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Fast Runge-Kutta methods for nonlinear convolution systems of Volterra integral equations. (English) Zbl 1116.65128
Summary: 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. I suggest to compare this technique with Adomian decomposition method very suitable for integral equations.

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
44A35 Convolution as an integral transform
44A10 Laplace transform
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[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. · Zbl 0686.65095 · doi:10.1093/imanum/10.1.103
[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. · Zbl 0629.65144 · doi:10.1137/0908068
[3] H. Brunner and E. Messina, Time-stepping methods for Volterra–Fredholm integral equations, Rend. Mat. Appl., VII. Ser., 23 (2003), pp. 329–342. · Zbl 1095.65117
[4] H. Brunner and P. J. van der Houwen, The numerical solution of Volterra equations, CWI Monographs, vol. 3, North-Holland, Amsterdam, 1986. · Zbl 0611.65092
[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. · Zbl 1092.65118 · doi:10.1016/j.cam.2005.05.018
[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. · Zbl 1104.65122 · doi:10.1016/j.cam.2005.10.018
[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. · Zbl 0675.65140 · doi:10.1007/BF01952681
[8] F. de Hoog, R. Weiss, Implicit Runge–Kutta methods for second kind Volterra integral equations, Numer. Math., 23 (1975), pp. 199–213. · Zbl 0313.65117
[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. · Zbl 0575.65144 · doi:10.1080/00949658408810751
[10] D. Givoli, Numerical methods for problems in infinite domains, Elsevier Science Publishers, Amsterdam, 1992. · Zbl 0788.76001
[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. · Zbl 0581.65095 · doi:10.1137/0906037
[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. · Zbl 1078.65126 · doi:10.1016/j.apnum.2004.10.010
[13] R. Hiptmair and A. Schädle, Non-reflecting boundary conditions for Maxwell’s equations, Computing, 71(3) (2003), pp. 265–292. · Zbl 1042.78010 · doi:10.1007/s00607-003-0026-2
[14] C. Lubich and A. Schädle, Fast convolution for non-reflecting boundary conditions, Siam. J. Sci. Comput., 24 (2002), pp. 161–182. · Zbl 1013.65113 · doi:10.1137/S1064827501388741
[15] C. Lubich, Convolution quadrature and discretized operational calculus II, Numer. Math., 52 (1988), pp. 413–425. · Zbl 0643.65094 · doi:10.1007/BF01462237
[16] R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin, Menlo Park, CA, 1971.
[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. · Zbl 0887.65133 · doi:10.1145/212066.212068
[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. · Zbl 0972.65119
[19] A. Talbot, The accurate numerical inversion of Laplace Transforms, J. Inst. Math. Appl., 23 (1979), pp. 97–120. · Zbl 0406.65054 · doi:10.1093/imamat/23.1.97
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