On the determination of a coefficient in a space-fractional equation with operators of Abel type

Abstract

We consider the inverse problem of recovering an unknown, spatially-dependent coefficient q(x) from the fractional order equation Lαu=f defined in a region of R2 from boundary information. Here Lα=Dxαx+Dyαy+q(x) where the operators Dxαx, Dyαy denote fractional derivative operators based on the Abel fractional integral. In the classical case this reduces to −△u+q(x)u=f and this has been a well-studied problem. We develop both uniqueness and reconstruction results and show how the ill-conditioning of this inverse problem depends on the geometry of the region and the fractional powers αx and αy.