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Algorithms for the fractional calculus: a selection of numerical methods. (English) Zbl 1119.65352

Summary: Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
26A33 Fractional derivatives and integrals
34A34 Nonlinear ordinary differential equations and systems
65D32 Numerical quadrature and cubature formulas
65D25 Numerical differentiation
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)

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QUADPACK; NAG; nag
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References:

[1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of mathematical functions: With formulas, graphs, and mathematical tables, NBS Applied Mathematics Series, vol. 55, National Bureau of Standards, Washington, DC, 1964, Second printing with corrections. Republished by Dover, New York, 1965; M. Abramowitz, I.A. Stegun (Eds.), Handbook of mathematical functions: With formulas, graphs, and mathematical tables, NBS Applied Mathematics Series, vol. 55, National Bureau of Standards, Washington, DC, 1964, Second printing with corrections. Republished by Dover, New York, 1965 · Zbl 0171.38503
[2] Bagley, R. L.; Torvik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27, 201-210 (1983) · Zbl 0515.76012
[3] Brass, H.; Fischer, J.-W.; Petras, K., The Gaussian quadrature method, Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 47, 115-150 (1996) · Zbl 0935.65013
[4] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent-II, Geophys. J. Roy. Astron. Soc., 13, 529-539 (1967)
[5] Caputo, M.; Mainardi, F., Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento, 1, 161-198 (1971)
[6] J.-T. Chern, Finite element modeling of viscoelastic materials on the theory of fractional calculus, Ph.D. thesis, The Pennsylvania State University, December 1993, University Microfilms No. 9414260; J.-T. Chern, Finite element modeling of viscoelastic materials on the theory of fractional calculus, Ph.D. thesis, The Pennsylvania State University, December 1993, University Microfilms No. 9414260
[7] Diethelm, K., Generalized compound quadrature formulae for finite-part integrals, IMA J. Numer. Anal., 17, 479-493 (1997) · Zbl 0871.41021
[8] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003
[9] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29, 3-22 (2002) · Zbl 1009.65049
[10] K. Diethelm, Detailed error analysis for a fractional Adams method, Numerical Algorithms, in press; K. Diethelm, Detailed error analysis for a fractional Adams method, Numerical Algorithms, in press · Zbl 1055.65098
[11] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, (Keil, F.; Mackens, W.; Voß, H.; Werther, J., Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties (1999), Springer: Springer Heidelberg), 217-224
[12] Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numer. Algorith., 16, 231-253 (1997) · Zbl 0926.65070
[13] S. Ehrich, Stieltjes polynomials and the error of Gauss-Kronrod quadrature formulas, in: W. Gautschi, G. Golub, G. Opfer (Eds.), Applications and Computation of Orthogonal Polynomials, International Series of Numerical Mathematics, no. 131, Birkhäuser, Basel, 1999, pp. 57-77; S. Ehrich, Stieltjes polynomials and the error of Gauss-Kronrod quadrature formulas, in: W. Gautschi, G. Golub, G. Opfer (Eds.), Applications and Computation of Orthogonal Polynomials, International Series of Numerical Mathematics, no. 131, Birkhäuser, Basel, 1999, pp. 57-77 · Zbl 0941.65021
[14] Elliott, D., An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals, IMA J. Numer. Anal., 13, 445-462 (1993) · Zbl 0780.65014
[15] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi (Eds.), Higher transcendental functions, Bateman manuscript project, vol. 2, McGraw-Hill, New York, 1955; A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi (Eds.), Higher transcendental functions, Bateman manuscript project, vol. 2, McGraw-Hill, New York, 1955
[16] Ford, N. J.; Simpson, A. C., The numerical solution of fractional differential equations: Speed versus accuracy, Numer. Algorith., 26, 333-346 (2001) · Zbl 0976.65062
[17] A.D. Freed, K. Diethelm, T.C. Doehring, M.A. Lillie, I. Vesely, A fractional-order viscoelastic (FOV) model for soft tissues: Tethers, in preparation; A.D. Freed, K. Diethelm, T.C. Doehring, M.A. Lillie, I. Vesely, A fractional-order viscoelastic (FOV) model for soft tissues: Tethers, in preparation
[18] A.D. Freed, K. Diethelm, Y. Luchko, Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus: First annual report, NASA/TM 2002-211914, NASA’s Glenn Research Center, Brook Park, Ohio, December 2002; A.D. Freed, K. Diethelm, Y. Luchko, Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus: First annual report, NASA/TM 2002-211914, NASA’s Glenn Research Center, Brook Park, Ohio, December 2002
[19] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88 (1991)
[20] Glöckle, W. G.; Nonnenmacher, T. F., A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68, 46-53 (1995)
[21] R. Gorenflo, Fractional calculus: Some numerical methods, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, no. 378, Springer, Wien, 1997, pp. 277-290; R. Gorenflo, Fractional calculus: Some numerical methods, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, no. 378, Springer, Wien, 1997, pp. 277-290
[22] Gorenflo, R.; Loutchko, I.; Luchko, Yu., Computation of the Mittag-Leffler function \(E_{α,β}(z)\) and its derivatives, Fract. Calculus Appl. Anal., 5, 491-518 (2002), erratum, 6 (2003) 111-112 · Zbl 1027.33016
[23] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, no. 378, Springer, Wien, 1997, pp. 223-276; R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, no. 378, Springer, Wien, 1997, pp. 223-276
[24] Gorenflo, R.; Rutman, R., On ultraslow and intermediate processes, (Rusev, P.; Dimovski, I.; Kiryakova, V., Transform Methods and Special Functions, Sofia 1994 (1995), Science Culture Technology: Science Culture Technology Singapore), 61-81 · Zbl 0923.34005
[25] Hart, J. F.; Cheney, E. W.; Lawson, C. L.; Maehly, H. J.; Mesztenyi, C. K.; Rice, J. R.; Thacher, H. G.; Witzgall, C., Computer approximations, The SIAM series in applied mathematics (1968), John Wiley & Sons: John Wiley & Sons New York · Zbl 0174.20402
[26] de Hoog, F.; Weiss, R., Asymptotic expansions for product integration, Math. Comput., 27, 295-306 (1973) · Zbl 0303.65023
[27] Kilbas, A. A.; Trujillo, J. J., Differential equations of fractional order: Methods, results and problems-I, Appl. Anal., 78, 153-192 (2001) · Zbl 1031.34002
[28] Kronrod, A. S., Integration with control of accuracy, Sov. Phys. Doklady, 9, 1, 17-19 (1964) · Zbl 0131.15003
[29] A.S. Kronrod, Nodes and weights for quadrature formulae: Sixteen place tables, Nauka, Moscow, 1964, in Russian (English translation: Consultants Bureau, New York, 1965); A.S. Kronrod, Nodes and weights for quadrature formulae: Sixteen place tables, Nauka, Moscow, 1964, in Russian (English translation: Consultants Bureau, New York, 1965)
[30] J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, Journal de l’École Polytechnique 13 (1832), no. cahier 21, 1-66; J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, Journal de l’École Polytechnique 13 (1832), no. cahier 21, 1-66
[31] J. Lützen, 2001, personal communication; J. Lützen, 2001, personal communication
[32] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, no. 378, Springer, Wien, 1997, pp. 291-348; F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, no. 378, Springer, Wien, 1997, pp. 291-348 · Zbl 0917.73004
[33] F. Mainardi, R. Gorenflo, The Mittag-Leffler function in the Riemann-Liouville fractional calculus, in: A.A. Kilbas (Eds.), Boundary Value Problems, Special Functions and Fractional Calculus. Proceedings of international conference dedicated to 90th birthday of Prof. F.D. Gakhov (Minsk), Belarusian State University, February 1996, pp. 215-225; F. Mainardi, R. Gorenflo, The Mittag-Leffler function in the Riemann-Liouville fractional calculus, in: A.A. Kilbas (Eds.), Boundary Value Problems, Special Functions and Fractional Calculus. Proceedings of international conference dedicated to 90th birthday of Prof. F.D. Gakhov (Minsk), Belarusian State University, February 1996, pp. 215-225
[34] Marks, R. J.; Hall, M. W., Differintegral interpolation from a bandlimited signal’s samples, IEEE Trans. Acoust. Speech. Signal Process., 29, 872-877 (1981) · Zbl 0525.65005
[35] D. Matignon, G. Montseny (Eds.), Fractional differential systems: Models, methods, and applications, ESAIM Proceedings, vol. 5, SMAI, Paris, 1998; D. Matignon, G. Montseny (Eds.), Fractional differential systems: Models, methods, and applications, ESAIM Proceedings, vol. 5, SMAI, Paris, 1998
[36] Metzler, R.; Schick, W.; Kilian, H.-G.; Nonnenmacher, T. F., Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995)
[37] Miller, K. S.; Ross, B., An introduction to the fractional calculus and fractional differential equations (1993), John Wiley & Sons: John Wiley & Sons New York · Zbl 0789.26002
[38] Mittag-Leffler, G., Sur la représentation analytique d’une branche uniforme d’une fonction monogène, Acta Math., 29, 101-168 (1904) · JFM 36.0469.02
[39] Nonnenmacher, T. F.; Metzler, R., On the Riemann-Liouville fractional calculus and some recent applications, Fractals, 3, 557-566 (1995) · Zbl 0868.26004
[40] Numerical Algorithms Group, Ltd., Oxford, NAG FORTRAN Library Mark 17, 1995; Numerical Algorithms Group, Ltd., Oxford, NAG FORTRAN Library Mark 17, 1995
[41] K.B. Oldham, J. Spanier, The fractional calculus, Mathematics in Science and Engineering, vol. 111, Academic Press, New York, 1974; K.B. Oldham, J. Spanier, The fractional calculus, Mathematics in Science and Engineering, vol. 111, Academic Press, New York, 1974 · Zbl 0292.26011
[42] Olmstead, W. E.; Handelsman, R. A., Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Rev., 18, 275-291 (1976) · Zbl 0323.45008
[43] R. Piessens, E. de Doncker-Kapenga, C. Überhuber, D.K. Kahaner, QUADPACK-A subroutine package for automatic integration, Springer Series in Computational Mathematics, no. 1, Springer, Berlin, 1983; R. Piessens, E. de Doncker-Kapenga, C. Überhuber, D.K. Kahaner, QUADPACK-A subroutine package for automatic integration, Springer Series in Computational Mathematics, no. 1, Springer, Berlin, 1983 · Zbl 0508.65005
[44] I. Podlubny, Fractional-order systems and fractional-order controllers, Tech. Report UEF-03-94, Institute for Experimental Physics, Slovak Academy of Sciences, 1994; I. Podlubny, Fractional-order systems and fractional-order controllers, Tech. Report UEF-03-94, Institute for Experimental Physics, Slovak Academy of Sciences, 1994 · Zbl 1056.93542
[45] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, 1999; I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, 1999 · Zbl 0924.34008
[46] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus Appl. Anal., 5, 367-386 (2002) · Zbl 1042.26003
[47] Podlubny, I.; Dorcak, L.; Misanek, J., Application of fractional-order derivatives to calculation of heat load intensity change in blast furnace walls, Trans. Tech. Univ. Košice, 5, 137-144 (1995)
[48] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical recipes in C: The art of scientific computing, 2nd ed., Cambridge University Press, Cambridge, 1999, Reprinted, corrected to software version 2.08; W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical recipes in C: The art of scientific computing, 2nd ed., Cambridge University Press, Cambridge, 1999, Reprinted, corrected to software version 2.08 · Zbl 0661.65001
[49] Yu.N. Rabotnov, Creep problems in structural members, North-Holland Series in Applied Mathematics and Mechanics, vol. 7, North-Holland Publishing Company, Amsterdam, 1969, Originally published in Russian as: Polzuchest Elementov Konstruktsii; Yu.N. Rabotnov, Creep problems in structural members, North-Holland Series in Applied Mathematics and Mechanics, vol. 7, North-Holland Publishing Company, Amsterdam, 1969, Originally published in Russian as: Polzuchest Elementov Konstruktsii
[50] Ross, B., The development of fractional calculus 1695-1900, Hist. Math., 4, 75-89 (1977) · Zbl 0358.01008
[51] Samko, S. G.; Kilbas, A.; Marichev, O. I., Fractional integrals and derivatives: Theory and applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
[52] Unser, M.; Blu, T., Fractional splines and wavelets, SIAM Rev., 42, 43-67 (2000) · Zbl 0940.41004
[53] Unser, M., Wavelet theory demystified, IEEE Trans. Signal Process., 51, 470-483 (2003) · Zbl 1369.42001
[54] Welch, S. W.J.; Ropper, R. A.L.; Duren, R. G., Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials, Mech. Time-Dependent Mater., 3, 279-303 (1999)
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