calculus document

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Comput. Methods Appl. Mech. Engrg. 194 (2005) 743773www.elsevier.com/locate/cmaAlgorithms for the fractional calculus: A selectionof numerical methods qK. Diethelm a, N.J. Ford b, A.D. Freedac,d,*, Yu. LuchkoeInstitut fur Angewandte Mathematik, Technische Universitat Braunschweig, Pockelsstrae 14, D-38106 Braunschweig, GermanybDepartment of Mathematics, Chester College, Parkgate Road, Chester CH1 4BJ, UKcPolymers Branch, Materials Division, MS 49-3, NASAs John H. Glenn Research Center at Lewis Field,21000 Brookpark Road, Cleveland, OH 44135, USAdDepartment of Biomedical Engineering, ND 20, Lerner Research Institute, Cleveland Clinic Foundation, 9500 Euclid Avenue,Cleveland, OH 44195, USAeDepartment of Business Informatics, European University Frankfurt, P.O. Box 1768, D-15207 Frankfurt (Oder), GermanyReceived 21 March 2003; received in revised form 12 February 2004; accepted 7 June 2004AbstractMany recently developed models in areas like viscoelasticity, electrochemistry, diusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection ofnumerical algorithms for the solution of the various problems arising in this context. We believe that this will givethe engineer the necessary tools required to work with fractional models in an ecient way.Published by Elsevier B.V.Keywords: RiemannLiouville integrals; Caputo derivatives; Caputo dieren

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