Abstract
In this paper, we consider an inverse problem of identifying the source term for a generalization of the time-fractional diffusion equation, where regularized hyper-Bessel operator is used instead of the time derivative. First, we investigate the existence of our source term; the conditional stability for the inverse source problem is also investigated. Then, we show that the backward problem is ill-posed; the fractional Landweber method and the fractional Tikhonov method are used to deal with this inverse problem, and the regularized solution is also obtained. We present convergence rates for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Finally, we present a numerical example to illustrate the proposed method.
Highlights
1 Introduction Fractional calculus has a long history in the mathematical theory and has attracted much attention in various fields of the applied science [3, 4, 13, 23]
Fractional differential equations have an important position in the mathematical modeling of different physical systems [1, 10, 21], in engineering [6, 18], [7], and finance [24], in physics, chemistry, medicine, and they describe anomalous diffusion [12, 16, 20]
The time-fractional diffusion is discussed in this paper as follows:
Summary
Fractional calculus has a long history in the mathematical theory and has attracted much attention in various fields of the applied science [3, 4, 13, 23]. The authors [2] considered two direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator. They established the existence and uniqueness of solutions to the problem and gave the explicit eigenfunction expansions. We apply the fractional Landweber method and fractional Tikhonov method to restore the unknown space source function F Both methods were studied by Klann and Ramlau [15] when they considered a linear ill-posed problem. 2.2 Solution for a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator. Applying the fractional Landweber method given by [15], we propose the following regularized solution with exact data H : Fm,θ(x).
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