Abstract
Volterra integral equation of the second kind with weakly singular kernel usually exhibits singular behavior at the origin, which deteriorates the accuracy of standard numerical methods. This paper develops a singularity separation Chebyshev collocation method to solve this kind of Volterra integral equation by splitting the interval into a singular subinterval and a regular one. In the singular subinterval, the general psi-series expansion for the solution about the origin or its Padé approximation is used to approximate the solution. In the regular subinterval, the Chebyshev collocation method is used to discretize the equation. The details of the implementation are also discussed. Specifically, a stable and fast recurrence procedure is derived to evaluate the singular weight integrals involving Chebyshev polynomials analytically. The convergence of the method is proved. We further extend the method to the nonlinear Volterra integral equation by using the Newton method. Three numerical examples are provided to show that the singularity separation Chebyshev collocation method in this paper can effectively solve linear and nonlinear weakly singular Volterra integral equations with high precision.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11075-023-01629-3/MediaObjects/11075_2023_1629_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11075-023-01629-3/MediaObjects/11075_2023_1629_Fig2_HTML.png)
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Gorenflo, R., Vessella, S.: Abel Integral Equations. Springer, Berlin (1991)
Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press, Cambridge (2017)
Lighthill, M.J.: Contributions to the theory of heat transfer through a laminar boundary layer. Proc. R. Soc. Lond. A 202, 359–377 (1950)
Diogo, T., Ma, J.T., Rebelo, M.: Fully discretized collocation methods for nonlinear singular Volterra integral equations. J. Comput. Appl. Math. 247,84–101 (2013). https://doi.org/10.1016/j.cam.2013.01.002
Huang, C., Wang, L.L.: An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media. Adv. Comput. Math. 45,707–734 (2019). https://doi.org/10.1007/s10444-018-9636-2
Levinson, N.: A nonlinear Volterra equation arising in the theory of superfluidity. J. Math. Anal. Appl. 1, 1–11 (1960)
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)
Cao, Y.Z., Herdman, T., Xu, Y.S.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41, 364–381 (2004). https://doi.org/10.1137/S0036142901385593
Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comp. 45, 417–437 (1985). https://doi.org/10.1090/S0025-5718-1985-0804933-3
Khater, A.H., Shamardan, A.B., Callebaut, D.K., Sakran, M.R.A.: Solving integral equations with logarithmic kernels by Chebyshev polynomials. Numer. Algorithms 47, 81–93 (2008). https://doi.org/10.1007/s11075-007-9148-5
Liang, H., Brunner, H.: The convergence of collocation solutions in continuous piecewise polynomial spaces for weakly singular Volterra integral equations. SIAM J. Numer. Anal. 57, 1875–1896 (2019). https://doi.org/10.1137/19M1245062
Kant, K., Nelakanti, G.: Approximation methods for second kind weakly singular Volterra integral equations. J. Comput. Appl. Math. 368, 112531 (2020). https://doi.org/10.1016/j.cam.2019.112531
Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20, 1106–1119 (1983). https://doi.org/10.1137/0720080
Rebelo, 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). https://doi.org/10.1016/j.cam.2010.01.034
Ma, J.J., Liu, H.L.: Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels. Numer. Algorithms 84, 743–760 (2020). https://doi.org/10.1007/s11075-019-00777-9
Huang, Q.M., Wang, M.: Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind. Comput. Appl. Math. 40, 71 (2021). https://doi.org/10.1007/s40314-021-01435-4
Eshaghi, J., Adibi, H., Kazem, S.: Solution of nonlinear weakly singular Volterra integral equations using the fractional-order Legendre functions and pseudospectral method. Math. Meth. Appl. Sci. 39, 3411–3425 (2016). https://doi.org/10.1002/mma.3788
Cai, H.T., Chen, Y.P.: A fractional order collocation method for second kind Volterra integral equations with weakly singular kernels. J. Sci. Comput. 75, 970–992 (2018). https://doi.org/10.1007/s10915-017-0568-7
Cai, H.T.: A fractional spectral collocation for solving second kind nonlinear Volterra integral equations with weakly singular kernels. J. Sci. Comput. 80, 1529–1548 (2019). https://doi.org/10.1007/s10915-019-00987-2
Hou, D.M., Lin, Y.M., Azaiez, M., Xu, C.J.: A Müntz-collocation spectral method for weakly singular Volterra integral equations. J. Sci. Comput. 81, 2162–2187 (2019). https://doi.org/10.1007/s10915-019-01078-y
Talaei, Y.: Chelyshkov collocation approach for solving linear weakly singular Volterra integral equations. J. Appl. Math. Comput. 60, 201–222 (2019). https://doi.org/10.1007/s12190-018-1209-5
Pedas, A., Vainikko, G.: Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integral equations. Computing 73, 271–293 (2004). https://doi.org/10.1007/s00607-004-0088-9
Zhao, J.J., Long, T., Xu, Y.: Super implicit multistep collocation methods for weakly singular Volterra integral equations. Numer. Math. Theor. Meth. Appl. 12, 1039–1065 (2019). https://doi.org/10.4208/nmtma.OA-2018-0084
Chen, Y.P., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comp. 79, 147–167 (2010). https://doi.org/10.1090/S0025-5718-09-02269-8
Liu, X., Chen, Y.P.: Convergence analysis for the Chebyshev collocation methods to Volterra integral equations with a weakly singular kernel. Adv. Appl. Math. Mech. 9, 1506–1524 (2017). https://doi.org/10.4208/aamm.OA-2016-0049
Allaei, S.S., Diogo, T., Rebelo, M.: The Jacobi collocation method for a class of nonlinear Volterra integral equations with weakly singular kernel. J. Sci. Comput. 69, 673–695 (2016). https://doi.org/10.1007/s10915-016-0213-x
Sohrabi, S., Ranjbar, H., Saei, M.: Convergence analysis of the Jacobi-collocation method for nonlinear weakly singular Volterra integral equations. Appl. Math. Comput. 299, 141–152 (2017). https://doi.org/10.1016/j.amc.2016.11.022
Li, X.J., Tang, T., Xu, C.J.: Numerical solutions for weakly singular Volterra integral equations using Chebyshev and Legendre pseudo-spectral Galerkin methods. J. Sci. Comput. 67, 43–64 (2016). https://doi.org/10.1007/s10915-015-0069-5
Baratella, P., Orsi, A.P.: A new approach to the numerical solution of weakly singular Volterra integral equations. J. Comput. Appl. Math. 163, 401–418 (2004). https://doi.org/10.1016/j.cam.2003.08.047
Baratella, P.: A Nyström interpolant for some weakly singular nonlinear Volterra integral equations. J. Comput. Appl. Math. 237, 542–555 (2013). https://doi.org/10.1016/j.cam.2012.06.024
Lü, T., Huang, Y.: Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind. J. Math. Anal. Appl. 324, 225–237 (2006). https://doi.org/10.1016/j.jmaa.2005.12.013
Wang, T.K., Qin, M., Zhang, Z.Y.: The Puiseux expansion and numerical integration to nonlinear weakly singular Volterra integral equations of the second kind. J. Sci. Comput. 82, 64 (2020). https://doi.org/10.1007/s10915-020-01167-3
Wang, T.K., Gu, Y.S., Zhang, Z.Y.: An algorithm for the inversion of Laplace transforms using Puiseux expansions. Numer. Algorithms 78, 107–132 (2018). https://doi.org/10.1007/s11075-017-0369-y
Wang, T.K., Qin, M., Lian, H.: The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform. Numer. Algorithms 85, 683–711 (2020). https://doi.org/10.1007/s11075-019-00832-5
Hemmi, M.A., Melkonian, S.: Convergence of psi-series solutions of nonlinear ordinary differential equations. Canad. Appl. Math. Q. 3, 43–88 (1995)
Wang, T.K., Liu, Z.F., Kong, Y.T.: The series expansion and Chebyshev collocation method for nonlinear singular two-point boundary value problems. J. Eng. Math. 126, 5 (2021). https://doi.org/10.1007/s10665-020-10077-0
Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)
Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms. Analysis and Applications. Springer, New York (2011)
Orsi, A.P.: Product integration for Volterra integral equations of the second kind with weakly singular kernels. Math. Comp. 65, 1201–1212 (1996). https://doi.org/10.1090/S0025-5718-96-00736-3
Allaei, S.S., Diogo, T., Rebelo, M.: Analytical and computational methods for a class of nonlinear singular integral equations. Appl. Numer. Math. 114, 2–17 (2017). https://doi.org/10.1016/j.apnum.2016.06.001
Huang, C., Stynes, M.: A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv. Comput. Math. 42, 1015–1030 (2016). https://doi.org/10.1007/s10444-016-9451-6
Wang, T.K., Liu, Z.F., Zhang, Z.Y.: The modified composite Gauss type rules for singular integrals using Puiseux expansions. Math. Comp. 86, 345–373 (2017). https://doi.org/10.1090/mcom/3105
Wang, T.K., Zhang, Z.Y., Liu, Z.F.: The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions. Adv. Comput. Math. 43, 319–350 (2017). https://doi.org/10.1007/s10444-016-9487-7
Brezinski, C., Van Iseghem, J.: A taste of Padé approximation. Acta Numerica 4, 53–103 (1995). https://doi.org/10.1017/S096249290000252X
Mason, J.C., Handscomb, D.: Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton (2003)
Li, C.P., Zeng, F.R., Liu, F.W.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, 383–406 (2012). https://doi.org/10.2478/s13540-012-0028-x
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)
Atkinson, K., Han, W.M.: Theoretical Numerical Analysis: A Functional Analysis Framework. Springer, New York (2009)
Polyanin, A.D., Manzhirov, A.V.: Handbook of Integral Equations, 2nd edn. Chapman & Hall/CRC, London (2008)
Lubich, C.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comp. 45, 463–469 (1985). https://doi.org/10.1090/S0025-5718-1985-0804935-7
Acknowledgements
The authors are very grateful to Editors and Referees for the valuable comments, which improve the quality of the paper significantly.
Funding
This work was supported by the National Natural Science Foundation of China under Grant No. 11971241 and the Program for Innovative Research Team in Universities of Tianjin under Grant No. TD13-5078.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. The manuscript was written by Tongke Wang. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Ethical approval and consent to participate
Not applicable.
Consent for publication
All of the material is owned by the authors, and no permissions are required.
Human and animal ethics
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, T., Lian, H. & Ji, L. Singularity separation Chebyshev collocation method for weakly singular Volterra integral equations of the second kind. Numer Algor 95, 1829–1854 (2024). https://doi.org/10.1007/s11075-023-01629-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-023-01629-3
Keywords
- Volterra integral equation of the second kind
- Kernel with algebraic and logarithmic singularity
- Singularity separation Chebyshev collocation method
- Convergence analysis