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Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations

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Abstract

In this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra integral equations of Hammerstein type are constructed. The coefficients of the methods are expressed in terms of the values of the Laplace transform of the kernel. These methods have been suitably constructed in order to be implemented in an efficient way, thus leading to a very low computational cost both in time and in space. The order of convergence of the constructed methods is studied. The numerical experiments confirm the expected accuracy and computational cost.

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References

  1. A. Bellen, Z. Jackiewicz, R. Vermiglio, and M. Zennaro, Stability analysis of Runge–Kutta methods for Volterra integral equations of second kind, IMA J. Numer. Anal., 10 (1990), pp. 103–118.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. G. Blom and H. Brunner, The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 806–830.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Brunner and E. Messina, Time-stepping methods for Volterra–Fredholm integral equations, Rend. Mat. Appl., VII. Ser., 23 (2003), pp. 329–342.

    MATH  MathSciNet  Google Scholar 

  4. H. Brunner and P. J. van der Houwen, The numerical solution of Volterra equations, CWI Monographs, vol. 3, North-Holland, Amsterdam, 1986.

    MATH  Google Scholar 

  5. A. Cardone, E. Messina, and E. Russo, A fast iterative method for Volterra–Fredholm integral equations, J. Comput. Appl. Math., 189(1–2) (2006), pp. 568–579.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Conte, I. Del Prete, Fast collocation methods for Volterra integral equations of convolution type, J. Comput. Appl. Math., 196(2) (2006), pp. 652–663.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. R. Crisci, E. Russo, and A. Vecchio, On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation, BIT, 29 (1989), pp. 258–269.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. de Hoog, R. Weiss, Implicit Runge–Kutta methods for second kind Volterra integral equations, Numer. Math., 23 (1975), pp. 199–213.

    Article  MATH  Google Scholar 

  9. Z. S. Deligonoul, S. Bilgen, Solution of the Volterra equation of renewal theory with the Galerkin technique using cubic spplines, J. Stat. Comput. Simulation, 20 (1984), pp. 37–45.

    Google Scholar 

  10. D. Givoli, Numerical methods for problems in infinite domains, Elsevier Science Publishers, Amsterdam, 1992.

    MATH  Google Scholar 

  11. E. Hairer, C. Lubich, and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 532–541.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Han, L. Zhu, H. Brunner, and J. Ma, The numerical solution of parabolic Volterra integro-differential equations on unbounded spatial domains, Appl. Numer. Math., 55 (2005), pp. 83–99.

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Hiptmair and A. Schädle, Non-reflecting boundary conditions for Maxwell’s equations, Computing, 71(3) (2003), pp. 265–292.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Lubich and A. Schädle, Fast convolution for non-reflecting boundary conditions, Siam. J. Sci. Comput., 24 (2002), pp. 161–182.

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Lubich, Convolution quadrature and discretized operational calculus II, Numer. Math., 52 (1988), pp. 413–425.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin, Menlo Park, CA, 1971.

    MATH  Google Scholar 

  17. M. Rizzardi, A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace Transform, ACM Trans. Math. Softw., 21(4) (1995), pp. 347–371.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. W. Sharp and J. H. Verner: Some extended explicit Bel’tyukov pairs for Volterra integral equations of the second kind, SIAM J. Numer. Anal., 38(2) (2000), pp. 347–359.

    Google Scholar 

  19. A. Talbot, The accurate numerical inversion of Laplace Transforms, J. Inst. Math. Appl., 23 (1979), pp. 97–120.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento S.T.A.T., Università degli Studi del Molise, Contrada Fonte Lappone, I-86090, Pesche (IS), Italy

    G. Capobianco

  2. Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte Don Melillo, I-84084, Fisciano (SA), Italy

    D. Conte

  3. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126, Napoli, Italy

    I. Del Prete & E. Russo

Authors
  1. G. Capobianco
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  2. D. Conte
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  3. I. Del Prete
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  4. E. Russo
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Corresponding author

Correspondence to I. Del Prete.

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AMS subject classification (2000)

65R20, 45D05, 44A35, 44A10

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Capobianco, G., Conte, D., Del Prete, I. et al. Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations . Bit Numer Math 47, 259–275 (2007). https://doi.org/10.1007/s10543-007-0120-5

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  • DOI: https://doi.org/10.1007/s10543-007-0120-5

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