Skip to main content

Comparison of Five Numerical Schemes for Fractional Differential Equations

  • Chapter

This paper presents a comparative study of the performance of five different numerical schemes for the solution of fractional differential equations.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Mainardi F (1997) Fractional calculus: some basic problem in continuum and statistical mechanics, In: Fractals and Fractional Calculus in Continuum Mechanics. Carpinteri, A. Mainardi, F (eds.), Springer, Wein, New York, pp. 291-348.

    Google Scholar 

  2. Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50:15-67.

    Article  Google Scholar 

  3. Podlubny I (1999) Fractional Differential Equations. Academic Press, New York.

    MATH  Google Scholar 

  4. Hilfer R (2000) Applications of Fractional Calculus in Physics. World Scientific, New Jersey.

    MATH  Google Scholar 

  5. West BJ, Bologna M, Grigolini P (2003) Physics of Fractal Operators. Springer, New York.

    Google Scholar 

  6. Magin RL (2004) Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32(1):1-104.

    Article  Google Scholar 

  7. Magin RL (2004) Fractional calculus in bioengineering - Part 2. Crit. Rev. Biomed. Eng. 32(2):105-193.

    Article  Google Scholar 

  8. Magin RL (2004) Fractional calculus in bioengineering - Part 3. Crit. Rev. Biomed. Eng. 32(3/4):194-377.

    Google Scholar 

  9. Miller KS, Ross B (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.

    MATH  Google Scholar 

  10. Gaul L, Klein P, Kempfle S (1989) Impulse response function of an oscillator with fractional derivative in damping description. Mech. Res. Commun. 16(5):4447-4472.

    Article  Google Scholar 

  11. Suarez LE, Shokooh A (1997) An eigenvector expansion method for the solution of motion containing fractional derivatives. ASME. J. Appl. Mech. 64:629-635.

    Article  MATH  MathSciNet  Google Scholar 

  12. Yuan L, Agrawal OP (2002) A numerical scheme for dynamic systems containing fractional derivatives. Transactions of the ASME, J. Vib. Acoust. 124:321-324.

    Article  Google Scholar 

  13. Machado JAT (2001) Discrete-time fractional-order controllers. FCAA J. 4:47-66.

    MATH  Google Scholar 

  14. Heleschewitz D, Matignon D (1998) Diffusive Realizations of Fractional inte- grodifferential Operators: Structural Analysis Under Approximation, in: Proceedings IFAC Conference System, Structure and Control, Nantes, France, 2:243-248.

    Google Scholar 

  15. Aoun M, Malti R, Levron F, Oustaloup A (2003) Numerical simulation of fractional systems, in: Proceedings of DETC2003, 2003 ASME Design Engineering Technical Conferences, September 2-6, Chicago, Illinois.

    Google Scholar 

  16. Poinot T, Trigeassou J (2003) Modeling and simulation of fractional systems using a non integer integrator, in: Proceedings of DETC2003, 2003 ASME Design Engineering Technical Conferences, September 2-6, Chicago, Illinois.

    Google Scholar 

  17. Padovan J (1987) Computational algorithms and finite element formulation involving fractional operators. Comput. Mech. 2:271-287.

    Article  MATH  Google Scholar 

  18. Gorenflo R (1997) Fractional calculus: some numerical methods In: Carpinteri A, Maincardi, F (eds.), Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wein, New York, pp. 277-290.

    Google Scholar 

  19. Ruge P, Wagner N (1999) Time-domain solutions for vibration systems with feding memory. European Conference of Computational Mechanics, Munchen, Germany, August 31 September 3.

    Google Scholar 

  20. Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J. Math. Anal. Appl. 265:229-248.

    Article  MATH  MathSciNet  Google Scholar 

  21. Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29(1-4): 3-22.

    Article  MATH  MathSciNet  Google Scholar 

  22. Diethelm K (2003) Efficient solution of multi-term fractional differential equation using P(EC)mE methods. Computing 71:305-319.

    Article  MATH  MathSciNet  Google Scholar 

  23. Diethelm K, Ford NJ (2004) Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154(3):621-640.

    Article  MATH  MathSciNet  Google Scholar 

  24. Diethelm K, Ford NJ, Freed AD, Luchko Y (2005) Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mechan. Eng. 194:743-773.

    Article  MATH  MathSciNet  Google Scholar 

  25. Agrawal OP (2004) Block-by-Block Method for Numerical Solution of Fractional Differential Equations, in: Proceedings of IFAC2004, First IFAC Workshop on Fractional Differentiation and Its Applications. Bordeaux, France, July 19-21.

    Google Scholar 

  26. Kumar P, Agrawal OP (2005) A Cubic Scheme for Numerical Solution of Fractional Differential Equations, in: Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven University of Technology, Eindhoven, The Netherland, August 7-12.

    Google Scholar 

  27. Kumar P, Agrawal OP (2005) Numerical Scheme for the Solution of Fractional Differential Equations, in: Proceedings of the 2005 ASME Design Engineering Technical Conferences and Computer and Information Engineering Conference, Long Beach, California, September 24-28.

    Google Scholar 

  28. Sabatier J, Malti R(2004) Simulation of Fractional Systems: A Benchmark, in: Proceedings of IFAC2004, First IFAC Workshop on Fractional Differentiation and Its Applications. Bordeaux, France, July 19-21.

    Google Scholar 

  29. Lorenzo CF, Hartley TT (2000) Initialized fractional calculus. Int. J. Appl. Math. 3:249-265.

    MATH  MathSciNet  Google Scholar 

  30. Achar BN, Lorenzo CF, Hartley TT (2005) Initialization issue of the Caputo fractional derivative, in: Proceedings of the2005 ASME Design Engineering Technical Conferences, Long Beach, California, September 24-28.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer

About this chapter

Cite this chapter

Agrawal, O.P., Kumar, P. (2007). Comparison of Five Numerical Schemes for Fractional Differential Equations. In: Sabatier, J., Agrawal, O.P., Machado, J.A.T. (eds) Advances in Fractional Calculus. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6042-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4020-6042-7_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-6041-0

  • Online ISBN: 978-1-4020-6042-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics