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

Abstract: 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.