On nonlocal terminal value problem for tempered fractional diffusion equation

Abstract

In this paper, we are interested in studying the tempered fractional diffusion equation subject to a nonlocal terminal condition. The primary equation incorporates the tempered Caputo derivative, which serves as a generalized form of the traditional Caputo derivative.  Our findings contribute by first establishing the well-posedness, highlighting the challenges added by the tempered kernel together with nonlocal conditions. Following this, we study the solution's continuity with respect to the tempered parameter, a crucial  consideration for modeling, due to the challenges in accurately measuring this index. Lastly, we propose convergence results as parameters $b \to 0^+$, $a \to 0^+$, and $\alpha \to 1^-$, linking the current terminal fractional approach with traditional cases.