Direct and inverse source problems for two-term time-fractional diffusion equation with Hilfer derivative

TL;DR

This paper studies direct and inverse source problems for a two-term time-fractional diffusion equation with Hilfer derivative, deriving solutions using spectral methods and special functions, and proving convergence under certain conditions.

Contribution

It introduces a spectral expansion approach to solve two-term fractional diffusion equations with Hilfer derivatives, including solution representations and convergence analysis.

Findings

Solution expressed via sinus and Mittag-Leffler functions

Proved uniform convergence of solution series

Established conditions for data to ensure convergence

Abstract

In this paper, we investigate direct and inverse source problems for the diffusion equation with two-term generalized fractional derivative (Hilfer derivative) in a rectangular domain. Using spectral expansion method, we derive two-term fractional differential equation together with appropriate initial condition (Cauchy problem). Based on solution of that Cauchy prob\-lem, we represent solution of formulated problems as a combination of sinus and multinomial Mittag-Leffler function of two variables. Imposing certain conditions to the given data, we prove uniform convergence of certain infinite series.