Numerical methods for fractional calculus. (English) Zbl 1326.65033

Li, Changpin; Zeng, Fanhai

Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series. Boca Raton, FL: CRC Press (ISBN 978-1-4822-5380-1/hbk; 978-1-4822-5381-8/ebook). xviii, 281 p. (2015).

The book provides a survey of many different methods for the numerical computation of Riemann-Liouville integrals of fractional order and of fractional derivatives of Riemann-Liouville, Caputo, and Weyl type. Algorithms for the solution of associated ordinary differential equations and certain special classes of partial differential equations are presented as well. The methods under consideration include the classical ones frequently used for such purposes such as, e.g., the (shifted) Grünwald-Letnikov, \(L1\), and \(L2\) formulas, fractional linear multistep methods, and the fractional Adams method. Moreover, for time- and/or space-fractional (sub-)diffusion equations in one or two space dimensions, finite difference methods are described. Finally, the authors discuss finite element methods for steady-state advection-dispersion equations and for time- and/or space-fractional diffusion equations. The fundamental properties of the numerical schemes are listed, but convergence results are not always given with a precise statement of the assumed conditions on the given data.

Reviewer: Kai Diethelm (Braunschweig)

MSC:

65D30	Numerical integration
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
65D25	Numerical differentiation
26A33	Fractional derivatives and integrals
26-02	Research exposition (monographs, survey articles) pertaining to real functions
65L05	Numerical methods for initial value problems
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
34A08	Fractional ordinary differential equations and fractional differential inclusions
35R11	Fractional partial differential equations
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

fractional derivative; fractional integral; numerical method; finite difference; finite element; Caputo derivative; Riemann-Liouville derivative; Weyl derivative; monograph; algorithm; multistep method; Adams method; diffusion equation; advection-dispersion equations