About a Problem for Loaded Parabolic-Hyperbolic Type Equation with Fractional Derivatives

Abstract

An existence and uniqueness of solution of local boundary value problem with discontinuous matching condition for the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals have been investigated. The uniqueness of solution is proved by the method of integral energy and the existence is proved by the method of integral equations. Let us note that, from this problem, the same problem follows with continuous gluing conditions (at $\lambda =\mathrm{1}$); thus an existence theorem and uniqueness theorem will be correct and on this case.