Inverse Problem for a Third-Order Parabolic-Hyperbolic Equation Involves Fractional Derivatives

In this article the questions of unique solvability of inverse problem for a third order parabolic-hyperbolic type equation with Caputo derivatives are considered. Inverse problem is studied by an additional boundary condition on the hyperbolic domain and integral gluing condition on the line y = 0. The problem is reduced to study a second kind Volterra type integral equation with a shift. Existence and uniqueness of solution are proved by the method of successive approximations of factorial law.