Abstract

For a mixed elliptic-hyperbolic type equation in a mixed domain, when the elliptic part is a sector of the circle, and the hyperbolic part consists of two characteristic triangles, the Bitsadze–Samarskii type nonlocal problem is investigated. A feature of such problems is that the boundary conditions in the hyperbolic parts of the boundary are determined by a first-order differential operator and pointwise link the values of the partial derivatives of the desired solution on the characteristics with the partial derivatives on arbitrary monotone curves lying inside the characteristic triangles of the equation. To solve the problem, the methods of the theory of partial differential equations and the theory of singular integral equations as well as the methods of energy integrals and complex analysis were used, with the help of which the existence and uniqueness theorem for the solution of the investigated problem is proved. Also, a method of reducing the investigated problem to an equivalent system of singular integral equations is shown, also, a method for solving this system is proposed, which allows to obtain a solution to the problem in explicit form.

中文翻译：

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区域内混合型方程的Bitsadze-Samarskii型问题，其特征存在偏差

摘要

对于混合域中的混合椭圆-双曲型方程，当椭圆部分是圆的扇形，而双曲部分由两个特征三角形组成时，研究了Bitsadze-Samarskii型非局部问题。这些问题的特征在于，边界的双曲部分的边界条件由一阶微分算子确定，并将特征上所需解的偏导数的值与任意单调曲线上的偏导数逐点联系起来。位于方程的特征三角形内。为了解决这个问题，我们使用了偏微分方程和奇异积分方程的理论方法以及能量积分和复杂分析的方法，在此基础上，证明了所研究问题解的存在性和唯一性定理。另外，示出了将研究的问题简化为奇异积分方程的等效系统的方法，并且，提出了一种用于求解该系统的方法，其允许以显式形式获得该问题的解。