Reconstruction of the initial function from the solution of the fractional wave equation measured in two geometric settings

Photoacoustic tomography (PAT) is a novel and rapidly developing technique in the medical imaging field that is based on generating acoustic waves inside of an object of interest by stimulating non-ionizing laser pulses. This acoustic wave was measured by using a detector on the outside of the object it was then converted into an image of the human body after several inversions. Thus, one of the mathematical problems in PAT is reconstructing the initial function from the solution of the wave equation on the outside of the object. In this study, we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We demonstrate a way to recover the initial function from the data, namely, the solution of the fractional wave equation, measured on the sphere and hyperplane.