Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation

Abstract

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper is devoted to a nonlocal inverse problem related to the space-time fractional equation . The existence of the solution for the inverse problem is proved by using quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input–output mapping is defined and continuity of the mapping is established. The uniqueness of the solution for the inverse problem is also proved by using eigenfunction expansion of the solution and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, steepest descent method and least squares approach is proposed for the solution of the inverse problem. The numerical method determines the exponents of the fractional time and space derivatives simultaneously. Numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.

Keywords:

• fractional derivative
• fractional Laplacian
• inverse problem
• existence and uniqueness
• minimization problem
• input–output map
• steepest descent method
• least squares approach

AMS Subject Classifications:

• 34A12
• 35R11
• 35R30
• 65F22

Acknowledgements

The authors thank the referees for their very careful reading and for pointing out several mistakes as well as for their useful comments and suggestions.

Notes

The research has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) through the project Nr 113F373, and also by the Zirve University Research Fund.