A Mollification Regularization Method for the Inverse Source Problem for a Time Fractional Diffusion Equation

Le Dinh Long,Yong Zhou,Nguyen Can,Tran Thanh Binh

doi:10.3390/math7111048

Abstract

We consider a time-fractional diffusion equation for an inverse problem to determine an unknown source term, whereby the input data is obtained at a certain time. In general, the inverse problems are ill-posed in the sense of Hadamard. Therefore, in this study, we propose a mollification regularization method to solve this problem. In the theoretical results, the error estimate between the exact and regularized solutions is given by a priori and a posteriori parameter choice rules. Besides, the proposed regularized methods have been verified by a numerical experiment.

Highlights

In this work, we study an inverse source problem for the time-fractional diffusion equation in a infinite domain as follows: β∂ u( x, t) = u xx ( x, t) + φ(t) f ( x ), ( x, t) ∈ R × (0, T ], ∂t β (1)u( x, 0) = 0, x ∈ R, u( x, T ) = g( x ), x ∈ R, where the fractional derivative ∂β u∂t β is the Caputo derivative of order β (0 < β < 1) as defined by d β f (t) = Γ (1 − β ) dt β Zt d f (s) ds, ds (t − s) β (2)
To our knowledge, in the case φ(t), dependent on time, the results of the inverse source problem for the time-fractional diffusion equation still has a limited achievement, if φ(t) 6= 0, we know Huy and his group investigated this problem by way of the Tikhonov regularization method, see Reference [13]
In the theoretical results, which we have shown, we obtained the error estimates of both a priori and a posteriori parameter choice rule methods based on a priori condition

Summary

Introduction

We study an inverse source problem for the time-fractional diffusion equation in a infinite domain as follows: β. To our knowledge, in the case φ(t), dependent on time, the results of the inverse source problem for the time-fractional diffusion equation still has a limited achievement, if φ(t) 6= 0, we know Huy and his group investigated this problem by way of the Tikhonov regularization method, see Reference [13]. In these regularization methods, the priori parameter choice rule depends on the noise level and the priori bound.

Some Auxiliary Results

The Priori Parameter Choice

The Discrepancy Principle

Numerical Experiments

Conclusions