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

Authors:
E. Cuesta;C. Palencia
Affiliations:
Departamento de Matemática Aplicada a la Técnica, Universidad de Valladolid, Valladolid, Spain;Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Venue:
Applied Numerical Mathematics
Year:
2003

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Abstract

The abstract evolutionary equation with fractional derivative Dαu(t) = Au(t) + f(t), 1 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.