An inverse source problem for a one-dimensional 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 deals with a nonlocal inverse source problem for a one-dimensional space–time fractional diffusion equation where and . At first we define and analyze the direct problem for the space–time fractional diffusion equation. Later we define the inverse source problem. Furthermore, we set up an operator equation and derive the relation between the solutions of the operator equation and the inverse source problem. We also prove some important properties of the operator . By using these properties and analytic Fredholm theorem, we prove that the inverse source problem is well posed, i.e. can be determined uniquely and depends continuously on additional data .

Keywords:

• fractional derivative
• fractional Laplacian
• inverse source problem
• Mittag-Leffler function

AMS Subject Classifications:

• 34A12
• 35R11
• 35R30
• 65F22

Acknowledgements

The research has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through the project Nr 113F373, also by the Zirve University Research Fund. The authors thank the referee for his/her very careful reading and for pointing out several mistakes as well as for the useful comments and suggestions.