Generalised solution of fractional diffusion-wave equation

Abstract

This article is devoted to establishing the existence and uniqueness of solutions to the fractional problem of diffusion waves in the following Colombeau algebra: \begin{cases} D_t^\alpha u(x, t) + \Delta_x u(x, t) = f(t, u(t, x)); & (x, t) \in \Omega \times [0, T] \\ u(0, x) = \psi_0(x) = \delta(x); & \\ \partial_t u(0, x) = \psi_1(x). & \end{cases} . Where D_t^\alpha is the fractionnal derivative with 1 < ? < 2, ? is the Laplace operator, \psi_0 , \psi_1 are generalized functions, ? is distributions and \Omega \subset \mathbb{R}^n . This study is based on the integral solution of this problem using the Gronwall?s lemma. Finally we study the association concept with the classical solution.