Journal of Scientific Computing

, Volume 69, Issue 2, pp 673–695 | Cite as

The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel



A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form \( x^{\beta }\, (z-x)^{-\alpha } \, g(y(x))\), where \(\alpha \in (0,1), \beta >0\) and g(y) is a nonlinear function. Typically, the kernel will contain both an Abel-type and an end point singularity. The solution to these equations will in general have a nonsmooth behaviour which causes a drop in the global convergence orders of numerical methods with uniform meshes. In the considered approach a transformation of the independent variable is first introduced in order to obtain a new equation with a smoother solution. The Jacobi collocation method is then applied to the transformed equation and a complete convergence analysis of the method is carried out for the \(\displaystyle L^{\infty }\) and the \(L^2\) norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.


Jacobi spectral collocation method Nonlinear Volterra integral equation Weakly singular kernel Convergence analysis 

Mathematics Subject Classification

65R20 45J05 



This work was partially supported by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology), through the projects Pest-OE/MAT/UI0822/2014 and PTDC/MAT/101867/2008. The research of the first author (S. Seyed Allaei) was also co-financed by the Hong Kong Research Grants Council (RGC Project HKBU 200113 and 1369648). The work of the third author was also partially supported by the FCT Project UID/MAT/00297/2013 (Centro de Matemática e Aplicações). The first author would like to thank Professor Hermann Brunner for his valuable suggestions and constructive discussions.


  1. 1.
    Seyed Allaei, S., Diogo, T., Rebelo, M.: Analytical and computational methods for a class of nonlinear singular integral equations (Submitted)Google Scholar
  2. 2.
    Baratella, P.: A Nyström interpolant for some weakly singular nonlinear Volterra integral equations. J. Comput. Appl. Math. 237, 542–555 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University press, Cambridge (2004)CrossRefMATHGoogle Scholar
  4. 4.
    Chambré, P.L.: Nonlinear heat transfer problem. J. Appl. Phys. 30, 1683–1688 (1959)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer-Verlag, Berlin (2006)MATHGoogle Scholar
  6. 6.
    Chambré, P.L., Acrivos, A.: Chemical surface reactions in laminar boundary layer flows. J. Appl. Phys. 27, 1322 (1956)CrossRefGoogle Scholar
  7. 7.
    Chen, Y., Li, X., Tang, T.: A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Math. 31, 47–56 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, Y., Tang, T.: Spectral methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diogo, T., Ma, J., Rebelo, M.: Fully discretized collocation methods for nonlinear singular Volterra integral equations. J. Comput. Appl. Math. 247, 84–101 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Diogo, T., McKee, S., Tang, T.: Collocation methods for second-kind Volterra integral equations with weakly singular kernels. Proc. R. Soc. Edinb. 124A, 199–210 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Diogo, T., Lima, P.M., Rebelo, M.S.: Comparative study of numerical methods for a nonlinear weakly singular Volterra integral equation. HERMIS J. 7, 1–20 (2006)MATHGoogle Scholar
  13. 13.
    Elnagar, G.N., Kazemi, M.: Chebyshev spectral solution of nonlinear Volterra–Hammerstein integral equations. J. Comp. Appl. Math. 76, 147–158 (1996)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guo, H., Cai, H., Zhang, X.: A Jacobi-collocation method for second kind Volterra integral equations with a smooth kernel, Abstr. Appl. Anal. 2014, (2014)Google Scholar
  15. 15.
    Li, X., Tang, T.: Convergence analysis of Jacobi spectral Collocation methods for Abel–Volterra integral equations of second-kind. J. Front. Math. China. 7, 69–84 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li, X., Tang, T., Xu, C.: Numerical solutions for weakly singular Volterra integral equations using Chebyshev and Legendre pseudo-spectral Galerkin methods. J. Sci. Comput. doi: 10.1007/s10915-015-0069-5
  17. 17.
    Lighthill, J.M.: Contributions to the theory of the heat transfer through a laminar boundary layer. Proc. R. Soc. Lond. 202(A), 359–377 (1950)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mann, W.R., Wolf, F.: Heat transfer between solids and gases under nonlinear boundary conditions. Quart. Appl. Math. 9, 163–184 (1951)MathSciNetMATHGoogle Scholar
  19. 19.
    Padmavally, K.: On a non-linear integral equation. J. Math. Mech. 7, 533–555 (1958)MathSciNetMATHGoogle Scholar
  20. 20.
    Ragozin, D.L.: Polynomial approximation on compact manifolds ans homogeneous spaces. Trans. Am. Math. Soc. 150, 41–53 (1970)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ragozin, D.L.: Polynomial approximation on spheres manifolds and projective spaces. Trans. Am. Math. Soc. 162, 157–170 (1971)MathSciNetMATHGoogle Scholar
  22. 22.
    Rebelo, S.M., Diogo, T.: A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel. J. Comput. Appl. Math. 234, 2859–2869 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shen, J., Tang, T., Wang, L.: Spectral Methods Algorithms. Analysis and Applications. Springer-Verlag, Berlin (2011)MATHGoogle Scholar
  24. 24.
    Tang, T., Xu, X., Chen, J.: On spectral methods for Volterra type integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)MathSciNetMATHGoogle Scholar
  25. 25.
    Vainikko, G.: Cordial Volterra integral equations 1. Numer. Funct. Anal. Optim. 30, 1145–1172 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Vainikko, G.: Spline collocation-interpolation method for linear and nonlinear cordial Volterra integral equations. Numer. Funct. Anal. Optim. 32, 83–109 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53, 414–434 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Yang, Y., Chen, Y., Huang, Y., Yang, W.: Convergence analysis of Legendre collocation methods for nonlinear Volterra type integro equations. Adv. Appl. Math. Mech. 7, 74–88 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong Baptist UniversityKowloon TongChina
  2. 2.Center for Computational and Stochastic Mathematics (CEMAT), Department of Mathematics, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.Centro de Matemática e Aplicações (CMA) and Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade NOVA de LisboaCaparicaPortugal

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