Direct and inverse coefficient problems for the fractional diffusion wave equation with the Riemann–Liouville time derivative

This article considers the inverse problem in the fractional wave equation with the Riemann–Louville derivative. In this case, the direct problem is an initial nonlocal boundary value problem for this equation with initial Cauchy type and nonlocal boundary conditions. As a redefinition condition, a nonlocal integral condition with respect to the direct solution of the problem is specified. Using the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we obtain an a priori estimate of the solution in terms of the unknown coefficient, which we will need to investigate for the inverse problem. The inverse problem is reduced to the equivalent integral of a Volterra type equation. To solve this equation, the contraction mapping principle is used. Local existence and global uniqueness have been proven.