Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

Abstract

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.

Keywords:

• Diffusion process
• fractional derivative
• backward problem
• Tikhonov regularization method

AMS Subject Classifications:

• 35K05
• 35K57
• 65J20

Acknowledgements

The authors thank to anonymous referees for their valuable comments and suggestions that have greatly improved the presentation of the paper.

Notes

No potential conflict of interest was reported by the authors.

Present Address: Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.

Present Address: Karataş District, 400. Street, Nizamkent Apartments, Building B, No: 24, 27470, Şahinbey, Gaziantep, Turkey.

Additional information

Funding

This first and the second authors have been supported by Institute for Computational Science and Technology Ho Chi Minh City under project named Fractional diffusion-wave equations and application to soil contaminant. The third author has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) through the [project number 113F373].