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 -