Abstract
The approximate solution of KdV-type partial differential equations of order seven is presented. The algorithm based on one-dimensional Haar wavelet collocation method is adapted for this purpose. One-dimensional Haar wavelet collocation method is verified on Lax equation, Sawada-Kotera-Ito equation and Kaup-Kuperschmidt equation of order seven. The approximated results are displayed by means of tables (consisting point wise errors and maximum absolute errors) to measure the accuracy and proficiency of the scheme in a few number of grid points. Moreover, the approximate solutions and exact solutions are compared graphically, that represent a close match between the two solutions and confirm the adequate behavior of the proposed method.
Highlights
The importance of the nonlinear phenomena of differential equations (DEs) in sciences is significant
To understand the complete physical phenomena, the mathematical modeling comes into account by means of partial differential equations (PDEs), that shows an exceptional performance in science and engineering
We identify the integer R~ 1⁄4 2M~ and S~ 1⁄4 2R~ 1⁄4 2M~ þ1, to define discrete Haar wavelet (HW) functions, where M~ is maximum resolution level
Summary
The importance of the nonlinear phenomena of differential equations (DEs) in sciences (biology, physics and chemistry) is significant. There exist many analytical and numerical schemes like He’s variation iteration method, pseudospectral method, Adomian decomposition method (ADM), Backlund 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, 14,15,16,17], some of them are given by: Arora and Sharma [1] approximated the seventh order KdV equations for example Sawada-Kotera-Ito equation, Lax equation and Kaup-Kuperschmidt equation by HAM They used a parameter namely h to control the convergence of the method and by fixing it, the computational results were compared with the analytical solutions. The aim of this research work is to choose an authentic approximation of seventh-order KdV-type equations (Lax, Sawada-Kotera-Ito and Kaup-Kuperschmidt equations) using onedimensional Haar wavelet collocation method (1-D HWCM), that provides high quality computed results in short time with a few grid points.
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