In this paper, we study periodic and S‐asymptotically periodic solutions for fractional diffusion equations (FDE). As we all know, there is no exact periodic solution to differential equations with Caputo or Riemann‐Liouville fractional derivatives. Even so, in this paper, periodic (S‐asymptotically periodic) mild or classical solutions for FDE with Weyl‐Liouville fractional derivatives could be obtained in various fractional power spaces. In addition, a numerical simulation example and a specific example of fractional diffusion equation are given to verify the main theoretical results.
In this paper, a generalized Gronwall inequality is demonstrated, playing an important role in the study of fractional differential equations. In addition, with the fixed-point theorem and the properties of Mittag–Leffler functions, some results of the existence as well as asymptotic stability of square-mean S-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion are obtained. In the end, an example of numerical simulation is given to illustrate the effectiveness of our theory results.
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