Summary. The procedure sivp presented in this paper calculates an approximate solution of Cauchy's initial value problem for hyperbolic systems of the form (t) (see below). The discretization which proceeds along the characteristics is performed using the midpoint rule started by Euler's method. To provide an algorithm of high accuracy the numerical solution is improved by step size extrapolation. This paper contains an ALGOL program completed by examples of the use and test results.
Zusammenfassung --AbstractNumerisehe Behandlung einer Volterraschen Integralgleichung. Die Arbeit behandelt eine verallgemeinerte nichtlineare Volterrasche Integralgleichung zweiter Art, in deren Kern die unbekannte Funktion mit zwei verschiedenen Argumenten vorkommt. Die Gleichung wird durch einen Kollokationsansatz mit stfickweisen Hermite-Polynomen gel6st. Bei Verwendung von Polynomen des Grades 2 m-1, m e N, und von geeigneten Quadraturformeln hat das Verfahren die Ordnung 2 m. Die Kollokationsstellen miissen im Einklang mit einer Stabilitiitsbedingung gewfihlt werden.Numerical Solution of a Volterra Integral Equation. This paper deals with a generalized nonlinear Volterra integral equation, whose kernel contains the unknown function at two different arguments. The equation is solved by collocation with piecewise Hermite-polynomials. The method has order 2 m, m E N, if polynomials of degree 2 m-1 and appropriate integration formulas are used. The collocation points must be chosen in accordance with a certain stability condition.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.