On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

Abstract

An inverse source problem for a non-automonous time fractional diffusion equation of order (0 < β < 1) is considered in a bounded Lipschitz domain in R d . The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe's method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from t = 0. Rothe's method on a uniform grid addresses the existence of a such a solution (non-smooth with t γ term where 1 > γ > β ) under low regularity assumptions, whilst Rothe's method over graded grids has the advantage to cope better with the behaviour at t = 0 (also here t β is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions

“…This naturally contradicts the behavior of the solution near t = 0 as a lack of smoothness occurs due to the singular kernel of time Caputo fractional derivative. The authors in [13] extended the results to possibly nonsmooth solutions. They showed that Rothe's method on a uniform grid addresses the existence of a such a solution (non-smooth solutions are admissible with t γ term where 1 > γ > β with β the order of the fractional derivative) under low regularity assumptions, and that Rothe's method over graded grids has the advantage to cope better with the behaviour at t = 0 (also here t β is included in the class of admissible solutions) for the considered problems.…”

“…These non-uniform meshes have the advantage to be flexible and reasonably convenient for practical implementation, however they can significantly complicate the analysis of the constructed Rothe schemes. As mentioned before, in [13], a similar problem is studied for M = 0 and f (x,t) = f (x). However, in this paper, we use the time-graded mesh properties side by side to obtain a crucial extension of Grönwall's inequalities for multiterm fractional operators, which is built on [14, Lemma 5.2] and an application of the classical discrete Grönwall inequality.…”

Inverse potential problem for a semilinear generalized fractional diffusion equation with spatio-temporal dependent coefficients

“…This type of measurement represents the weighted average of u over the domain Ω. And there are many literatures on inverse problems by this type measured data for fractional diffusion (wave) equations at present (see [9,32,39] and the references therein).…”

Determination of an Energy Source Term for Fractional Diffusion Equation

“…These mathematical problems have become a significant tool in simulating many real-life difficulties. For recent developments, one can see Hendy, and Bockstal in [33] for reconstruction of a solely time-dependent source in a time-fractional equation. Ansari et al in [34] study a class of distributed order fractional diffusion equation.…”