Simultaneous Determination of a Source Term and Diffusion Concentration for a Multi-Term Space-Time Fractional Diffusion Equation

Abstract

An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

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

“…By equation (21), we can deduce that the series of functions representing g(x), as expressed in equation (10), is bounded by a convergent series. Therefore, by applying the Weierstrass M-test (WMT), we can confidently assert the continuity of the function g(x)…”

“…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.…”

“…Ali et al [20] employed the operational calculus method to investigate an ISP focused on determining both the temperature concentration and the source function for a two-term TFDE. Similarly, Malik et al [21] addressed an ISP concerning the determination of diffusion concentration and the source function, specifically for a multi-term Fractional Differential equation (FDE) with an integral-type over-specified condition. Asim et al [22] presented results regarding the existence and ill-posedness of Inverse Problems (IPs) related to an evolution equation involving multi-term FDs in time and an involution term.…”

On the Inverse Problems for a Family of Integro-Differential Equations

“…Ismailov et al Liao et al [15] studied an IP of recovering a fractional order and a space dependent source term in a multi-dimensional time fractional diffusion wave equation by the final time measurement data. Malik et al [18] considered an IP of the determination of the source term and diffusion concentration for a multi-term FDE with integral type over-specified condition. Kinash et al [12] presented two IPs for a generalized subdiffusion equation with final over-determination condition.…”

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

“…Ali et al [17], using the operational calculus approach, investigated IP of identifying the temperature concentration and source term for a two‐term TFDE. Malik et al [18] considered an IP for the determination of source term and diffusion concentration for a multi‐term fractional diffusion equation with integral type over‐specified condition. Asim et al [19] studied two IPs for a multi‐term time fractional evolution equation with an involution with appropriate over‐specified conditions.…”