A New Modified Approach For Solving Seven-order Sawada-kotara Equations

Abstract

Herein, Reconstruction of Variational Iteration Method (RVIM) is used for computing solutions of the seventh-order Sawada-Kotera equation (sSK) and a Lax's seventh order KdV equations (LsKdV). The results are compared with the Adomian decomposition method (ADM) and the known analytical solutions. Results obtained expose effectiveness and capability of this method to solve the seven-order Sawada-Kotera (sSK) and a Lax's seven-order KdV (LsKdV) equations.

On classification of high-order integrable nonlinear partial differential equations

“…But there was no information on general structure of these equations. We discovered, apart from known completely integrable cases (Lax and SK hierarchies, [3,5,4,8,9,10,11,12]), one hierarchy of two-soliton equations (new) and countable family of isolated two-soliton equations (tail).…”

“…These selections keep symmetry property of α 12 , so its zeros are reflected poles by Im(z) axis. This principle allows equation (E n p ) α12 (u) with two-soliton solution in form (8) to be determined explicitly. It is uniqely defined by α 12 .…”

A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven

“…In literature, there exist many analytical and numerical schemes like He’s variation iteration method, pseudospectral method, Adomian decomposition method (ADM), Bäcklund transformations to solve such problems, finite difference method (FDM), finite element method (FEM), finite volume method (FVM), homotopy analysis method (HAM), Fourier spectral method (FSM) and variation iteration method (VIM) to solve such problems [ 1 , 2 , 14 – 17 ], some of them are given by:…”

“…Saravi et al . [ 14 ] reconstructed the variational iteration method (VIM) for the numerical solution of Lax’s and Sawada-Kotera equations of order seven. They compared the results with the Adomian decomposition method (ADM) and existing analytical solutions.…”

New Implementation Of Reproducing Kernel Hilbert Space Method For Solving A Class Of Third-order Differential Equations

“…This technique has successfully been treated singular linear two-point boundary value problems [11,12],singular nonlinear second-order boundary value problems [13,14,15,16], nonlinear system of boundary value problems [17],third-order boundary value problems [18,19], fifth-order boundary value problems [20], and nonlinear partialdifferential equations [21] in recent years. This paper investigates the approximate solution of the following third-order boundary value problem using new implementation of reproducing kernel Hilbertspace method [23,24,25]. As we known, Gram-Schmidt orthogonalization process isnumerically unstable and in addition it may take a lot oftime to produce numerical approximation.…”