In this paper, a new computational scheme based on operational matrices (OMs) of two‐dimensional wavelets is proposed for the solution of variable‐order (VO) fractional partial integro‐differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two‐dimensional Legendre wavelets. By implementing two‐dimensional wavelets approximations and the OMs of integration and variable‐order fractional derivative (VO‐FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples.

中文翻译：

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二维小波运算矩阵法求解变阶分数阶偏微分方程的新方法

本文提出了一种基于二维小波运算矩阵（OMs）的新计算方案，用于求解变阶（VO）分数阶部分积分微分方程（PIDE）。为了实现此方法，已使用二维Legendre小波导出了积分的第一OM和VO分数导数（FD）。通过实现二维小波逼近以及积分和可变阶分数阶导数（VO-FD）以及配置点的OM，VO分数部分PIDE被简化为代数方程组。除此之外，还讨论了一些有用的定理，以建立所提出数值技术的收敛性分析和误差估计。此外，