A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces

The abstract evolutionary equation with fractional derivative D(alpha)u(t) = Au(t) + f(t), 1 < alpha < 2, written in its integro-differential format, is considered. The linear operator A : D(A) subset of X --> X is assumed to be sectorial in a Banach space X. This equation is discretized in time by means of a method based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a standard way, a fractional quadrature rule, constructed again from the trapezoidal rule, is used to approximate the integral term. The resulting scheme is shown to be stable and convergent of second order. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.