Abstract

In the work, considered an inhomogeneous equation with partial derivatives of the fourth order with a singular coefficient, for which a boundary problem with initial conditions was investigated in a rectangle. The study of the considered problem was carried out by the a method of spectral analysis. On the basis of the property of completeness of systems of eigenfunctions of one-dimensional spectral problem, i.e. the system of sinus functions, the uniqueness theorem was proved. The solution of the initial-boundary problem was constructed in the form of series with respect to the system of eigenfunctions of a one-dimensional spectral problem. For proving uniform convergence of the constructed series, it was used estimates for trigonometric functions and Bessel-Clifford functions. On the basis of them, estimates were obtained for each member of a series that made it possible to prove the convergence of the obtained series and its derivatives to the fourth order inclusive, as well as the existence theorem in the class of regular solutions.

中文翻译：

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矩形区域中具有奇异系数的四阶方程的边界问题

摘要

在工作中，考虑了具有奇异系数的具有四阶偏导数的不均匀方程，对于该方程，在矩形中研究了具有初始条件的边界问题。所考虑的问题的研究是通过频谱分析的方法进行的。基于一维频谱问题本征函数系统（即窦函数系统）的完备性，证明了唯一性定理。相对于一维频谱问题的本征函数系统，初始边界问题的解决方案以级联形式构建。为了证明所构造序列的一致收敛性，使用了三角函数和Bessel-Clifford函数的估计。在他们的基础上，