An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

Abstract

In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

An efficient wavelet‐based approximation method for solving nonlinear fractional‐time long wave equations: An operational matrix approach

“…where γ is level of resolution τ is translation parameter r is order of the Gegenbauer polynomials Hence, the Gegenbauer wavelets on the interval [0, 1] are defined by [38]…”

“…Also, Singh [32][33][34][35][36][37] established the approximate algorithms for solving nonlinear and fractional differential equations arising in engineering. Secer and Ozdemir [38] established the Gegenbauer wavelet method for solving the time-fractional KdV types of PDEs. Srivastava et al [39] used the Gegenbauer wavelet method for the approximate solutions of the fractional Bagley-Torvik equation.…”

Wavelet-based Numerical Approaches for Solving the Korteweg-de Vries (KdV) Equation

“…Up to now, a huge of papers have focused on this topic. Some of these methods are the Gegenbauer wavelets methods [6,26,27,31], the Legendre wavelet operational matrix method [33], the Münz wavelets collocation method [3], the Jacobi wavelets method [32], wavelet-Taylor-Galerkin methods [4], Genocchi wavelet method [9] and the discontinuous Legendre wavelet Galerkin method [39].…”

The second elliptic equation method for non-linear equations with variable coefficients

“…Eq. 1is also called as KdV-Burgers -Kuramoto(KBK) equation, and has been widely studied, many solution properties have been revealed [3,4,5]. However, much literature focused on constant coefficient, and its partner with variable coefficients was rarely studied: (2) This equation can model many real problems arising in plasma physics, fluid dynamics and thermodynamics, and this paper aims at solving Eq.…”