INverse Source Problem for a Space-Time Fractional Diffusion Equation

Abstract

For a space-time fractional diffusion equation, an inverse problem of determination of a space dependent source term along with the solution is considered. The fractional derivatives in time and space are defined in the sense of Caputo. Due to an over-specified data at final time say T, we proved that there exists a unique solution of the inverse source problem. We use the eigenfunction expansion method to prove our main results. Several special cases of space-time fractional diffusion equations are discussed and results are interpolated from generalized results. Some examples are provided.

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

“…Remark 5. For σ 1 = 0, σ 2 = σ 3 = ... = σ m = 0, m ∈ N the above Lemma reduces to the following result proved in [3] |g(τ Proof. By taking the Laplace transform on the both sides, we obtained the result.…”

“…An inverse problem involving generalized fractional derivative in diffusion and wave equations for time and space dependent source terms are discussed in [15]. In [4], for a Space-Time Fractional Diffusion Equation (STFDE) inverse problem of determining a temporal component in the source term from the total energy of the system is considered and the recovery of a space dependent source term from final data is discussed in [3]. Inverse problems of identifying the space and time dependent source terms for a STFDE are considered in [5].…”

An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

“…The ISP of determination of a time‐dependent source term has been considered by many, for example, see other studies 6–9 . Inverse problems (IPs) for PDEs involving arbitrary order integrals and derivatives are a new subject when compared with their integer order theory of IPs.…”

Some inverse problems for time‐fractional diffusion equation with nonlocal Samarskii‐Ionkin type condition

“…19 Existence, uniqueness, and regularity of the weak solution for a semilinear time-fractional diffusion equation is addressed in Slodicka and Siskova. 20 Determination of the spatial and temporal component of source term for the space-time-fractional diffusion equation is considered in Ali et al, 21,22 respectively. Generalized Tikhonov regularization method is used to construct the solution of the inverse source problem for time-fractional diffusion equation.…”