On a two-point problem for a second-order partial differential equation with constant coefficients
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.
scientific article; zbMATH DE number 1541448
scientific article