Advertisement

Spectral Collocation Methods for Nonlinear Volterra Integro-Differential Equations with Weakly Singular Kernels

  • Yin Yang
  • Yanping Chen
Article

Abstract

A spectral Jacobi-collocation approximation is proposed and analyzed for nonlinear integro-differential equations of Volterra type with weakly singular kernel, and a rigorous error analysis is provided for the spectral methods to show both the errors of approximate solutions and the errors of approximate derivatives of the solutions decaying exponentially in infinity-norm and weighted \(L^2\)-norm. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.

Keywords

Spectral collocation method Nonlinear Volterra integro-differential equations 

Notes

Acknowledgements

The work was supported by NSFC Project (11671342, 91430213, 11671157), and Hunan Province Natural Science Fund (2016JJ3114).

References

  1. 1.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brunner, H., Pedas, A., Vainikko, G.: Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 39(3), 957–982 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gu, Z., Chen, Y.: Piecewise Legendre spectral-collocation method for Volterra integro-differential equations. LMS J. Comput. Math. 18(1), 231–249 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Tang, T.: Superconvergence of numerical solutions to weakly singular Volterra integrodifferential equations. Numer. Math. 61(1), 373–382 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Tarang, M.: Stability of the spline collocation method for second order Volterra integrodifferential equations. Math. Model. Anal. 9(1), 79–90 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, Y., Gu, Z.: Legendre spectral-collocation method for Volterra integral differential equations with non-vanishing delay. Commun. Appl. Math. Comput. Sci. 8(1), 67–98 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gu, Z., Chen, Y.: Legendre spectral-collocation method for Volterra integral equations with non-vanishing delay. Calcolo 51(1), 151–174 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4(1), 181–193 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yang, Y.: Jacobi spectral Galerkin methods for fractional integro-differential equations. Calcolo 52(4), 519–542 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yang, Y.: Jacobi spectral Galerkin methods for Volterra integral equations with weakly singular kernel. Bull. Korean Math. Soc. 53(1), 247–262 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel. Math. Comput. 79(269), 147–167 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yang, Y., Chen, Y., Huang, Y., Yang, W.: Convergence analysis of Legendre-collocation methods for nonlinear Volterra type integral Equations. Adv. Appl. Math. Mech. 7(1), 74–88 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yang, Y., Chen, Y., Huang, Y.: Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations. Acta Math. Sci. 34B(3), 673–690 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yang, Y., Chen, Y., Huang, Y.: Spectral-collocation method for fractional Fredholm integro-differential equations. J. Korean Math. Soc. 51(1), 203–224 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yang, Y., Chen, Y., Huang, Y., Wei, H.: Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis. Comput. Math. Appl. 73, 1218–1232 (2017)Google Scholar
  16. 16.
    Bhrawy, A., Alghamdi, M.A.: A shifted Jacobi–Gauss–Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals. Bound. Value Probl. 1(62), 1–13 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer, Berlin (2006)zbMATHGoogle Scholar
  18. 18.
    Guo, B., Wang, L.: Jacobi interpolation approximations and their applications to singular differential equations. Adv. Comput. Math. 14, 227–276 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Samko, S.G., Cardoso, R.P.: Sonine integral equations of the first kind in Lp(0, b). Fract. Calc. Appl. Anal. 6(3), 235–258 (2003)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Mastroianni, G., Occorsto, D.: Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey. J. Comput. Appl. Math. 134(1–2), 325–341 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1989)Google Scholar
  22. 22.
    Ragozin, D.L.: Polynomial approximation on compact manifolds and homogeneous spaces. Trans. Am. Math. Soc. 150, 41–53 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ragozin, D.L.: Constructive polynomial approximation on spheres and projective spaces. Trans. Am. Math. Soc. 162, 157–170 (1971)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Colton, D., Kress, R.: Inverse Coustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 2nd edn. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  25. 25.
    Nevai, P.: Mean convergence of Lagrange interpolation: III. Trans. Am. Math. Soc. 282(2), 669–698 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kufner, A., Persson, L.E.: Weighted Inequalities of Hardy Type. World Scientific, New York (2003)CrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceXiangtan UniversityXiangtanPeople’s Republic of China
  2. 2.School of Mathematical SciencesSouth China Normal UniversityGuangzhouPeople’s Republic of China

Personalised recommendations