Time discretization and convergence to superdiffusion equations via Poisson distribution

Let $ A be a closed linear operator defined on complex Banach space X $. We show novel representation, using strongly continuous families of bounded operators \mathbb{N}_0 $, for the unique solution following time-stepping scheme \begin{document}$ \begin{eqnarray*} (*)\left\{ \begin{array}{rcl} \,_C \nabla^{\alpha} u^n& = &Au^{n}+f^{n}, \quad n\geq 2;\\ u^0& &u_0; \\ u^1& &u_1; \end{array} \right. \end{eqnarray*} $\end{document} as well its convergence with rates to abstract fractional Cauchy problem$ \partial_t^\alpha u(t) & Au(t)+f(t), t>0;\\ u(0)& u'(0)& $in superdiffusive case 1<\alpha <2. Here, u^n is Caputo-like difference order \alpha.