Advertisement

A Fractional Spectral Collocation for Solving Second Kind Nonlinear Volterra Integral Equations with Weakly Singular Kernels

  • Haotao CaiEmail author
Article

Abstract

The classical integer-order Jacobi spectral methods for solving second kind nonlinear Volterra integral equations with weakly singular kernels may cause a low-order accuracy in numerically approximating the exact solution. To overcome the shortcomings, we in this paper present a fractional spectral collocation method for solving weakly singular nonlinear Volterra integral equations. Based on the behavior of the original solution near the initial point of integration, we construct the fractional interpolation basis in the collocation method, and then develop an easily implementing technique to approximate the entry with one-fold integral in the resulting nonlinear system produced by the fractional spectral method. Consequently, we establish that both the semi-discrete and the fully discrete nonlinear systems have a unique solution for sufficiently large n, respectively, where \(n+1\) denotes the dimension of the approximate space. We also ensure that two approximate solutions produced by both the semi-discrete and the fully discrete method arrive at the quasi-optimal convergence order in the infinite norm. At last, numerical examples are given to confirm the theoretical results.

Keywords

Nonlinear Volterra integral equations Fractional spectral collocations method Convergence analysis 

Mathematical Subject Classification

45E05 65R20 

Notes

Acknowledgements

This work is supported by National Science Foundation of Shandong Province (ZR2014JL003). The authors thank the referees for very helpful suggestions, which help us improve this paper.

References

  1. 1.
    Atkinson, K.E.: The Numerical Solution of Integral Equations of Second Kind. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brunne, H., Pedas, A., Vainikko, G.: The piecewise polynomial collocation method for weakly singular Volterra integral equations. Math. Comput. 68, 1079–1095 (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20, 1106–1119 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brunner, H.: Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 6, 221–239 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations Methods. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  6. 6.
    Cai, H.: A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels. Sci. China Math. 57, 2163–2178 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cai, H., Chen, Y.: A fractional order collocation method for second kind Volterra integral equations with weakly singular kernels. J. Sci. Comput. 75, 970–992 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cai, H., Qi, J.: A Legendre–Galerkin method for solving general Volterra functional integral equations. Numer. Algorithm 73, 1159–1180 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cao, Y., Xu, Y.: Singularity preserving Galerkin methods for weakly singular Fredholm integral equations. J. Int. Equ. Appl. 6, 303–334 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41, 364–381 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cao, Y., Huang, M., Liu, L., Xu, Y.: Hybrid collocation methods for Fredholm integral equations with weakly singular kernels. Appl. Numer. Math. 57, 549–561 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations. Math. Comput. 79, 147–167 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, J., Chen, Z., Zhang, Y.: Fast singularity preserving methods for integral equations with non-smooth solutions. J. Int. Equ. Appl. 24, 213–240 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, S., Shen, J., Wang, L.: Generalized Jacobi functions and their applications to fractional differrential equations. Math. Comput. 85, 1603–1638 (2016)CrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, S., Shen, J., Mao, Z.: Efficient and accurate spectral methods using general Jacobi Functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165–181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gautschi, W.: Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38, 77–81 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Huang, C., Stynesz, M.: A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv. Comput. 42, 1015–1030 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Huang, C., Stynesz, M.: Spectral Galerkin methods for a weakly singular Volterra integral equation of the second kind. IMA. J. Numer. Anal. 37, 1411–1436 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Huang, C., Tang, T., Zhang, Z.: Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions. J. Comput. Math. 29, 698–719 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Huang, C., Jiao, Y., Wang, L., Zhang, Z.: Optimal fractional integration preconditioning and error analysis of fractional collocation method using nodal generalized Jacobi functions. SIAM J. Numer. Anal. 54, 3357–3387 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kress, R.: Linear Integral Equations. Springer, Berlin (2001)zbMATHGoogle Scholar
  22. 22.
    Li, X., Tang, T.: Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind. Front. Math. China. 7, 69–84 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Li, X., Tang, T., Xu, C.: Parallel in time algorithm with spectral-subdomain enhancement for volterra integral equations. SIAM J. Numer. Anal. 51, 1735–1756 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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. 67, 43–64 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Liang, H., Stynes, M.: Collocation methods for general caputo two-point boundary value problems. J. Sci. Comput. 76, 390–425 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Monegato, G., Scuderi, L.: High order methods for weakly singular integral equations with nonsmooth input functions. Math. Comput. 224, 1493–1515 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ragozin, D.: Constructive polynomial approximation on spheres and projective spaces. Trans. Am. Math. Soc. 162, 157–170 (1971)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ragozin, D.: Polynomial approximation on compact manifolds and homogeneous spaces. Trans. Am. Math. Soc. 150, 41–53 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  30. 30.
    Shen, J., Sheng, C.T., Wang, Z.Q.: Generalized Jacobi spectral- Galerkin method for nonlinear Volterra integral equations with weakly singular kernels. J. Math. Study 48(4), 315–329 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sheng, C., Wang, Z., Guo, B.: Multistep Legendre–Gauss spectral collocation method for nonlinear Volterra integra equations. SIAM J. Numer. Anal. 52, 1953–1980 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Shi, X., Wei, Y., Huang, F.: Spectral collocation methods for nonlinear weakly singular Volterra integro differential equations. Numer. Methods Differ. Equ. 63, 576–596 (2018)Google Scholar
  33. 33.
    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)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Wei, Y., Chen, Y.: Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations. Numer. Math. Theory. Methods Appl. 50, 419–438 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Wei, Y., Chen, Y.: Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation. Appl. Numer. Math. 81, 15–29 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53, 414–434 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yang, Y., Chen, Y.: Spectral collocation methods for nonlinear Volterra integro-differential equations with weakly singular kernels. Bull. Malays. Math. Sci. Soc. 3, 1–18 (2017)Google Scholar
  38. 38.
    Yi, L., Guo, B.: An h-p Version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM J. Numer. Anal. 53, 2677–2704 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zayernouri, M., Karniadakis, G.: Fractional Sturm-Liouville eigen-problems: theory and numerical approximations. J. Comput. Phys. 47, 2108–2131 (2013)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Zayernouri, M., Karniadakis, G.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, 40–62 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Quantitative EconomicsShandong University of Finance and EconomicsJinanPeople’s Republic of China

Personalised recommendations