Conformable Cosine Family and Nonlinear Fractional Differential Equations

Abstract

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.