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.

本文的目的是介绍一种新的小波方法，用于呈现由现象建模产生的多类型变阶 (VO) 分数阶偏微分方程的近似解。具体而言，本文重点研究了VO-分数阶移动-不动对流-弥散方程、Klein Gordon方程和Burgers方程的数值解。在二元 Genocchi 小波函数、它们的运算矩阵和可变阶分数 Caputo 导数算子的帮助下，这些方程被转换为代数方程组。此外，我们提出了一种新技术来获得积分和 VO 分数阶导数的运算矩阵。用于求解所提出方程的修改后的运算矩阵是强大而有效的。以便，这些矩阵的准确性直接影响实施过程。最后，我们考虑数值例子来证实该方案的优越性，并且对于每个例子，通过图表和表格展示结果。