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Piecewise continuous solutions of pseudoparabolic equations in two space dimensions†

Published online by Cambridge University Press:  14 November 2011

Heinrich Begehr  and
Robert P. Gilbert
Heinrich Begehr
Affiliation:
Free University, Berlin
Robert P. Gilbert
Affiliation:
University of Delaware
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Synopsis

Here the Riemann boundary value problem-well known in analytic function theory as the problem to find entire analytic functions having a prescribed jump across a given contour-is solved for solutions of a pseudoparabolic equation which is derived from the complex differential equation of generalized analytic function theory. The general solution is given by use of the generating pair of the corresponding class of generalized analytic functions which gives rise to a representation for special bounded solutions of the pseudoparabolic equation. These solutions are obtained by linear integral equations one of which is given by a development of the generalized fundamental kernels of generalized analytic functions and which leads to a Cauchy-type integral representation. The bounded solutions are needed to transform the general boundary value problem (of non-negative index) with arbitrary initial data into a homogeneous problem which can easily be solved by the Cauchy-type integral (if the index is zero).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 1-2 , 1978 , pp. 153 - 173
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Begehr, H. and Gilbert, R. P.Randwertaufgaben ganzzahliger Charakteristik für verallgemeinerte hyperanalytische Funktionen. Applicable Anal. 6 (1977), 189–205.CrossRefGoogle Scholar
2Begehr, H. and Gilbert, R. P. On Riemann boundary value problems for certain linear elliptic systems in the plane J. Differential Equations, to appear.Google Scholar
3Bers, L.Theory of pseudoanalytic functions. Lecture Notes. New York University, 1953.Google Scholar
4Gakhov, F. D.Boundary value problems (Oxford: Pergamon, 1966).CrossRefGoogle Scholar
5Gilbert, R. P. and Schneider, M.Generalized meta and pseudo parabolic equations in the plane, to be published.Google Scholar
6Muskhelishvili, N. I.Singular integral equations. (Groningen: Noordhoff, 1953).Google Scholar
7Vekua, I. N.Generalized analytic functions (London: Pergamon, 1962).Google Scholar
8Zabreyko, P. P., Koshelev, A. I., Krasnosels'kii, M. A., Mikhlin, S. G., Rakovshchik, L. S. and Stet'senko, V. Ya.Integral equations – a reference text (Leyden: Noordhoff, 1975).CrossRefGoogle Scholar