An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives

Numerical Algorithms

Volumn 47, Issue 4, 2008, Pages 361-390

  Diethelm, Kai ^a,b  

^a TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

^b GNS Gesellschaft für Numerische Simulation MbH ^*  (Germany)

Author keywords

Caputo derivative; Fractional derivatives; Fractional differential equation; Numerical approximation; Yuan Agrawal method

Indexed keywords

EID: 42149141688     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-008-9193-8     Document Type: Article

References (24)

1. Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304-311 (1991)
2. Capobianco, M.R., Criscuolo, G.: Some remarks on the numerical computation of integrals on an unbounded interval. Numer. Algorithms 45, 37-48 (2007)
3. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529-539 (1967) (reprinted in Fract. Calc. Appl. Anal. 11, 4-14 (2008))
4. Chatterjee, A.: Statistical origins of fractional derivatives in viscoelasticity. J. Sound Vib. 284, 1239-1245 (2005)
5. Davis, P.J., Rabinowitz, P.: Some geometrical theorems for abscissas and weights of Gauss type. J. Math. Anal. Appl. 2, 428-437 (1961)
6. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic, Orlando (1984)
7. Diethelm, K.: Generalized compound quadrature formulae for finite-part integrals. IMA J. Numer. Anal. 17, 479-493 (1997)
8. Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Voß, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, pp. 217-224. Springer, Heidelberg (1999)
9. Elliott, D.: An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals. IMA J. Numer. Anal. 13, 445-462 (1993)
10. Ford, N.J., Simpson, A.C.: The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26, 333-346 (2001)
11. Gautschi, W.: Quadrature formulae on half-infinite intervals. BIT 31, 438-446 (1991)
12. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223-276, Springer, Wien (1997)
13. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. 2nd revised edn. Springer, Berlin (1996)
14. Lu, J.-F., Hanyga, A.: Wave field simulation for heterogeneous porous media with singular memory drag force. J. Comput. Phys. 208, 651-674 (2005)
15. Lubich, C.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comp. 45, 463-469 (1985)
16. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704-719 (1986)
17. Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)
18. Ruge, P., Trinks, C.: Consistent modelling of infinite beams by fractional dynamics. Nonlinear Dyn. 38, 267-284 (2004)
19. Schmidt, A., Gaul, L.: On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems. Mech. Res. Commun. 33, 99-107 (2006)
20. Schwarz, H.R.: Numerical Analysis. Wiley, Chichester (1989)
21. Singh, S.J., Chatterjee, A.: Galerkin projections and finite elements for fractional order derivatives. Nonlinear Dyn. 45, 183-206 (2006)
22. Szego, G.: Orthogonal Polynomials. 4th edn. American Mathematical Society, Providence (1975)
23. Trinks, C., Ruge, P.: Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals. Comput. Mech. 29, 471-476 (2002)
24. Yuan, L., Agrawal, O.P.: A numerical scheme for dynamic systems containing fractional derivatives. J. Vib. Acoust. 124, 321-324 (2002)