Uniqueness for an inverse coefficient problem for a one-dimensional time-fractional diffusion equation with non-zero boundary conditions

Abstract

We consider initial boundary value problems for one-dimensional diffusion equation with time-fractional derivative of order $\alpha \in (0,1)$ which are subject to non-zero Neumann boundary conditions. We prove the uniqueness for an inverse coefficient problem of determining a spatially varying potential and the order of the time-fractional derivative by Dirichlet data at one end point of the spatial interval. The imposed Neumann conditions are required to be within the correct Sobolev space of order α. Our proof is based on a representation formula of solution to an initial boundary value problem with non-zero boundary data. Moreover, we apply such a formula and prove the uniqueness in the determination of boundary value at another end point by Cauchy data at one end point.

Keywords:

• Inverse coefficient problem
• fractional diffusion equation
• uniqueness

AMS subject classifications:

• 35R30
• 35R11

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported in part by the National Science Foundation through award dms-2111020. The second author was supported by Grant-in-Aid for Scientific Research (S) 15H05740 and Grant-in-Aid (A) 20H00117 of Japan Society for the Promotion of Science and by The National Natural Science Foundation of China (no. 11771270, 91730303). This paper has been supported by the RUDN University Strategic Academic Leadership Program.