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:
$$
\left\{ \begin{array}{ccl}
\mathrm{D}_{t}^{\alpha}u(x,t) +\Delta_{x} u(x,t)& = & f(t,u(t,x)); \hspace{1.2cm} (x,t)\in\Omega \times [0,T] \\
u(0,x) & = &\psi_{0}(x) = \delta(x); \\
\partial_{t}u(0,x) & = & \psi_{1}(x).
\end{array}\right.$$
Where $\mathrm{D}_{t}^{\alpha}$ is the fractionnal derivative with $1<\alpha<2$, $\Delta$ is the Laplace operator, $\psi_{0}, \ \psi_{1}$ are generalized functions, $\delta$ 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.