Advertisement

Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels

  • Junjie Ma
  • Huilan LiuEmail author
Original Paper

Abstract

We discuss the numerical solution to a class of weakly singular Volterra integral equations in this paper. Firstly, the fractional Lagrange interpolation is applied to deal with the singularity of the solution, and efficient fractional collocation boundary value methods are developed. Secondly, local convergence estimates are derived from examining the asymptotic property of the solution and the interpolation remainder. We find that the second kind Volterra integral equation with a weakly singular kernel can be efficiently solved on a uniform grid. Finally, several numerical examples are given to illustrate the performance of fractional collocation boundary value methods.

Keywords

Collocation Boundary value method Numerical integration Volterra integral equation Weakly singular 

Notes

Funding information

This work is supported by NSF of China (No. 11761020), Scientific Research Foundation for Young Talents of Department of Education of Guizhou Province (No. 2016125), Major Scientific and Technological Special Project of Guizhou Province (No. 20183001), and Science and Technology Foundation of Guizhou Province (No. QKH[2017]5788).

References

  1. 1.
    Roberts, C.A., Lasseigne, D.G., Olmstead, W.E.: Volterra equations which model explosion in a diffusive medium. J. Integr. Eq. Appl. 5(4), 531–546 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Olmstead, W.E., Roberts, C.A.: Explosion in a diffusive strip due to a source with local and nonlocal features. Methods Appl. Anal. 3(3), 345–357 (1996)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press, Cambridge (2017)zbMATHGoogle Scholar
  4. 4.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Baumann, G., Stenger, F.: Fractional calculus and Sinc methods. Fract. Calc. Appl. Anal. 14(4), 568–622 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Diethelm, K., Ford, N.J.: Volterra integral equations and fractional calculus: do neighboring solutions intersect? J. Integr. Eq. Appl. 24(1), 25–37 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Esmaeili, S., Shamsi, M., Dehghan, M.: Numerical solution of fractional differential equations via a Volterra integral equation approach. Central Eur. J. Phys. 11(10), 1470–1481 (2013)Google Scholar
  8. 8.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  9. 9.
    Diogo, T.: Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations. J. Comput. Appl. Math. 229(2), 363–372 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Zhao, J., Xiao, J., Ford, N.J.: Collocation methods for fractional integro-differential equations with weakly singular kernels. Numer. Algorithm. 65(4), 723–743 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)zbMATHGoogle Scholar
  12. 12.
    Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53(2), 414–434 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Huang, C., Stynes, M.: A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv. Comput. Math. 42(5), 1015–1030 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Brunner, H., Linz, P.: Analytical and numerical methods for Volterra equations. Math. Comput. 48(178), 841 (1987)Google Scholar
  15. 15.
    Berrut, J.P., Hosseini, S.A., Klein, G.: The linear barycentric rational quadrature method for Volterra integral equations. SIAM J. Sci. Comput. 36(1), A105–A123 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Li, M., Huang, C.: The linear barycentric rational quadrature method for auto-convolution Volterra integral equations. J. Sci. Comput. 78(1), 549–564 (2019)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lubich, C. h.: Runge-Kutta theory for Volterra and Abel integral equations of the second kind. Math. Comput. 41(163), 87–102 (1983)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Crisci, M.R., Jackiewicz, Z., Russo, E., Vecchio, A.: Global stability analysis of the Runge-Kutta methods for Volterra integral and integro-differential equations with degenerate kernels. Computing 45(4), 291–300 (1991)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Garrappa, R.: Order conditions for Volterra RungeC̈Kutta methods. Appl. Numer. Math. 60(5), 561–573 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    van der Houwen, P.J., Riele, H.J.J.: Linear multistep methods for Volterra integral equations of the second kind. Queueing Systems (1982)Google Scholar
  21. 21.
    Lubich, C. h.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 45(172), 463–469 (1985)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Garrappa, R.: On some explicit Adams multistep methods for fractional differential equations. J. Comput. Appl. Math. 229(2), 392–399 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 45(172), 417–437 (1985)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Diogo, T., McKee, S., Tang, T.: Collocation methods for second-kind Volterra integral equations with weakly singular kernels. Proc. R. Soc. Edinb. 124, 199–210 (1994)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20(6), 1106–1119 (1983)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41(1), 364–381 (2003)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Rebelo, M., Diogo, T.: A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel. J. Comput. Appl. Math. 234(9), 2859–2869 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ford, N.J., Morgado, M.L., Rebelo, M.: Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract. Calc. Appl. Anal. 16(4), 874–891 (2013)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ford, N.J., Morgado, M.L., Rebelo, M.: A nonpolynomial collocation method for fractional terminal value problems. J. Comput. Appl. Math. 275(1), 392–402 (2015)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Pedas, A., Vainikko, G.: Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integral equations. Computing 73(3), 271–293 (2004)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Diogo, T., Lima, P.M., Pedas, A., Vainikko, G.: Smoothing transformation and spline collocation for weakly singular Volterra integro-differential equations. Appl. Numer. Math. 114, 63–76 (2017)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Hou, D., Xu, C.: A fractional spectral method with applications to some singular problems. Adv. Comput. Math. 43(5), 911–944 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Pedas, A., Tamme, E., Vikerpuur, M.: Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems. J. Comput. Appl. Math. 317, 1–16 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Cai, H., Chen, Y.: A fractional order collocation method for second kind Volterra integral equations with weakly singular kernels. J. Sci. Comput. 75(2), 970–992 (2018)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Brugnano, L., Trigiante, D.: Solving differential problems by multistep initial and boundary value methods. Cordon and Breach Science Publishers (1998)Google Scholar
  36. 36.
    Lopez, L., Trigiante, D.: Boundary value methods and BV-stability in the solution of initial value problems. Appl. Numer. Math. 11(1), 225–239 (1993)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218(6), 2619–2630 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Ma, J., Xiang, S.: A collocation boundary value method for linear Volterra integral equations. J. Sci. Comput. 71(1), 1–20 (2017)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40–A62 (2014)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Olver, F., Lozier, D., Boisvert, R., Clark, C.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Guizhou Provincial Key Laboratory of Public Big DataGuizhou UniversityGuiyangPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsGuizhou UniversityGuiyangPeople’s Republic of China

Personalised recommendations