Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time

Abstract

We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

“…In Ahmad et al, there are good references to publications on related issues. We note from recent papers close to the theme of our article. In these papers, different variants of direct and inverse initial‐boundary value problems for evolutionary equations are considered, including problems with nonlocal boundary conditions and problems for equations with fractional derivatives.…”

“…The mathematical problem to for a = 0 was considered in Torebek and Tapdigoglu, and for a = β = 0 in Kirane et al…”

A new approach for determination of boundary function for diffusion equation by fourier method

“…These sets form a biorthogonal system for the space L 2 (0, 1) and by using the biorthogonal system we expand the unknown functions into series to solve the inverse problem. By using the biorthogonal system, the inverse problem of determining unknown functions has been already considered in the literature, see for example [12,13,19].…”

A Point Source Identification Problem for a Time Fractional Diffusion Equation

“…In [33], when additional data is given on the partial boundary, the uniqueness in identifying a source term independent of time is established for one-dimensional time fractional diffusion equation. In [30], if the final time temperature distribution is known, the existence and uniqueness results are proved. Murio and Mejía [31] propose a mollification regularization technique to reconstruct the unknown forcing term ( , ).…”

“…For elliptic-type differential equation, one can refer to [27][28][29], though the source identification problem has been well discussed in the classic framework, yet, to the best of the authors' knowledge, there are rare researches in the aspect of the source identification problem associated with fractional differential equation in spite of the physical and practical importance. As indicated in [30][31][32][33], the source identification problem associated with the time fractional diffusion equation is also ill-posed. That means the solution does not depend continuously on the given data and any small perturbation in the given data may cause large change to the solution.…”