Numerical Approximations of Fractional Derivatives with Applications

Abstract

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains fractional derivatives into a classical problem in which only derivatives of integer order are present. Corresponding approximations provide useful numerical tools to compute fractional derivatives of functions. Application of such approximations to fractional differential equations and fractional problems of the calculus of variations are discussed. Illustrative examples show the advantages and disadvantages of each approximation.

This is a preprint of a paper whose final and definite form will be published in: Asian Journal of Control. Submitted 13-Oct-2011; revised 11-Apr-2012; accepted 10-Aug-2012

References in corpus (4)

• Fractional conservation laws in optimal control theory
• 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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• A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems
• Discrete and continuous fractional persistence problems - the positivity property and applications
• Solving Abel integral equations of first kind via fractional calculus
• A numerical scheme to solve fractional optimal control problems
• An expansion formula with higher-order derivatives for fractional operators of variable order
• A Fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives
• Fractional derivative order determination from harmonic oscillator damping factor