Existence and uniqueness results for a multi-parameters nonlocal diffusion equation

On the solvability of direct and inverse problems for a generalized diffusion equation

“…Based on the relationships described in (22) and (24), it can be deduced that u x t , xx ( ), represented by equation (23), is confined within the bounds of a convergent series. Analogously, within the same set of assumptions, an analogous line of reasoning can be established to confirm the uniform convergence of the series related to u x t , g t 0 , ( )…”

“…Multi-term FDs come into play when different powers of t become predominant as t → 0 + and t → ∞ within the kernel. This property allows them to effectively model both accelerating and retarding sub (super) diffusion phenomena, as extensively discussed in references such as [21][22][23]. On the other hand, FDs with exponential tempering are employed to describe the gradual transition from anomalous diffusion to normal diffusion, offering advantages both in mathematical modeling and practical applications.…”

Identifying temperature distribution and source term for generalized diffusion equation with arbitrary memory kernel

“…This is achieved by allowing different powers of $$$ t $$$ to predominate as $$$ t\to 0 $$$ and as $$$ t\to \infty $$$ in the kernel. Equation () is the generalization of Equation () in the sense that we can have Equation () from Equation () and multi‐term Caputo fractional derivatives with diffusion equations have been used, see [16, 17, 24–26].…”

An inverse problem for a two‐dimensional diffusion equation with arbitrary memory kernel

“…A number of researchers investigate inverse problems for fractional diffusion equations involving the Caputo fractional derivative with respect to time [23, 24]. Multi‐term fractional derivatives enable to model accelerating and retarding sub(super) diffusion, since different powers of $$$ t $$$ dominate as $$$ t\to 0+ $$$ and $$$ t\to \infty $$$ in the kernel [19, 25]. Tempered fractional derivatives are used to describe slow transition of anomalous diffusion to a normal one.…”