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Numerical methods for fractional calculus. (English) Zbl 1326.65033
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.

##### 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
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