Advertisement

Convergence Analysis of Variational Iteration Method for Caputo Fractional Differential Equations

  • Zhiwu Wen
  • Jie Yi
  • Hongliang Liu
Part of the Communications in Computer and Information Science book series (CCIS, volume 324)

Abstract

In this paper, the variational iteration method is applied to solve initial value problems of Caputo fractional differential equations. The convergence of the variational iteration method for solving the initial value problems of the kind of equation has been proved. The numerical examples show the efficiency of the variational iteration method for solving the initial value problems of the kind of equation.

Keywords

Fractional differential equation Variational iteration method Caputo derivatives Convergence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265, 229–248 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Xu, L.: Variational iteration method for solving integral equations. Computers and Mathematics with Applications 54, 1071–1078 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ghorbani, A., Nadjufi, J.S.: An effective modification of He’s variational iteration method. Nonlinear Analysis: Real World Applications 10, 2828–2833 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped strucures. Journal of Guidance, Control and Dynamics 14, 304–311 (1991)CrossRefGoogle Scholar
  5. 5.
    Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics 51, 299–307 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Pudlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)Google Scholar
  7. 7.
    Saha Ray, S., Bera, R.K.: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Applied Mathematics and Computation 167, 561–571 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Momani, S., Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Applied Mathematics and Computation 162, 1351–1365 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yu, Q.: Numerical Approximation of the Fractional Reaction-diffusion Equation. Master Paper, Xiamen University (2007)Google Scholar
  10. 10.
    Momani, S., Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons and Fractals 31(5), 1248–1255 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Momani, S., Odibat, Z.: Numerical approach to differential equations of fractional order. Journal of Computational and Applied Mathematics 207, 96–110 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equation. Nonlinear Dynamics 29(14), 3–22 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yu, Z.H.: Variational iteration method for solving the multi-pantograph delay equation. Physics Letters A 372, 6475–6479 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhang, Y.: Application for some Fractional Ordinary Differential Equation sbout new iteration method.  Master Paper, Xiangtan University (2009)Google Scholar
  15. 15.
    Salkuyeh, D.K.: Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients. Computers Math. Applic. 56, 2027–2033 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Odibat, Z.: A study on the convergence of variational iteration method. Mathematics and Computer Modelling 51, 1181–1192 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Nauka, Moscow (1979)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Zhiwu Wen
    • 1
  • Jie Yi
    • 1
  • Hongliang Liu
    • 1
  1. 1.School of Mathematics and Computational ScienceXiangtan UniversityHunanChina

Personalised recommendations