Estimates for a general fractional relaxation equation and application to an inverse source problem

TL;DR

This paper analyzes a broad class of fractional relaxation equations with convolutional derivatives, deriving fundamental solutions and bounds, and applies these results to prove uniqueness and stability in an inverse source problem for fractional diffusion.

Contribution

It introduces new estimates for a general fractional relaxation equation and applies these to establish stability results for an inverse source problem.

Findings

Derived fundamental and impulse-response solutions.

Established analyticity and subordination properties.

Proved uniqueness and stability for the inverse source problem.

Abstract

A general fractional relaxation equation is considered with a convolutional derivative in time introduced by A. Kochubei (Integr. Equ. Oper. Theory 71 (2011), 583-600). This equation generalizes the single-term, multi-term and distributed-order fractional relaxation equations. The fundamental and the impulse-response solutions are studied in detail. Properties such as analyticity and subordination identities are established and employed in the proof of an upper and a lower bound. The obtained results extend some known properties of the Mittag-Leffler functions. As an application of the estimates, uniqueness and conditional stability are established for an inverse source problem for the general time-fractional diffusion equation on a bounded domain.