Forward and Inverse Problems for Fractional Telegraph Equation

The paper studies the Cauchy problem and the inverse problem of determining the right-hand side of an abstract fractional telegraph equation. In this case, an arbitrary self-adjoint operator with a complete orthonormal system of eigenfunctions is taken as the elliptic part, and the highest time derivative is the fractional Caputo derivative of order \(2\rho\) . When solving the Cauchy problem by the Fourier method, a new representation of the solution for the corresponding ordinary fractional differential equation is established. An additional condition for solving the inverse problem has the form \(u(t_{0})=\Phi\) . In the paper, it is proved that with a special choice of \(t_{0}\) , the existence of a solution to the inverse problem can be achieved, but the solution to the inverse problem is not unique.