Inverse coefficient problem for the time-fractional diffusion equation

We study the inverse problem of determining the time depending reaction diffu- sion coefficient in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x = 0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is used and properties of this solution are investigated. The fundamental solution contains the Fox’s H− functions widely used in fractional calculus. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient which will be used in study inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. Also the stability estimate is obtained.