Abstract

We consider an initial-boundary value problem for the heat equation with an inhomogeneous initial condition and boundary conditions. One of the boundary conditions contains a mixed derivative. When solving this problem by the method of separation of variables, a spectral problem arises. A system of eigenfunctions of this spectral problem and a biorthogonally conjugate system, are constructed explicitly. Also we obtain an asymptotic formula for the eigenvalues. In this paper we formulate theorems on the properties of the system of eigenfunctions of the spectral problem and a theorem about representing the solution of the initial initial-boundary value problem in the form of a Fourier series in the system of eigenfunctions. Thus, the existence of a solution is shown if the initial condition belongs to the Holder class. However, it has been shown that the solution is not unique. We show that additional condition guarantees the uniqueness of the solution. The unique solution of this problem is also obtained in the article.

中文翻译：

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边界条件下带混合导数的热方程混合问题的正确性

摘要

我们考虑具有非齐次初始条件和边界条件的热方程的初始边界值问题。边界条件之一包含混合导数。当用分离变量的方法解决这个问题时，就会出现一个谱问题。明确地构造了该谱问题的本征函数系统和双正交共轭系统。我们还获得了特征值的渐近公式。在本文中，我们制定了关于谱问题的本征函数系统性质的定理和关于在本征函数系统中以傅立叶级数的形式表示初初边值问题的解的定理。因此，如果初始条件属于 Holder 类，则表明存在解。然而，已经证明该解决方案不是唯一的。我们证明了附加条件保证了解的唯一性。文章中也得到了这个问题的唯一解。