Gegenbauer wavelet collocation method for the extended Fisher‐Kolmogorov equation in two dimensions

Abstract

Gegenbauer wavelets operational matrices play an important role in the numeric solution of differential equations. In this study, operational matrices of rth integration of Gegenbauer wavelets are derived and used to obtain an approximate solution of the nonlinear extended Fisher-Kolmogorov (EFK) equation in two-space dimension. Nonlinear EFK equation is converted into the linear partial differential equation by quasilinearization technique. Numerical examples have shown that present method is convergent even in the case of a small number of grid points. The results of the presented method are in a good agreement with the results in literature. K E Y W O R D S approximate solution, extended Fisher-Kolmogorov equation, Gegenbauer wavelets, quasilinearization technique

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].…”

Gegenbauer wavelet quasi‐linearization method for solving fractional population growth model in a closed system

“…This gives us a profound inspiration to present and apply the Gegenbauer wavelets in solving differential equations. For more details about Gegenbauer wavelets, the reader is referred to previous works 24,25 …”

“…For more details about Gegenbauer wavelets, the reader is referred to previous works. 24,25 In the present work, we develop a novel wavelets collocation method by utilizing the Gegenbauer polynomials and wavelets as their basic functions along with a quasi-linearization technique for solving the population growth model of fractional order. The main idea behind this joint venture lies in the fact that the quasilinearization technique is first used to linearize the non-linear population growth model and then, the updation of the numerical solution at each iteration is performed by the Gegenbauer wavelet method.…”