Chebyshev collocation treatment of Volterra–Fredholm integral equation with error analysis

  • Y. H. YoussriEmail author
  • R. M. Hafez
Open Access


This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integral equation (V-FIE), using shifted Chebyshev collocation (SCC) method. Some properties of the shifted Chebyshev polynomials are presented. These properties together with the shifted Gauss–Chebyshev nodes were then used to reduce the Volterra–Fredholm integral equation to the solution of a matrix equation. Nextly, the error analysis of the proposed method is presented. We compared the results of this algorithm with others and showed the accuracy and potential applicability of the given method.

Mathematics Subject Classification

65R20 65M70 42C10 



We are indebted to the anonymous reviewers for their instructive comments. Thanks are due to Prof. N. Tatar for helping in a final draft proofreading.


  1. 1.
    Abd-Elhameed, W.M.; Youssri, Y.H.: Numerical solutions for Volterra–Fredholm–Hammerstein integral equations via second kind Chebyshev quadrature collocation algorithm. Adv. Math. Sci. Appl. 24, 129–141 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abd-Elhameed, W.M.; Youssri, Y.H.: Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations. Comp. Appl. Math. 37(3), 2897–2921 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abd-Elhameed, W.M.; Doha, E.H.; Youssri, Y.H.; Bassuony, M.A.: New Tchebyshev-Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations. Numer. Methods Partial Diff. Equ. 32(6), 1553–1571 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brunner, H.: On the numerical solution of Volterra–Fredholm integral equation by collocation methods. SIAM J. Numer. Anal. 27(4), 87–96 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cardone, A.; Messina, E.; Vecchio, A.: An adaptive method for Volterra–Fredholm integral equations on the half line. J. Comput. Appl. Math. 228, 538–547 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Delves, L.M.; Mohamed, J.L.: Computational Methods for Integral Equations. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  7. 7.
    Dickman, O.: Thresholds and traveling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1978)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Doha, E.H.; Youssri, Y.H.: On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain. Nonlinear Eng. (2018).
  9. 9.
    Doha, E.H.; Youssri, Y.H.; Zaky, M.A.: Spectral solutions for differential and integral equations with varying coefficients using classical orthogonal polynomials. Bull. Iran. Math. Soc. (2018).
  10. 10.
    Guoqiang, H.; Liqing, Z.: Asymptotic expansion for the trapezoidal Nystrom method of linear Volterra–Fredholm equation. J. Comput. Appl. Math. 51, 339–348 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hacia, L.: On approximate solution for integral equation of mixed type. ZAMM Z. Angew. Math. Mech. 76, 415–428 (1996)zbMATHGoogle Scholar
  12. 12.
    Hafez, R.M.: Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method. Comp. Appl. Math. 37(4), 5253–5273 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hafez, R.M.; Youssri, Y.H.: Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation. Comp. Appl. Math. 37(4), 5315–5333 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hsiao, Chun-Hui: Hybrid function method for solving Fredholm and Volterra integral equations of the second kind. J. Comput. Appl. Math. 230, 59–68 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Maleknejad, K.; Hadizadeh, M.: A new computational method for Volterra–Fredholm integral equations. Comput. Math. Appl. 37, 1–8 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Maleknejad, K.; Mahmoudi, Y.: Taylor polynomial solution of high-order nonlinear Volterra–Fredholm integro-differential equations. Appl. Math. Comput. 145, 641–653 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Nemati, S.: Numerical solution of Volterra–Fredholm integral equations using Legendre collocation method. J. Comput. Appl. Math. 278, 29–36 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Owolabi, K.M., Patidar, K.C.: Numerical simulations of multicomponent ecological models with adaptive methods, Theor. Biol. Med. Model., 13(1) (2016)
  19. 19.
    Owolabi, K.M.: Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology. Appl. Math. Comput. 240, 30–50 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Owolabi, K.M.: Robust IMEX schemes for solving two-dimensional reaction-diffusion models. Int. J. Nonlinear Sci. Numer. 16, 271–284 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Owolabi, K.M.: Mathematical study of two-variable systems with adaptive Numerical methods. Numer. Anal. Appl. 19, 281–295 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Owolabi, K.M.: Mathematical study of multispecies dynamics modeling predator-prey spatial interactions. J. Numer. Math. 25, 1–16 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Reihani, M.H.; Abadi, Z.: Rationalized Haar functions method for solving Fredholm and Volterra integral equations. J. Comput. Appl. Math. 200, 12–20 (2007)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shali, J.A.; Darania, P.; Akbarfam, A.A.J.: Collocation method for nonlinear Volterra–Fredholm integral equations. Open J. Appl. Sci. 2, 115–121 (2012)CrossRefGoogle Scholar
  25. 25.
    Wang, K.Y.; Wang, Q.S.: Lagrange collocation method for solving Volterra–Fredholm integral equations. Appl. Math. Comput. 219, 10434–10440 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Wang, K.; Wang, Q.: Taylor collocation method and convergence analysis for the Volterra–Fredholm integral equations. J. Comput. Appl. Math. 260, 294–300 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wazwaz, A.M.: A reliable treatment for mixed Volterra–Fredholm integral equations. Appl. Math. Comput. 127, 405–414 (2002)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Yousefi, S.A.; Lotfi, A.: Mehdi Dehghan, He’s variational iteration method for solving nonlinear mixed Volterra–Fredholm integral equations. Comput. Math. Appl. 58, 2172–2176 (2009)Google Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
  2. 2.Department of Mathematics, Alwagjh University CollegeUniversity of TabukTabukSaudi Arabia
  3. 3.Department of Basic ScienceInstitute of Information Technology, Modern AcademyCairoEgypt

Personalised recommendations