Inverse problem of determining the right-hand side of a one-dimensional fractional diffusion equation with variable coefficients

TL;DR

This paper addresses an inverse problem for a one-dimensional fractional diffusion equation with variable coefficients, aiming to determine a time-dependent multiplier in the right-hand side using integral overdetermination conditions.

Contribution

It introduces an explicit solution method for the inverse problem with variable coefficients and proves its correctness within the class of regular solutions.

Findings

Explicit solution to the inverse problem was constructed.

The solution's correctness was rigorously proven.

The method applies to fractional diffusion equations with variable coefficients.

Abstract

In this paper, we study the inverse problem of finding a time-dependent multiplier of the right-hand side of a time-fractional one-dimensional diffusion equation with variables coefficients in the case where the usual Cauchy, homogeneous Dirichlet boundary, and an integral overdetermination conditions are given. The overdetermination condition has the form of an integral with a weight over a spatial segment from the solution of the direct problem, in which the weight function is a spatially dependent known factor of the right-hand side of the equation. This made it possible to construct a solution to the inverse problem in explicit form and prove its correctness in the class of regular solutions