Generalised solution of fractional diffusion-wave equation

This article is devoted to establishing the existence and uniqueness of solutions to the fractional problem of diffusion waves in the following Colombeau algebra: {cases} D_t^α u(x, t) + Δ_x u(x, t) = f(t, u(t, x)); & (x, t) ∈ Ω × [0, T] \\ u(0, x) = ψ_0(x) = δ(x); & \\ ∂_t u(0, x) = ψ_1(x). & {cases} . Where D_t^α is the fractionnal derivative with 1 < ? < 2, ? is the Laplace operator, ψ_0 , ψ_1 are generalized functions, ? is distributions and Ω ⊂ 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.