Modified wavelet method for solving multitype variable-order fractional partial differential equations generated from the modeling of phenomena

Abstract

The aim of this paper is to introduce a new wavelet method for presenting approximate solutions of multitype variable-order (VO) fractional partial differential equations arising from the modeling of phenomena. In specific, this paper focuses on the numerical solution of the VO-fractional mobile-immobile advection-dispersion equation, Klein Gordon equation and Burgers equation. These equations are converted into a system of algebraic equations with the assistance of the bivariate Genocchi wavelet functions, their operational matrices, and the variable-order fractional Caputo derivative operator. Also, we present a new technique to get the operational matrix of integration and VO-fractional derivative. The modified operational matrices for solving the proposed equations are powerful and effective. So that, the accuracy of these matrices directly affects the implementation process. Finally, we consider numerical examples to confirm the superiority of the scheme, and for each example, exhibit the results through graphs and tables.

An Accurate Numerical Scheme for Three-Dimensional Variable-Order Time-Fractional Partial Differential Equations in Two Types of Space Domains

“…In the meantime, numerical methods for solving VO-fractional differential equations worked very powerfully. Therefore, several numerical methods have been introduced, such as fractionalorder Taylor wavelets [24], spline finite difference scheme [13], modified wavelet method [6], Genocchi collocation method [5], etc. This paper presents the numerical framework based on discrete shifted Hahn polynomials for solving 3D-VO time-fractional partial differential equations.…”

A mathematical model for precise predicting microbial propagation based on solving variable‐order fractional diffusion equation

“…In recent years, many researchers have employed the VO fractional calculus in the fields of mechanics, physics, chemistry, and biology [39][40][41]. In order to avoid probability difficulty in achieving analytical solutions to such problems, many numerical techniques have been used to solve VO fraction problems [42][43][44].…”