Inverse source problem for a time-fractional heat equation with generalized impedance boundary condition

TL;DR

This paper addresses the inverse problem of identifying a time-dependent source and temperature distribution in a one-dimensional time-fractional heat equation with a generalized impedance boundary, establishing well-posedness through spectral and integral equation methods.

Contribution

It introduces a novel approach combining spectral expansion and Volterra integral equations to prove the well-posedness of the inverse source problem for a fractional heat equation with complex boundary conditions.

Findings

Proved the well-posedness of the inverse problem.

Developed a method using eigenfunction expansion.

Analyzed integral equations with singular kernels.

Abstract

The paper considers an inverse source problem for a one-dimensional time-fractional heat equation with the generalized impedance boundary condition. The inverse problem is the time dependent source parameter identification together with the temperature distrubution from the energy measurement. The well-posedness of the inverse problem is shown by applying the Fourier expansion in terms of eigenfunctions of a spectral problem which has the spectral parameter also in the boundary condition and by using the results on Volterra type integral equation with the kernel may have a diagonal singularity.