In this work, we consider multidimensional diffusion–reaction equations with time-fractional partial derivatives of the Caputo type and orders of differentiation in $(0,1)$. The models are extensions of various well-known equations from mathematical physics, biology, and chemistry. In the present manuscript, we will impose initial–boundary data on a closed and bounded spatial multidimensional domain. Single-term and multi-term fractional systems are considered in this work. In the first stage, we show that the fractional models possess energy-like functionals which are dissipated in $L^2\left(\Omega \right)$ with respect to time. The systems are investigated rigorously from the analytical point of view, and dissipative numerical models to approximate their solutions are proposed and rigorously analyzed. Our discretizations will make use of the uniform $L1$ approximation scheme to estimate the time-fractional derivatives, and the usual central-difference operators to approximate the spatial Laplacian. To that end, various results of the literature will be crucial, including some useful discrete forms of Paley–Wiener inequalities. Some numerical examples are included to show the asymptotic behavior of the numerical methods and, ultimately, their dissipative character.

在这项工作中，我们考虑了具有 Caputo 型时间分数偏导数和微分阶数的多维扩散反应方程 $(0,1)$. 这些模型是数学物理、生物学和化学中各种著名方程的扩展。在本手稿中，我们将在封闭和有界空间多维域上施加初始边界数据。在这项工作中考虑了单项和多项分数系统。在第一阶段，我们表明分数模型具有耗散在${升}^2\left(\Omega \right)$关于时间。从分析的角度对系统进行了严格的研究，并提出并严格分析了近似其解的耗散数值模型。我们的离散化将利用统一$升1$估计时间分数导数的近似方案，以及用于近似空间拉普拉斯算子的常用中心差分算子。为此，文献的各种结果将是至关重要的，包括一些有用的 Paley-Wiener 不等式的离散形式。包括一些数值示例以显示数值方法的渐近行为以及最终的耗散特性。