An inverse source problem for a two-parameter anomalous diffusion with local time datum

TL;DR

This paper addresses an inverse problem for a two-parameter fractional diffusion equation, using local time data and bi-orthogonal bases to uniquely determine the space-dependent source term, with analytical and numerical validation.

Contribution

It introduces a novel method utilizing local time data and bi-orthogonal bases to solve the inverse source problem for a two-parameter fractional diffusion equation.

Findings

Unique determination of the source term under given conditions

Development of a series solution using bi-orthogonal bases

Validation through analytical and numerical examples

Abstract

We determine the space-dependent source term for a two-parameter fractional diffusion problem subject to nonlocal non-self-adjoint boundary conditions and two local time-distinct datum. A bi-orthogonal pair of bases is used to construct a series representation of the solution and the source term. The two local time conditions spare us from measuring the fractional integral initial conditions commonly associated with fractional derivatives. On the other hand, they lead to delicate $2\times 2$ linear systems for the Fourier coefficients of the source term and of the fractional integral of the solution at $t=0$. The asymptotic behavior and estimates of the generalized Mittag-Leffler function are used to establish the solvability of these linear systems, and to obtain sufficient conditions for the existence of our construction. Analytical and numerical examples are provided.