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Fast collocation methods for Volterra integral equations of convolution type. (English) Zbl 1104.65122
The authors study fast collocation methods for Volterra integral equations of convolution type. They construct fast discrete collocation methods which involve evaluations of the Laplace transform of the kernel. The exact collocation methods are generalization of those by C. Lubich and A. Schädle [SIAM J. Sci. Comput. 24, No. 1, 161–182 (2002; Zbl 1013.65113)]. Numerical results are presented. The methods presented in this paper are highly parallelizable.

MSC:
65R20 Numerical methods for integral equations
65Y05 Parallel numerical computation
45G10 Other nonlinear integral equations
Software:
Algorithm 689
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References:
[1] Atkinson, K.E., Introduction to numerical analysis, (1989), Wiley New York · Zbl 0718.65001
[2] Blom, J.G.; Brunner, H., The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods, SIAM J. sci. statist. comput., 8, 5, 806-830, (1987) · Zbl 0629.65144
[3] Blom, J.G.; Brunner, H., Algorithm 689, discretized collocation and iterated collocation for nonlinear Volterra integral equations of the second kind, ACM trans. math. software, 17, 2, 167-177, (1991) · Zbl 0900.65382
[4] Brunner, H., Collocation methods for Volterra integral and related functional equations, (2004), Cambridge University Press Cambridge · Zbl 1059.65122
[5] H. Brunner, P.J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3, CWI Monographs, North-Holland, Amsterdam, 1986. · Zbl 0611.65092
[6] Crisci, M.R.; Kolmanovskii, Vb.; Russo, E.; Vecchio, A., Stability of discrete Volterra equations of Hammerstein type, J. differential equations appl., 6, 2, 127-145, (2000) · Zbl 0951.65148
[7] Hairer, E.; Lubich, Ch.; Schlichte, M., Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. sci. statist. comput., 6, 532-541, (1985) · Zbl 0581.65095
[8] Hiptmair, R.; Schädle, A., Non-reflecting boundary conditions for Maxwell’s equations, Computing, 71, 3, 265-292, (2003) · Zbl 1042.78010
[9] Lubich, Ch., Convolution quadrature and discretized operational calculus II, Numer. math., 52, 413-425, (1988) · Zbl 0643.65094
[10] Lubich, C.; Schädle, A., Fast convolution for non-reflecting boundary conditions, SIAM J. sci. comput., 24, 161-182, (2002) · Zbl 1013.65113
[11] Rizzardi, M., A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace transform, ACM trans. math. software, 21, 4, 347-371, (1995) · Zbl 0887.65133
[12] Talbot, A., The accurate numerical inversion of Laplace transforms, J. inst. math. appl., 23, 97-120, (1979) · Zbl 0406.65054
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