Approximation of fractional integrals by means of derivatives

Abstract

We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n\in \mathbb{N}$, and allow us to develop suitable numerical approximations with known estimations for the error. The usefulness of the obtained results, in solving fractional integral equations and fractional problems of the calculus of variations, is illustrated.

Submitted 15-10-2011; revised 24-01-2012; accepted 25-01-2012; for publication in Computers and Mathematics with Applications

References in corpus (5)

• Fractional conservation laws in optimal control theory
• Calculus of variations with fractional derivatives and fractional integrals
• The generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative
• Modified Optimal Energy and Initial Memory of Fractional Continuous-Time Linear Systems
• Leitmann's direct method for fractional optimization problems

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• Optimality Conditions for Fractional Variational Problems with Dependence on a Combined Caputo Derivative of Variable Order
• A numerical scheme to solve fractional optimal control problems
• Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
• Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense
• An expansion formula with higher-order derivatives for fractional operators of variable order
• Generalized Transversality Conditions in Fractional Calculus of Variations
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• On the Study of Chaos and Memory Effects in the Bonhoeffer-van der Pol Oscillator with a Non-Ideal Capacitor
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• Uniform approximation of fractional derivatives and integrals with application to fractional differential equations
• Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control
• A Commutative Alternative to Fractional Calculus on Continuous Functions