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Zbl 0644.35005
Kracht, Manfred; Kreyszig, Erwin
Methods of complex analysis in partial differential equations with applications. (English)
[B] Canadian Mathematical Society Series of Monographs and Advanced Texts; Wiley-Interscience Publication. New York etc: Wiley. xiv, 394 p. \sterling 47.50 (1988). ISBN 0-471-83091-7

This book is a self-contained presentation of the function theoretical method in partial differential equations. It describes some methods for constructing integral operators which transform analytic functions into solutions to certain partial differential equations. \par Such operators were introduced by S. Bergman (1937) and by I. N. Vekua (1937). The plan of the book is explained in the introduction. In Chapters 2-7 the authors develop various methods for constructing the integral operators and study integral representations of solutions to various PDE. Construction of Riemann's function is discussed in detail. \par In Chapters 8-10, the authors study: 1) singularities of solutions to some PDE in terms of the properties of the coefficients of the series which represent the solutions; 2) approximation of solutions to PDE by some special solutions (Runge's, Mergelyan's and Walsh's type theorems); and 3) value distribution theory for solutions to some PDE. \par In Chapters 11-13, various applications are given. In particular, the Bauer-Peschl equation, Chaplygin's equation, and Tricomi's equation are treated. \par The book has an extensive bibliography and author, subject and symbol indices. The authors are active contributors to the field.
[A.G.Ramm]

MSC 2000:: *35-02 Research monographs (partial differential equations)
35C15 Integral representations of solutions of PDE

Keywords: function theoretical method; integral operators; transform; integral representations; Riemann's function; singularities; approximation of solutions; special solutions; value distribution theory; Bauer-Peschl equation; Chaplygin's equation; Tricomi's equation; bibliography

Cited in: Zbl 0954.35013 Zbl 0685.35025

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