TY - JOUR

T1 - Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels

AU - Hoppensteadt, F. C.

AU - Jackiewicz, Zdzislaw

AU - Zubik-Kowal, B.

N1 - Funding Information:
★ Received March 20, 2006. Accepted February 4, 2007. Communicated by Timo Eirola. The work of this author was partially supported by NSF grant DMS–0509597.

PY - 2007/6

Y1 - 2007/6

N2 - Variable stepsize algorithms for the numerical solution of nonlinear Volterra integral and integro-differential equations of convolution type are described. These algorithms are based on an embedded pair of Runge-Kutta methods of order p=5 and p=4 proposed by Dormand and Prince with interpolation of uniform order p=4. They require O(N) number of kernel evaluations, where N is the number of steps. The cost of the algorithms can be further reduced for equations that have rapidly vanishing convolution kernels, by using waveform relaxation iterations after computing the numerical approximation by variable stepsize algorithm on some initial interval.

AB - Variable stepsize algorithms for the numerical solution of nonlinear Volterra integral and integro-differential equations of convolution type are described. These algorithms are based on an embedded pair of Runge-Kutta methods of order p=5 and p=4 proposed by Dormand and Prince with interpolation of uniform order p=4. They require O(N) number of kernel evaluations, where N is the number of steps. The cost of the algorithms can be further reduced for equations that have rapidly vanishing convolution kernels, by using waveform relaxation iterations after computing the numerical approximation by variable stepsize algorithm on some initial interval.

KW - Embedded Runge-Kutta methods

KW - Numerical simulation of linear and nonlinear time invariant systems

KW - Volterra integral equation of convolution type

KW - Waveform relaxation iterations

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U2 - 10.1007/s10543-007-0122-3

DO - 10.1007/s10543-007-0122-3

M3 - Article

AN - SCOPUS:34250831524

VL - 47

SP - 325

EP - 350

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 2

ER -