Mathematical Analysis of a Cauchy Problem for the Time-Fractional Diffusion-Wave Equation with α∈ (0 , 2)

Abstract

This paper deals with a theoretical mathematical analysis of a Cauchy problem for the time-fractional diffusion-wave equation in the upper half-plane, x∈ R, t∈ R+, where the Caputo fractional derivative of order α∈ (0 , 2) is considered. An explicit solution to this Cauchy problem is obtained via separation of variables. A first proof of the validity of the obtained results is provided for a certain kind of initial conditions. Throughout this work a new expression of the solution to this problem and its utility for carrying out rigurous proofs are presented. Finally, several new properties of the solution are obtained.