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Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels. (English) Zbl 1120.65135
Summary: 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 J. R. Dormand and P. J. Prince [J. Comput. Appl. Math. 6, 19–26 (1980; Zbl 0448.65045)] 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.

MSC:
65R20 Numerical methods for integral equations
45G10 Other nonlinear integral equations
45J05 Integro-ordinary differential equations
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