Journal of Scientific Computing

, Volume 75, Issue 2, pp 970–992 | Cite as

A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Article
  • 151 Downloads

Abstract

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n, where \(n+1\) denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order \(\mathcal{O}(n^{-m}\log n)\) in the infinite norm and \(\mathcal{O}(n^{-m})\) in the weighted square norm. In addition, we prove that for sufficiently large n, the infinity-norm condition number of the coefficient matrix corresponding to the linear system is \(\mathcal{O}(\log ^2 n)\) and its spectral condition number is \(\mathcal{O}(1)\). Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Keywords

A fractional order collocations spectral method Second kind Volterra integral equations with weakly singular kernels Stability analysis Convergence analysis Condition number 

Mathematics Subject Classification

45E05 65R20 

Notes

Acknowledgements

This work is supported by National Science Foundation of China ( 11671157, 91430104) and 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)CrossRefMATHGoogle Scholar
  2. 2.
    Ali, I., Brunner, H., Tang, T.: Spectral methods for pantograph-type differential and integral equations with multiple delays. Front. Math. China 4, 49–61 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations Methods. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  4. 4.
    Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20, 1106–1119 (1983)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brunner, H.: Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels. IMA J. Numer. Anal. 6, 221–239 (1986)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cao, Y., Xu, Y.: Singularity preserving Galerkin methods for weakly singular Fredholm integral equations. J. Int. Equ. Appl. 6, 303–334 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, S., Shen, J., Wang, L.: Generalized Jacobi functions and their applications to fractional differrential equations. Math. Comput. 85, 1603–1638 (2016)CrossRefMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Chen, Y., Li, X., Tang, T.: A note on Jacobi-collocation method for weakly singular Volterra integral equations. J Comput. Math. 1, 47–56 (2013)CrossRefMATHGoogle Scholar
  15. 15.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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. 7, 1411–1436 (2017)MathSciNetGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Huang, M., Xu, Y.: Superconvergence of the iterated hybrid collocation method for weakly singular Volterra integral equations. J. Integral Equ. Appl. 18, 83–116 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kress, R.: Linear Integral Equations. Springer, Berlin (2001)MATHGoogle Scholar
  22. 22.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equatio. SIAM J. Numer. Anal. 47, 2108–2131 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lin, T., Lin, Y., Rao, M., Zhang, S.: Petrov–Galerkin methods for linear Volterra integro-differential equations. SIAM J. Numer. Anal. 38, 937–963 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lin, T., Lin, Y., Luo, P., Zhang, S.: Petrov–Galerkin methods for nonlinear Volterra integro-differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. B 8, 405–426 (2009)MathSciNetMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Mikhlin, S., Prossdorf, S.: Singular Integral Operators. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  29. 29.
    Ragozin, D.: Constructive polynomial approximation on spheres and projective spaces. Trans. Am. Math. Soc. 162, 157–170 (1971)MathSciNetMATHGoogle Scholar
  30. 30.
    Ragozin, D.: Polynomial approximation on compact manifolds and homogeneous spaces. Trans. Am. Math. Soc. 150, 41–53 (1979)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Springer, New York (2011)CrossRefGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tang, T.: A note on collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 13, 93–99 (1993)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tang, T., Yuan, W.: The numerical solution of second-order weakly singular Volterra integro-differential equations. J. Comput. Math. 8, 307–320 (1990)MathSciNetMATHGoogle Scholar
  35. 35.
    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
  36. 36.
    Wei, Y., Chen, Y.: Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Adv. Appl. Math. Mech. 121, 1–20 (2012)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Wei, Y., Chen, Y.: Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations. Numer. Math. Theor. Methods Appl. 50, 419–438 (2011)MathSciNetMATHGoogle Scholar
  38. 38.
    Wei, Y., Chen, Y.: Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation. Appl. Numer. Math. 81, 15–29 (2014)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    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
  40. 40.
    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)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zayernouri, M., Karniadakis, G.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, 40–62 (2014)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Zayernouri, M., Karniadakis, G.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximations. J. Comput. Phys. 47, 2108–2131 (2013)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesSouth China Normal UniversityGuangzhouPeople’s Republic of China
  2. 2.School of Mathematics and Quantitative EconomicsShandong University of Finance and EconomicsJinanPeople’s Republic of China
  3. 3.School of Mathematics and Computational ScienceXiangtan UniversityXiangtanPeople’s Republic of China

Personalised recommendations