The Cauchy Problem
The Cauchy Problem
复制标题
DOI：
10.1007/978-3-662-05558-8_5
复制
发表时间：
1986
期刊：
影响因子：
--
通讯作者：
V. S. Vladimirov
中科院分区：
文献类型：
--
作者：
V. S. Vladimirov
我们从平面上的柯西问题开始。方程a (x, y) uxx + 2b (x, y) uxy+ c (x, y) uy+ d (x, y) ux + e (x, y) uy+ t (x, y) U = F (x, y)（12.1）边界条件full '= uo (x, y),~~！R = u1 (x, y)（12.2）包含如下式。假设在区域D中，我们有一个双曲型方程（12.1）（b' ' ' -ac> 0），并且在曲线r上属于D或构成D边界的一段，给出两个函数U o （x, y）和U 1 (x, y)，以及方向向量l （x, y）。我们必须找到一个函数U (x, y)，它是方程（12.1）在D和r上满足p边界条件（12.2）的解。如果曲线r的切线与特征不重合，那么在以经过r端点的特征为界的D中，只要Eq.（12.1）的系数和（12.2）中的数据足够光滑，则柯尼问题（12.1）、（12.2）只有一个解。
We start with the Cauchyproblem in the plane. The Cauchy problem for the equation a (x, y) U xx+ 2b (x, y) uxy+ c (x, y) uyy+ d (x, y) U x+ e (x, y) uy+ t (x, y) u= F (x, y)(12.1) with the boundary conditions ull'= uo (x, y),~~! r= u1 (x, y)(12.2) consists in the following. Suppose that in a region D we have an equation (12.1) of the hyperbolic type (b'"-ac> 0) and that on a curve r that belongs to D or constitutes a section of the boundary of D two functions, U o (x, y) and U 1 (x, y), and the direction vector l (x, y) are given. We must find a function U (x, y) that is a solution of Eq.(12.1) in D and on r satisfies thp boundary conditions (12.2). Jf at eachpoint oi curve r the direction vector l does not He on the tangent to rand if the tangents to curve r do not coincide with the characteristics, then in D, which is bounded by characteristics passing through the ends of r, there is only one solution of the Caucny problem (12.1),(12.2), provided the coefficients oi Eq.(12.1) amI the data in (12.2) are sufficiently smooth.