplitudes of waves at t = 25 are given in Table 5.…”

“…The invariants I 1 ,I 2 and I 3 [18] are monitored at the computations to check the conservation of the numerical scheme.…”

“…In Table 3, we give a comparison of our results with ref. [18] for ∆t = 0.01, λ = 0.5, a = −0.125, b = −3 and 2M = 1024. Numerical results of the present method are comparable with the other methods.…”

“…A quadratic B-spline Galerkin approach applied by Kutluay and Ucar [17] for solving cKdV system. Ismail and Ashi [18] used a Petrov-Galerkin method and product approximation technique to solve numerically the Hirota-Satsuma cKdV equation. In this paper, for obtaining numerical solutions of systems (1), we have employed Haar wavelet method.…”

Two‐level linearized and local uncoupled difference schemes for the two‐component evolutionary Korteweg‐de Vries system

“…The accuracy of this difference scheme was checked in L 2 and L ∞ norm. In , a Petrov‐Galerkin method and product approximation technique were used to solve numerically the Hirota‐Satsuma coupled KdV equation. The resulting schemes were of second‐order accuracy in both space and time directions.…”