DISCRETE APPROXIMATION OF SOLUTIONS OF THE CAUCHY PROBLEM FOR A LINEAR HOMOGENEOUS DIFFERENTIAL-OPERATOR EQUATION WITH A CAPUTO FRACTIONAL DERIVATIVE IN A BANACH SPACE

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order [alpha] [member of] (0,1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented. Keywords and phrases: Cauchy problem, Caputo derivative, Banach space, finite-difference scheme, error estimate, Mittag-Leffler function, hypergeometric function, sectorial operator. AMS Subject Classification: 47N40, 65J08, 35R11
1. Statement of the problem. The papers [13, 15-17] are devoted to the study of the class of finite-difference methods of the form [Please download the PDF to view the [...]