Recovering Multiple Fractional Orders in Time-Fractional Diffusion in an Unknown Medium

Abstract

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g., diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Further, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach for small time $t$.

arXiv comment20 pages, 2 figures, to appear at Proceedings of the Royal Society A

References in corpus (7)

• Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients
• Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation
• Inverse Problems of Determining Sources of the Fractional Partial Differential Equations
• Uniqueness in inverse boundary value problems for fractional diffusion equations
• On time-fractional diffusion equations with space-dependent variable order
• Inverse Problems of Determining Parameters of the Fractional Partial Differential Equations
• Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data