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Zbl 0643.65094
Lubich, C.
Convolution quadrature and discretized operational calculus. II. (English)
[J] Numer. Math. 52, No.4, 413-425 (1988). ISSN 0029-599X; ISSN 0945-3245/e

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 $\int\sp{x}\sb{0}f(x-t)g(t)dt$ by $\sum\sp{n}\sb{j=0}\omega\sb{n-j}(h)g(jh),$ $n=0,1,...,N$. The weights $\{\omega\sb n\}$ are given by $\sum\sp{n}\sb{j=0}\omega\sb j(h)\zeta\sp j=F(\delta (\zeta)/h),$ 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. \par 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 $(1/\epsilon\sb 1)k\sb 1(t/\epsilon\sb 1)+...+(1/\epsilon\sb m)k\sb m(t/\epsilon\sb m)$ where $0<\epsilon\sb m<...<\epsilon\sb 1\le 1$, $\epsilon\sb m<<\epsilon\sb 1$. 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 $\epsilon\sb i$. 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 2000:: *65R20 Integral equations (numerical methods)
65D32 Quadrature formulas (numerical methods)
41A55 Approximate quadratures
45D05 Volterra integral equations

Keywords: operational quadrature rules; convolution integral; Laplace transform; Volterra integral equations with convolution kernels; algebraic- logarithmic singularities; weights; absorption-diffusion problem; boundary integral equation; multistep methodquential map grammars; parallel map generating systems; fission/fusion systems; graph grammars; edge-label-control; 0L-systems; BPM0L-systems

Citations: Zbl 0637.65016

Cited in: Zbl 1138.74365 Zbl 1109.74374 Zbl 0989.74036 Zbl 0913.73075 Zbl 0768.65090 Zbl 0709.65119 Zbl 0693.65097

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