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The numerical solution of linear multi-term fractional differential equations: Systems of equations. (English) Zbl 1019.65048
The present paper deals with the analysis of a numerical method for linear scalar fractional differential equations of arbitrary order. To discretize fractional derivatives a linear combination of convolution weights is used. Moreover, a convergence theorem for the proposed numerical scheme is presented. Finally, some explicit computations including the well known Bagley Torvik equation modelling the motion of a rigid plate immersed in a Newtonian fluid are presented.

MSC:
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
26A33 Fractional derivatives and integrals
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
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[1] L. Blank, Numerical treatment of differential equations of fractional order, Numerical Analysis Report 287, Manchester Centre for Computational Mathematics, 1996.
[2] Caputo, M., Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. royal astron. soc., 13, 529-539, (1967)
[3] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, in: Electronic Transactions on Numerical Analysis, Vol. 5, Kent State University, ETNA 5 (1997) pp. 1-6. · Zbl 0890.65071
[4] Diethelm, K., Numerical approximation of finite-part integrals with generalised compound quadrature formulae, IMA J. numer. anal., 17, 479-493, (1997) · Zbl 0871.41021
[5] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[6] Diethelm, K.; Ford, N.J., Numerical solution of the bagley – torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067
[7] K. Diethelm, N.J. Ford, The numerical solution of linear and nonlinear fractional differential equations involving fractional derivatives of several orders, Numerical Analysis Report 379, Manchester Centre for Computational Mathematics, 2001.
[8] K. Diethelm, Y. Luchko, Numerical solution of linear multi-term differential equations of fractional order, J. Comput. Anal. Appl., to appear. · Zbl 1083.65064
[9] Ford, N.J.; Simpson, A.C., The numerical solution of fractional differential equationsspeed versus accuracy, Numer. algorithms, 26.4, 333-346, (2001) · Zbl 0976.65062
[10] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (), 223-276
[11] Lambert, J.D., Numerical methods for ordinary differential systems, (1991), Wiley New York · Zbl 0745.65049
[12] Lubich, C., Discretized fractional calculus, SIAM J. math. anal., 17, 3, 704-719, (1986) · Zbl 0624.65015
[13] Lubich, C., Convolution quadrature and discretized operational calculus, I, Numer. math., 52, 129-145, (1988) · Zbl 0637.65016
[14] Mainardi, F., Some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[15] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993, pp. 209-217. · Zbl 0789.26002
[16] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[17] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[18] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives, (1993), Gordon and Breach Science Publishers London · Zbl 0818.26003
[19] Torvik, P.J.; Bagley, R.L., On the appearance of the fractional derivative in the behavior of real materials, J. appl. mech., 51, 294-298, (1984) · Zbl 1203.74022
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