Conformable Cosine Family and Nonlinear Fractional Differential Equations

Abdelmjid Benmerrous, Lalla saadia Chadli, Abdelaziz Moujahid, M'hamed Elomari, Said Melliani


This paperis focuses to the existence and uniqueness solution to the following problem:\begin{equation}\left\{\begin{array}{l}D^{(\alpha)} f(t, y)+A f(t, y)=F(t,f(t,y)) \quad y \in \mathbb{R}, \quad t \geq 0 \\f(0,y)=u_0(y), \quad D^{(\alpha)} f(0, y)=v_0(y)\end{array}\right.\end{equation} where$D^{(\alpha)}$is the conformable derivation for $1<\alpha<2$which we will prove to be inside Colombeau algebra, $u_0$ and $v_0$ are singular distibution and F provides$L^{\infty}$logarithmictype, the operator A isdefined in Colombeau'salgebra. Nets ofconformable cosine family$(C^\alpha_\epsilon)_{\epsilon}$with polynomial development in $\epsilon$ as $\epsilon \rightarrow 0$ are defined for thefirst time and used for solvingirregular fractional problems.


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