Skip to main content
SpringerLink
Log in

Jacobi spectral Galerkin methods for fractional integro-differential equations

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

We propose general spectral and pseudo-spectral Jacobi–Galerkin methods for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide rigorous error analysis for spectral and pseudo-spectral Jacobi–Galerkin methods, which show that the errors of the approximate solution decay exponentially in \(L^\infty \) norm and weighted \(L^2\)-norm. The numerical examples are given to illustrate the theoretical results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Bagley, R.L., Trovik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)

    Article  MATH  Google Scholar 

  2. Bhrawy, A.H., Alofi, A.S.: The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. 26, 25–31 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bhrawy, A.H., Alshomrani, M.: A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Adv. Differ. Eqs. 2012, 1–8 (2012)

    Article  Google Scholar 

  4. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  5. Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Colton, D., Kress, R.: Inverse Coustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, 2nd edn. Springer, Heidelberg (1998)

    Book  Google Scholar 

  7. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35, 5662–5672 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Douglas, J., Dupont, T., Wahlbin, L.: The stability in Lq of the L2-Projection into finite element function spaces. Numer. Math. 23, 193–198 (1975)

    Article  MATH  Google Scholar 

  9. Khader, M.M., Hendy, A.S.: The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudo-spectral method. Int. J. Pure Appl. Math. 74, 287–297 (2012)

    MATH  MathSciNet  Google Scholar 

  10. Li, Y.: Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonlinear Sci. Numer. Simul. 15, 2284–2292 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mastroianni, G., Occorsto, D.: Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey. J. Comput. Appl. Math. 134, 325–341 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Momani, S.M.: Local and global existence theorems on fractional integro-differential equations. J. Fract. Calc. 18, 81–86 (2000)

    MATH  MathSciNet  Google Scholar 

  13. Odibat, Z., Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals 36, 167–174 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ragozin, D.L.: Polynomial approximation on compact manifolds and homogeneous spaces. Trans. Am. Math. Soc. 150, 41–53 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ragozin, D.L.: Constructive polynomial approximation on spheres and projective spaces. Trans. Am. Math. Soc. 162, 157–170 (1971)

    MathSciNet  Google Scholar 

  16. Tao, X., Xie, Z., Zhou, X.: Spectral Petrov–Galerkin methods for the second kind Volterra type integro-differential equations. Numer. Math. Theory Methods Appl. 4, 216–236 (2011)

    MATH  MathSciNet  Google Scholar 

  17. Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for volterra type Integral equations. J. Sci. Comput. 53, 414–434 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The work was supported by NSFC Project (11301446), China Postdoctoral Science Foundation Grant (2013M531789), Program for Changjiang Scholars and Innovative Research Team in University (IRT1179), Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2013RS4057) and the Research Foundation of Hunan Provincial Education Department (13B116). The author would like to express their sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Author information

Authors and Affiliations

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Yin Yang

Authors
  1. Yin Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yin Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Y. Jacobi spectral Galerkin methods for fractional integro-differential equations. Calcolo 52, 519–542 (2015). https://doi.org/10.1007/s10092-014-0128-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10092-014-0128-6

Keywords

Mathematics Subject Classification

Search

Navigation