Local Solvability for an Inverse Source Problem for Nonlocal Diffusion Equations With Variable Order

ABSTRACT This paper investigates the inverse problem of determining spatially dependent source term in diffusion equations with variable‐order time fractional derivatives from the knowledge of integral‐type measurements. We first construct an integral equation that combines unknown sources and integral‐type observations. With the aid of Mittag‐Leffler functions, we will use the Fredholm alternative for compact operators to study the existence, uniqueness, and regularity estimates of the solutions to the forward problem. Finally, we developed an algorithm utilizing Tikhonov regularization technique and validated its superior performance in accuracy and efficiency through a series of numerical experiments.