Lukankin, G. L.; Latyshev, A. V.; Ryndina, S. V. Analytical solution of a general boundary value problem for a BKW-equation. (Russian) Zbl 1057.35025 Vladikavkaz. Mat. Zh. 4, No. 4, 33-38 (2002). The article is devoted to the study of the one-dimensional stationary linearized BKW-like equation \[ \xi_x\frac{\partial h}{\partial x} = \xi\int_{-1}^1d\mu'\int_0^{\infty} \exp(-\xi^{\prime 2})k(\mu,\xi;\mu',\xi')h(x,\xi',\xi')\,d\xi' - \xi h(x,\mu',\xi'), \] where \(k(\mu,\xi;\mu',\xi') = 1 + (3/2)\mu\xi\mu'\xi' + (1/2)(\xi^{2} - 2) (\xi^{\prime 2} - 2)\) is the kernel of the integral operator. The function \(h\) satisfies the following boundary conditions: \[ \begin{gathered} h(0,\mu,\xi) = h_0(\mu,\xi),\quad 0 < \mu < 1,\\ h(x,\mu,\xi) = h_{as}(x,\mu,\xi) + o(1),\quad x\to\infty,\quad -1 < \mu < 0. \end{gathered} \] Here \(h_{as}\) is a linear combination of discrete solutions to the equation under consideration. The authors obtain an eigenfunction expansion for a general solution to the above boundary value problem. The solvability conditions make it possible to determine both unknown coefficients of this expansion and free parameters of a solution to the vector Riemann-Hilbert problem. Reviewer: V. Grebenev (Novosibirsk) MSC: 35Q15 Riemann-Hilbert problems in context of PDEs 35P05 General topics in linear spectral theory for PDEs 35C05 Solutions to PDEs in closed form Keywords:BKW-equation; boundary value problem; analytical solution; Riemann-Hilbert problem; discrete spectrum; continuous spectrum PDFBibTeX XMLCite \textit{G. L. Lukankin} et al., Vladikavkaz. Mat. Zh. 4, No. 4, 33--38 (2002; Zbl 1057.35025) Full Text: EuDML Link