Estimates for a general fractional relaxation equation and application to an 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, multiterm, 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.