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Pitfalls in fast numerical solvers for fractional differential equations. (English) Zbl 1078.65550
Summary: We consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form \(x^{(\alpha)}(t)=f(t,x(t)), x(0)=x_{0}\), where \(x^{(\alpha)}\) is the derivative of \(x\) of order \(\alpha\) in the sense of Caputo and \(0<\alpha < 1\). We review some of the existing methods and explain their respective strengths and weaknesses. We identify and discuss potential problems in the development of generally applicable schemes.

65L05 Numerical methods for initial value problems involving ordinary differential equations
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[1] C.T.H. Baker, M.S. Derakhshan, Stability barriers to the construction of \(\{\rho, \sigma \}\)-reducible and fractional quadrature rules, in: H. Braß, G. Hämmerlin (Eds.), Numerical Integration III, no. 85 in Internat. Ser. Numer. Math., Birkhäuser, Basel, 1988, pp. 1-15.
[2] Björck, Å., Solving linear least squares problems by Gram-Schmidt orthogonalization, Nordisk tidskr. informations-behandling, 7, 1-21, (1967) · Zbl 0183.17802
[3] Björck, Å.; Pereyra, V., Solution of Vandermonde systems of equations, Math. comput., 24, 893-903, (1970) · Zbl 0221.65054
[4] Caputo, M., Linear models of dissipation whose Q is almost frequency independent, II, Geophys. J. R. astron. soc., 13, 529-539, (1967)
[5] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electr. trans. numer. anal., 5, 1-6, (1997) · Zbl 0890.65071
[6] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[7] Diethelm, K.; Ford, N.J., Numerical solution of the bagley-torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067
[8] Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solution, Appl. math. comput., 154, 621-640, (2004) · Zbl 1060.65070
[9] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear dynamics, 29, 3-22, (2002) · Zbl 1009.65049
[10] Diethelm, K.; Ford, N.J.; Freed, A.D., Detailed error analysis for a fractional Adams method, Numer. algorithms, 36, 31-52, (2004) · Zbl 1055.65098
[11] K. Diethelm, A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, in: S. Heinzel, T. Plesser (Eds.), Forschung und wissenschaftliches Rechnen 1998, no. 52 in GWDG-Berichte, Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen, 1999, pp. 57-71.
[12] Diethelm, K.; Freed, A.D., On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, (), 217-224
[13] Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numer. algorithms, 16, 231-253, (1997) · Zbl 0926.65070
[14] Ford, N.J.; Simpson, A.C., The numerical solution of fractional differential equations: speed versus accuracy, Numer. algorithms, 26, 333-346, (2001) · Zbl 0976.65062
[15] A.D. Freed, K. Diethelm, Y. Luchko, Fractional-order viscoelasticity (FOV): Constitutive developments using the fractional calculus: First annual report, Technical Memorandum, TM-2002-211914, NASA Glenn Research Center, Cleveland, 2002.
[16] Gorenflo, R., Afterthoughts on interpretation of fractional derivatives and integrals, (), 589-591
[17] Greenbaum, A., Iterative methods for solving linear systems, (1997), SIAM Philadelphia, PA · Zbl 0883.65022
[18] Hairer, E.; Lubich, C.; Schlichte, M., Fast numerical solution of weakly singular Volterra equations, J. comput. appl. math., 23, 87-98, (1988) · Zbl 0654.65091
[19] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I: nonstiff problems, (1993), Springer Berlin · Zbl 0789.65048
[20] Higham, N.J., Error analysis of the björck – pereyra algorithms for solving Vandermonde systems, Numer. math., 50, 613-632, (1987) · Zbl 0595.65029
[21] Joulin, G., Point-source initiation of Lean spherical flames of light reactants: an asymptotic theory, Combust. sci. technol., 43, 99-113, (1985)
[22] Lambert, J.D., Computational methods in ordinary differential equations, (1973), Wiley London · Zbl 0258.65069
[23] Lederman, C.; Roquejoffre, J.-M.; Wolanski, N., Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames, C.R., math., acad. sci. Paris, 334, 569-574, (2002) · Zbl 0998.80008
[24] Levinson, N., A nonlinear Volterra equation arising in the theory of superfluidity, J. math. anal. appl., 1, 1-11, (1960) · Zbl 0094.08501
[25] Lubich, C., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. comput., 45, 463-469, (1985) · Zbl 0584.65090
[26] Lubich, C., Discretized fractional calculus, SIAM J. math. anal., 17, 704-719, (1986) · Zbl 0624.65015
[27] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[28] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[29] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[30] Rall, L.B., Automatic differentiation: techniques and applications, (1981), Springer Berlin · Zbl 0473.68025
[31] Robbin, J.W.; Salamon, D.A., The exponential Vandermonde matrix, Linear algebra appl., 317, 225-226, (2000) · Zbl 0967.15014
[32] Saad, Y., Iterative methods for sparse linear systems, (2003), SIAM Philadelphia · Zbl 1002.65042
[33] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[34] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[35] 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
[36] Walker, H.F., Implementation of the GMRES method using Householder transformations, SIAM J. sci. statist. comput., 9, 152-163, (1988) · Zbl 0698.65021
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