Problem of Determining the Time Dependent Coefficient in the Fractional Diffusion-Wave Equation

In this article the inverse problem of determining the time depending coefficient in the Cauchy problem for a time-fractional diffusion-wave equation with a single observation at the point $$x=0$$ is studied. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion-wave equation is used and properties of this solution are investigated. The fundamental solution contains a Wright function, which is widely used in the theory of diffusion-wave equation. 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. This estimate by norm is used in further in studying inverse problem. The inverse problem is reduced to the equivalent integral equation. In the proof of one valued solvability of this integral equation the contracted mapping principle is applied. The local existence and global uniqueness of the solution of the inverse problem are proven. The stability estimate also is obtained.