A Robin boundary sub-diffusion equation is considered with fractional partial derivatives of the Caputo type. The model is an extension of various well-known equations from mathematical physics, biology, and chemistry. Initial–boundary data are imposed upon a closed and bounded spatial domain. We state and prove two main theorems in differential and difference settings to ensure the algebraic decay rate of the long-time behavior for that kind of problem. The dissipation of the continuous solution for such a problem is discussed in the first theorem based on energy inequalities and by the aid of Grönwall inequalities. It demonstrates that with an $L^2\left(\Omega \right)$-bounded absorbing set, the solution is dissipated with respect to time. The numerical dissipativity is proved in the second theorem by using discrete energy inequalities and the discrete Paley–Wiener inequality. Finally, an example is provided to illustrate the main outcomes.

Robin 边界子扩散方程被考虑与 Caputo 类型的分数偏导数。该模型是数学物理、生物学和化学中各种著名方程的扩展。初始边界数据被强加在一个封闭且有界的空间域上。我们陈述并证明了微分和差分设置中的两个主要定理，以确保此类问题的长期行为的代数衰减率。在基于能量不等式并借助 Grönwall 不等式的第一定理中讨论了此类问题的连续解的耗散。它证明了用${升}^2\left(\Omega \right)$有界吸收集，解随时间消散。通过使用离散能量不等式和离散 Paley-Wiener 不等式，在第二个定理中证明了数值耗散性。最后，提供了一个例子来说明主要结果。