Computational algorithms for computing the fractional derivatives of functions

Mathematics and Computers in Simulation

Volumn 79, Issue 7, 2009, Pages 2013-2020

  Odibat, Zaid M ^a  

^a AL BALQA APPLIED UNIVERSITY   (Jordan)

Author keywords

Caputo fractional derivative; Caputo fractional derivative rule; Modified trapezoidal rule

Indexed keywords

ALGORITHMS; ERROR ANALYSIS;

CAPUTO FRACTIONAL DERIVATIVE; CAPUTO FRACTIONAL DERIVATIVE RULE; CAPUTO SENSE; COMPUTATIONAL ALGORITHMS; FRACTIONAL DERIVATIVES; FUNCTION VALUES; ILLUSTRATIVE EXAMPLES; MODIFIED TRAPEZOIDAL RULE; NUMERICAL RESULTS; WEIGHTED SUMS;

DERIVATIVES;

EID: 61449085393     PISSN: 03784754     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matcom.2008.08.003     Document Type: Article

References (36)

1. Bagley R., and Torvik P. A theortical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27 3 (1983) 201-210
2. Beyer H., and Kempfle S. Defintion of physically consistent damping laws with fractional derivatives. Z. Angew Math. Mech. 75 (1995) 623-635
3. Caputo M. Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc. 13 (1967) 529-539
4. Diethelm K. An algorithm for the numerical solution for differential equations of fractional order. Elec. Transact. Numer. Anal. 5 (1997) 1-6
5. Diethelm K., and Ford N. Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002) 229-248
6. Diethelm K., Ford N., and Freed A. A Predictor-Corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29 (2002) 3-22
7. Diethelm K., Ford N.J., and Freed A.D. Detailed error analysis for a fractional Adams Method. Numer. Algorithms 36 (2004) 31-52
8. Diethelm K., Ford N.J., Freed A.D., and Luchko Yu. Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194 6-8 (2005) 743-773
9. Gerald C., and Wheatley P. Applied Numerical Analysis (2004), Addison-Wesley, USA
10. 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, Springer Verlag, Wien and New York, 1997, pp. 223-276.
11. He J.H. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167 (1998) 57-68
12. He J.H. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15 2 (1999) 86-90
13. Hilfer R. Foundations of fractional dynamics. Fractals 3 3 (1995) 549-556
14. Huang F., and Liu F. The fundamental solution of the spcae-time fractional advection-dispersion equation. J. Appl. Math. Comput. 18 2 (2005) 339-350
15. Lubich C. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comp. 45 172 (1985) 463-469
16. Lubich C. Discretized fractional calculus. SIAM J. Math. Anal. 17 (1986) 704-719
17. Y. Luchko, R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivatives, Preprint Series A08-98, Fachbreich Mathematik and Informatik, Freic Universitat Berlin, 1998.
18. Mainardi F. Fractional relaxation-oscilation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7 (1996) 1461-1477
19. Mainardi F. Fractional calculus: 'Some basic problems in continuum and statistical mechanics'. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional calculus in Continuum Mechanics (1997), Springer-Verlag, New York 291-348
20. Mathews J., and Fink K. Numerical Methods Using Mathlab (1990), Prentice-Hall, USA
21. Meerschaert M.M., Scheffler H.P., and Tadjeran C. Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211 (2006) 249-261
22. Metzler R., and Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37 (2004) R161-208
23. Miller K.S., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons, Inc., New York
24. Momani S. Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons Fractals 28 4 (2006) 930-937
25. Momani S., and Odibat Z. Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Appl. Math. Comput. 177 (2006) 488-494
26. Momani S., and Odibat Z. Numerical comparison of methods for solving linear differential equations of fractional order, Chaos. Solitons Fractals 31 (2007) 1248-1255
27. Momani S., and Odibat Z. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365 (2007) 345-350
28. Odibat Z. Approximations of fractional integral and Caputo fractional derivatives. Appl. Math. Comput. 178 (2006) 527-533
29. Odibat Z., and Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlin. Sci. Numer. Simulat. 7 1 (2006) 15-27
30. Odibat Z., and Momani S. Numerical methods for solving nonlinear partial differential equations of fractional order. Appl. Math. Modell. 32 1 (2008) 28-39
31. Odibat Z., and Momani S. Generalized differential transform method for linear partial differential equations of fractional order. Appl. Math. Lett. 21 2 (2008) 194-199
32. Oldham K.B., and Spanier J. The Fractional Calculus (1974), Academic Press, New York
33. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego, CA
34. Podlubny I. Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calculus Appl. Anal. 5 (2002) 367-386
35. Shawagfeh N. Analytical approximate solutions for nonlinear fractional differential equations. J. Math. Anal. Appl. 131 (2002) 517-529
36. Yuan L., and Agrawal O.P. A numerical scheme for dynamic systems containing fractional derivatives. J. Vibrat. Acoust. 124 (2002) 321-324