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

The abstract evolutionary equation with fractional derivative Dαu(t)=Au(t)+f(t), 1<α<2, written in its integro-differential format, is considered. The linear operator A:D(A)⊂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.