Skip to main content
SpringerLink
Log in

The spectral method for solving systems of Volterra integral equations

  • Applied mathematics
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This paper presents a high accurate and stable Legendre-collocation method for solving systems of Volterra integral equations (SVIEs) of the second kind. The method transforms the linear SVIEs into the associated matrix equation. In the nonlinear case, after applying our method we solve a system of nonlinear algebraic equations. Also, sufficient conditions for the existence and uniqueness of the Linear SVIEs, in which the coefficient of the main term is a singular (or nonsingular) matrix, have been formulated. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods. All of the numerical computations have been performed on a PC using several programs written in MAPLE 13.

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.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Aminikhah, H., Biazar, J.: A new analytical method for solving systems of Volterra integral equations. Int. J. Comput. Math. 87, 1142–1157 (2010)

    Article  MATH  Google Scholar 

  2. Berenguer, M.I., Gamez, D., Garralda-Guillem, A.I., Ruiz Galan, M., Serrano Perez, M.C.: Biorthogonal systems for solving Volterra integral equation systems of the second kind. J. Comput. Appl. Math. 235, 1873–1875 (2011)

    Article  MathSciNet  Google Scholar 

  3. Biazar, J.: Solving system of integral equations by Adomian decomposition method. Ph.D. thesis, Teaching Training University, Iran (2002)

  4. Biazar, J., Babolian, E., Islam, R.: Solution of a system of Volterra integral equations of the first kind by Adomian method. Appl. Math. Comput. 139, 249–258 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biazar, J., Ebrahimi, H.: Chebyshev wavelets approach for nonlinear systems of Volterra integral equations. Comput. Math. Appl. 63, 608–616 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biazar, J., Ghazvini, H.: He’s homotopy perturbation method for solving system of Volterra integral equations of the second kind. Chaos Solitons Fractals 39, 770–777 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2000)

    Google Scholar 

  8. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  9. Bulatov, M.V.: Regularization of singular systems of Volterra integral equations. Comput. Math. Math. Phys. 42, 315–320 (2002)

    MathSciNet  Google Scholar 

  10. Bulatov, M.V.: Transformation of differential-algebraic systems of equations. Comput. Math. Math. Phys. 34, 301–331 (1994)

    MathSciNet  Google Scholar 

  11. Bulatov, M.V., Chistyakov, V.F.: The properties of differential-algebraic systems and their integral analogs. Preprint, Memorial University of Newfoundland (1997)

  12. Bulatov, M.V., Lima, P.M.: Two-dimensional integral-algebraic systems: analysis and computational methods. J. Comput. Appl. Math. 236, 132–140 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  15. Chen, Y.P., Tang, T.: Spectral methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chistyakov, V.F.: Differential-Algebraic Operators with Finite Dimensional Kernel. Nauka, Novosibirk (1996) (in Russian)

    Google Scholar 

  17. Delves, L.M., Mohamed, J.L.: Computational Methods for Integral Equations. Cambridge University Press, London (1985)

    Book  MATH  Google Scholar 

  18. Elnagar, G.N., Kazemi, M.: Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations. J. Comput. Appl. Math. 76, 147–158 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hadizadeh, M., Ghoreishi, F., Pishbin, S.: Jacobi spectral solution for integral algebraic equations of index-2. Appl. Numer. Math. 61, 131–148 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Han, H., Liu, Y., Lau, T., He, X.: New algorithm for the system of nonlinear weakly singular Volterra integral equations of the second kind and integro-differential equations. J. Inf. Comput. Sci. 7, 1229–1235 (2010)

    Google Scholar 

  21. Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  22. Jiang, Y.-J.: On spectral methods for Volterra-type integro-differential equations. J. Comput. Appl. Math. 230, 333–340 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Katani, R., Shahmorad, S.: Block by block method for the systems of nonlinear Volterra integral equations. Appl. Math. Model. 34, 400–406 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kauthen, J.P.: The numerical solution of integral-algebraic equations of index-1 by polynomial spline collocation methods. Math. Comput. 70, 1503–1514 (2001)

    MathSciNet  MATH  Google Scholar 

  25. Kress, R.: Linear Integral Equations. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  26. Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985)

    Book  MATH  Google Scholar 

  27. Maleknejad, K., Salimi Shamloo, A.: Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices. Appl. Math. Comput. 195, 500–505 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nadjafi, J.S., Samadi, O.R.N., Tohidi, E.: Numerical solution of two dimensional integral equations via aspectral Galerkin method. J. Appl. Math. Biol. 1, 343–359 (2011)

    Google Scholar 

  29. Rabbani, M., Maleknejad, K., Aghazadeh, N.: Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method. Appl. Math. Comput. 187, 1143–1146 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Saberi-Nadjafi, J., Tamamgar, M.: The variational iteration method: a highly promising method for solving the system of integro differential equations. Comput. Math. Appl. 56, 346–351 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sahin, N., Yuzbasi, S., Gulsu, M.: A collocation approach for solving systems of linear Volterra integral equations with variable coefficients. Comput. Math. Appl. 62, 755–769 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sorkun, H.H., Yalkinbas, S.: Approximate solutions of linear Volterra integral equation systems with variable coefficients. Appl. Math. Model. 34, 3451–3464 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Tohidi, E.: Legendre approximation for solving linear HPDEs and comparison with Taylor and Bernoulli matrix methods. Appl. Math. 3, 410–416 (2012). doi:10.4236/am.2012.35063

    Article  Google Scholar 

  35. Tohidi, E., Samadi, O.R.N.: Optimal control of nonlinear Volterra integral equations via Legendre polynomials. IMA J. Math. Control Inf. (2012). doi:10.1093/imamci/DNS014

    Google Scholar 

  36. Tohidi, E., Samadi, O.R.N., Farahi, M.H.: Legendre approximation for solving a class of nonlinear optimal control problems. J. Math. Finance 3, 8–13 (2011). doi:10.4236/jmf.2011.11002

    Article  Google Scholar 

  37. Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4, 181–193 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yusufoglu, E.: Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules. Int. J. Comput. Math. 84, 33–140 (2007)

    Article  MathSciNet  Google Scholar 

  39. Yuzbasi, S., Sahin, N., Sezer, M.: A Bessel collocation method for numerical solution of generalized pantograph equations. Numer. Methods Partial Differ. Equ. (2010). doi:10.1002/num.20660

    Google Scholar 

  40. Zarebnia, M., Rashidinia, J.: Approximate solution of systems of Volterra integral equations with error analysis. Int. J. Comput. Math. 87, 3052–3062 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are very much indebted to the all unknown reviewers, specially the sixth reviewer, for comments and remarks which led to improved presentation and quality of this paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    O. R. Navid Samadi & Emran Tohidi

Authors
  1. O. R. Navid Samadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emran Tohidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emran Tohidi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Samadi, O.R.N., Tohidi, E. The spectral method for solving systems of Volterra integral equations. J. Appl. Math. Comput. 40, 477–497 (2012). https://doi.org/10.1007/s12190-012-0582-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-012-0582-8

Keywords

Mathematics Subject Classification

Search

Navigation