Abstract

In this article, the existence and uniqueness of solution of the conjugation problem in a rectangular domain for a third-order partial differential equation is proved, when the characteristic equation has 3 multiple roots for \(y>0\), and it has 1 simple and 2 multiple roots for \(y<0\). Using the Green’s functions and the method of integral equations, the solution of the problem is equivalently reduced to solving the boundary value problem for the trace of the desired function at \(y=0\), and then to solving the Fredholm integral equation of the 2nd kind. The one-valued solvability of Fredholm integral equation of the 2nd kind is proved by the method of successive approximations. The solution of the problem for \(y>0\) is constructed by the Green’s function method, and for \(y<0\) by reducing to the problem of a two-dimensional Volterra integral equation of the 2nd kind.

中文翻译：

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三阶抛物双曲型混合方程的边值问题

摘要

本文证明了当特征方程有3个重根\(y>0\)时，三阶偏微分方程在矩形域内共轭问题解的存在唯一性，且有1 \(y<0\)的简单根和 2 个多重根。利用格林函数和积分方程的方法，问题的求解等效地简化为求解期望函数在\(y=0\)处的迹的边值问题，然后求解Fredholm积分方程第2种。用逐次逼近的方法证明了第二类Fredholm积分方程的一值可解性。对于\(y>0\)的问题的解是通过格林函数方法构造的，对于\(y<0\)的问题的解是通过简化为第二类二维Volterra积分方程问题来构造的。