Abstract
Starting from an existence and uniqueness theorem for a generalized nonsingular second kind Volterra equation existence and uniqueness for the solution of the nonlinear, weakly singular first kind Volterra equation is examined. A new type of numerical method is developed. A basic lemma concerning the boundedness of a special difference inequality is given and order two or three convergence of the method is shown. Two numerical examples illustrate the theoretical results.
Zusammenfassung
Ausgehend von einem Existenz- und Eindeutigkeitssatz für die Lösung einer verallgemeinerten Volterra-Integralgleichung zweiter Art wird die Existenz und Eindeutigkeit der Lösung der hier behandelten nichtlinearen, schwach singulären Volterra-Integralgleichung erster Art untersucht. Es wird ein numerisches Verfahren der Ordnung 2 bzw. 3 angegeben. Der Konvergenzbeweis basiert in beiden Fällen auf einem Lemma über die Beschränktheit einer speziellen Differenzenungleichung. An zwei numerischen Beispielen werden die theoretischen Ergebnisse demonstriert.
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This paper contains results of the author's dissertation [3]. Some of the proofs had to be shortened or omitted, so for the details we have to refer to the original paper [3], which will be send on to interested readers.
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Branca, H.W. The nonlinear Volterra equation of Abel's kind and its numerical treatment. Computing 20, 307–324 (1978). https://doi.org/10.1007/BF02252379
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DOI: https://doi.org/10.1007/BF02252379