The determination of an unknown boundary condition in a fractional diffusion equation

Abstract

In this article we consider an inverse boundary problem, in which the unknown boundary function is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

Keywords:

• inverse boundary
• time-fractional diffusion equation
• fixed point theory

AMS Subject Classifications::

• 35R11
• 35R30
• 65M32

Acknowledgements

The authors acknowledge partial support from National Science Foundation grants DMS-0715060 and DMS-0900889 and Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).