turkmath.org

We consider the inverse problem of recovering an unknown, spatially-dependent coefficient $q(x)$ from the fractional order equation $\mathbb{L}_\alpha u=f$ defined in a region of $\mathbb{R}^2$ from boundary information. Here $\mathbb{L_\alpha}={D}^{\alpha_x}_x+{D}^{\alpha_y}_y+q(x)$ where the operators ${D}^{\alpha_x}_x$, ${D}^{\alpha_y}_y$ denote fractional derivative operators based on the Abel fractional integral. In the classical case this reduces to $-\triangle u+q(x)u=f$ 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 $\alpha_x$ and $\alpha_y$.