Tricomi方程极点在椭圆区域的基本解

发布时间：2018-12-27 07:53

【摘要】：关于自变量x,y的二阶微分方程Tu=yu_(xx)+u_(yy)=0称为Tricomi方程,它是混合型偏微分方程的经典例子,称T为Tricomi算子.这个方程在上半平面y＞0上是椭圆型的;在x轴y=0上是抛物型的;在下半平面y＜0上是双曲型的.混合型方程是重要的偏微分方程,它在数学、物理和气体动力学等方面有广泛的应用[1-50].例如光滑定常跨音速气流满足一混合型方程.但对于一致椭圆方程和一致双曲方程,混合型方程有待更进一步的研究.Tricomi算子在(3,2)-尺度变换下不变.按照物理学家们的通常做法,Tricomi算子有对应于此尺度变换齐次的解.Barros-Neto和Gelfand[6]利用特征方法和Tricomi算子的齐次性得到Tricomi算子极点在退化线上的基本解F+和F-.本文第一章介绍其证明的背景知识和主要思路.Barros-Neto和Gelfand[7]还利用特征方法得到Tricomi算子极点在椭圆区域的基本解.本文第二章介绍其证明的主要方法.特别地,当极点趋于退化线时,基本解的极限是(10)F和-F的线性组合.实部系数的和为1,虚部系数的和相抵消,故极限仍是Tricomi算子T的基本解.本文第三章我们利用Barros-Neto和Cardoso[3]的级数展开方法得到Tricomi算子极点在椭圆区域新形式的基本解,当极点趋于退化线时,我们得到的基本解的极限行为与Barros-Neto和Gelfand[7]得到的基本解的极限行为不同:基本解的极限也是F+和F-的线性组合,但实部系数的和不为1,虚部系数的和不相抵消,故极限不再是Tricomi算子T的基本解.本文所做的工作为今后进一步研究混合型方程的边值问题和Cauchy问题等提供了帮助.
[Abstract]:The second order differential equation Tu=yu_ (xx) u _ (yy) = 0 for the independent variable xy is called Tricomi equation. It is a classical example of mixed partial differential equation, and T is called Tricomi operator. The equation is elliptic on the upper half plane y > 0, parabolic on the x axis y 0 and hyperbolic on the lower half plane y < 0. Hybrid equations are important partial differential equations, which are widely used in mathematics, physics and gas dynamics [1-50]. For example, smooth steady transonic flow satisfies a mixed equation. But for the uniform elliptic equation and the uniform hyperbolic equation, the mixed equation needs to be further studied. The Tricomi operator is invariant under the (3 ~ 2) -scale transformation. According to the usual practice of physicists, the Tricomi operator has a homogeneous solution corresponding to this scale transformation. Barros-Neto and Gelfand [6] obtain the fundamental solutions F and F of the Tricomi operator poles on the degenerate line by using the characteristic method and the homogeneity of the Tricomi operator. In the first chapter, the background knowledge and the main ideas of the proof are introduced. Barros-Neto and Gelfand [7] also obtain the basic solutions of the Tricomi operator poles in the elliptic region by using the characteristic method. The second chapter introduces the main methods of its proof. In particular, when the poles tend to degenerate, the limit of the fundamental solution is a linear combination of (10) F and -F. The sum of real part coefficients is 1, and the sum of imaginary part coefficients counteracts, so the limit is still the basic solution of Tricomi operator T. In the third chapter, we use the series expansion method of Barros-Neto and Cardoso [3] to obtain the basic solution of the pole of Tricomi operator in the new form of elliptic domain, when the pole tends to degenerate. The limit behavior of the basic solution is different from that obtained by Barros-Neto and Gelfand [7]: the limit of the fundamental solution is also a linear combination of F and F-, but the sum of the real part coefficient is not 1, the sum of the imaginary part coefficient is not offset. So the limit is no longer the basic solution of Tricomi operator T. The work in this paper will be helpful for the further study of the boundary value problem and Cauchy problem of the mixed equation in the future.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175.28

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