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The Grünwald-Letnikov method for fractional differential equations. (English) Zbl 1228.65121
Summary: This paper is devoted to the numerical treatment of fractional differential equations. Based on the Grünwald-Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given.

##### MSC:
 65L12 Finite difference and finite volume methods for ordinary differential equations 34A08 Fractional ordinary differential equations and fractional differential inclusions 26A33 Fractional derivatives and integrals 45J05 Integro-ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations
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