In this paper, we consider an important kind of fractional partial differential equations, namely multi-term time-fractional mixed sub-diffusion and diffusion-wave equation. The crucial importance of the considered equation is due to the fact that it generalizes some substantial types of fractional differential equations that can be widely used in describing many real-life phenomena, some of these equations are the time-fractional sub-diffusion, time-fractional diffusion-wave and time-fractional diffusion equations. In this study, the 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation is transformed to its integrated form with respect to time. An extension of the operational matrix of second-order derivative to the 2D case is used in combination with the operational matrix of fractional-order integrals and the time-space spectral collocation method to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation via a spectral approach. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

在本文中，我们考虑了一类重要的分数阶偏微分方程，即多项式时间分数混合次扩散和扩散波方程。所考虑方程式的至关重要性在于，它概括了一些基本类型的分数阶微分方程式，这些分数阶微分方程式可广泛用于描述许多现实生活中的现象，其中一些方程式是时间分数次扩散，时间离散。分数扩散波和时间分数扩散方程。在这项研究中，二维多维时间分数混合子扩散和扩散波方程相对于时间被转换为其积分形式。将二阶导数运算矩阵扩展到2D情况，结合分数阶积分运算矩阵和时空谱配点方法，将此类方程简化为代数方程组，可使用任何合适的求解器。据作者所知，这是首次尝试通过频谱方法处理2D多项式时间分数混合亚扩散和扩散波方程。提供了数值示例，以强调该方法的收敛速度和灵活性。我们的结果证实，非局部数值方法最适合离散化分数阶微分方程，因为它们自然地考虑了解决方案的整体特性。