Inverse problem for a two-dimensional wave equation with a fractional Riemann-Liouville time derivative

Abstract

In this paper, we consider direct and inverse problems for a two-dimensional fractional wave equation with the Riemann-Liouville time fractional derivative. The direct problem is the initial-boundary problem for this equation with nonlocal boundary conditions. In inverse problem it is required to find time variable coefficient at the lower term of equation. Using the method of separation of variables, a classical solution of direct problem was found in the form of a bi orthogonal series in terms of eigenfunctions and associated functions. A nonlocal integral condition is used as the overdetermination condition with respect to the direct problem solution. Using the Fourier method, direct problem is reduced to equivalent integral equations. Then, using the estimates for MittagLeffler function and the generalized singular Gronwall inequality, we obtain an a priori estimate of the solution through an unknown coefficient, which we will need to study the inverse problem. The inverse problem is reduced to a Volterra integral equation of the second kind. Based on the unique solvability of this equation in the class of continuous functions, theorems on the unique solvability of direct and inverse problems are proven. Stability estimate is also obtained.