On a two-point problem for a second-order partial differential equation with constant coefficients
type: Item
instance of: scholarly article
title: On a two-point problem for a second-order partial differential equation with constant coefficients
zbMATH Open document ID: 1006.35020
author: A. B. Mikhaĭlov
published in: Russian Mathematics
publication date: 2001-03-07T00:00:00Z
review text: This paper is devoted to the following second-order equation: \[ {\partial^2 w\over\partial z^2_2}-{\partial^2 w \over \partial z^2_1} a^2+b{\partial w\over \partial z_1}+ c{\partial w\over \partial z_2}+ dw=0.\tag{1} \] Here \(w(z_1,z_2)\) stands for the desired complex-valued function, and \(z_1,z_2\in\mathbb{C}\), \(a,b,c\) and \(d\) are complex values. The author seeks a solution of (1), which satisfies the conditions \[ \begin{cases}\alpha w(z_1,0)+ \beta{\partial w\over\partial z_2} (z_1,0)=f_1(z_1),\\ \gamma w(z_1, \ell)+\delta {\partial w\over\partial z_2}(z_1,\ell) =f_2(z_1),\end{cases}\tag{2} \] where \(|\alpha|+|\beta|\neq 0\), \(|\gamma|+|\delta |\neq 0\), \(\ell\in\mathbb{C}\), \(\ell\neq 0\). The functions \(f_1(z_1)\), \(f_2(z_1)\) are assumed to be analytic in a convex domain \(Q\subset\mathbb{C}\). He seeks a solution of (1), (2) in the form of a series of exponents, which is absolutely convergent in some suitable Fréchet space of analytic functions.
Mathematics Subject Classification ID: 35E20
Mathematics Subject Classification ID: 35G15
zbMATH DE Number: 1541448
zbMATH Keywords: convex domain
zbMATH Keywords: Fréchet space of analytic functions
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1541448