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Convolution quadrature and discretized operational calculus. II. (English) Zbl 0643.65094
In the first part [ibid. 52, 129-145 (1988; Zbl 0637.65016)] the author derived ‘operational’ quadrature rules for the approximate evaluation of the convolution integral $$\int^{x}_{0}f(x-t)g(t)dt$$ by $$\sum^{n}_{j=0}\omega_{n-j}(h)g(jh),$$ $$n=0,1,...,N$$. The weights $$\{\omega_ n\}$$ are given by $$\sum^{n}_{j=0}\omega_ j(h)\zeta^ j=F(\delta (\zeta)/h),$$ F is the Laplace transform of f and $$\delta =\rho (\zeta)/\sigma (\zeta)$$, where $$\rho$$ and $$\sigma$$ are associated (with the usual notation) with a linear multistep method. The rules are useful in problems in numerical integration and the numerical solution of Volterra integral equations with convolution kernels. These kernels may have algebraic or algebraic-logarithmic singularities, and indeed any kernel which has a known and simple Laplace transform.
The author discusses the problem of the calculation of the weights which have themselves to be calculated approximately. A section is devoted to kernels of the form $$(1/\epsilon_ 1)k_ 1(t/\epsilon_ 1)+...+(1/\epsilon_ m)k_ m(t/\epsilon_ m)$$ where $$0<\epsilon_ m<...<\epsilon_ 1\leq 1$$, $$\epsilon_ m<<\epsilon_ 1$$. The methods of the paper are said to be more appropriate to Volterra equations with kernels of this form than the usual methods based on pointwise quadrature formulae. In fact it is shown that accuracy improves with decreasing $$\epsilon_ i$$. A section is devoted to an examination of an absorption- diffusion problem. This is replaced by a boundary integral equation where the Laplace transform of the kernel is known rather than the kernel itself. It is shown that when the associated multistep methodquential map grammars, most of the recent work has been on parallel map generating systems. The main types of these are: (1) binary fission/fusion systems with labeling and interactions of the regions, (2) the map interpretations of parallel graph grammars (such as propagating graph 0L- systems) with node (region) labeling, and (3) the edge-label-controlled binary propagating map 0L-systems (BPM0L-systems). Of the latter systems various classes are defined: with single or double edge labels, and with edge insertion controlled by circular words or by markers. Properties and applicability to cell division pattern of these families of systems are compared.

##### MSC:
 65R20 Numerical methods for integral equations 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures 45D05 Volterra integral equations
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##### References:
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