On a Difference Scheme for Solving Cauchy Problems wuth the Caputo Fractional Derivative in a Banach Space

We construct and study a time–semidiscretization scheme for the Cauchy problem associated with a linear homogeneous differential equation with the Caputo fractional time derivative of order $\alpha \in (0,1)$ and a spatial sectorial operator in a Banach space. For this scheme, we obtain rate–of–convergence and error estimates in terms of the discretization step. We use properties of Mittag–Leffler functions, hypergeometric functions, and the calculus of sectorial operators in a Banach space. Results of numerical experiments are also reported.