In this manuscript, we present a new numerical technique based on two-dimensional Müntz–Legendre hybrid functions to solve fractional-order partial differential equations (FPDEs) in the sense of Caputo derivative, arising in applied sciences. First, one-dimensional (1D) and two-dimensional (2D) Müntz–Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann–Liouville operational matrix of 2D Müntz–Legendre hybrid functions is presented. Then, applying this operational matrix and collocation method, the considered equation transforms into a system of algebraic equations. Examples display the efficiency and superiority of the technique for solving these problems with a smooth or non-smooth solution over previous works.

中文翻译：

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二维Müntz–Legendre混合函数：求解分数阶偏微分方程的理论和应用

在本手稿中，我们提出了一种基于二维Müntz-Legendre混合函数的新数值技术，用于解决应用科学中从Caputo导数意义上的分数阶偏微分方程（FPDE）。首先，构造一维（1D）和二维（2D）Müntz-Legendre混合函数，并提供它们的性质。接下来，介绍了二维Müntz-Legendre混合函数的Riemann-Liouville操作矩阵。然后，应用此运算矩阵和并置方法，将考虑的方程式转换为代数方程式系统。实例显示了采用平滑或非平滑解决方案解决这些问题的技术的效率和优越性。