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→Classification
== Classification ==
== Classification ==
分类
Some linear, second-order partial differential equations can be classified as [[parabolic partial differential equation|parabolic]], [[hyperbolic partial differential equation|hyperbolic]] and [[elliptic partial differential equation|elliptic]]. Others, such as the [[Euler–Tricomi equation]], have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions.
Some linear, second-order partial differential equations can be classified as [[parabolic partial differential equation|parabolic]], [[hyperbolic partial differential equation|hyperbolic]] and [[elliptic partial differential equation|elliptic]]. Others, such as the [[Euler–Tricomi equation]], have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions.
Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic and elliptic. Others, such as the Euler–Tricomi equation, have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions.
Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic and elliptic. Others, such as the Euler–Tricomi equation, have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions.
一些线性二阶偏微分方程可分为抛物型方程、双曲型方程和椭圆型方程。其他的方程，如欧拉-特里科米方程，在不同的区域有不同的类型。这种分类为适当的初始和边界条件以及解的光滑性提供了指导。
一些线性二阶偏微分方程可分为抛物型方程、双曲型方程和椭圆型方程。其他的方程，如欧拉-特里科米方程，在不同的区域有不同的类型。这种分类有助于选择适当的初始和边界条件以及提高解的平滑性。
=== Equations of first order ===
=== Equations of first order ===
一阶方程
=== Linear equations of second order ===
=== Linear equations of second order ===
二阶线性方程
Assuming {{math|''u<sub>xy</sub>'' {{=}} ''u<sub>yx</sub>''}}, the general linear second-order PDE in two independent variables has the form
Assuming {{math|''u<sub>xy</sub>'' {{=}} ''u<sub>yx</sub>''}}, the general linear second-order PDE in two independent variables has the form
Assuming u<sub>yx</sub>}}, the general linear second-order PDE in two independent variables has the form
Assuming u<sub>yx</sub>}}, the general linear second-order PDE in two independent variables has the form
假设 u 子 yx / sub } ，两个独立变量的一般线性二阶偏微分方程具有
假设 {{math|''u<sub>xy</sub>'' {{=}} ''u<sub>yx</sub>''}}，含有两个独立变量的一般的线性二阶偏微分方程具有如下这样的形式
<math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0,</math>
<math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0,</math>
where the coefficients , , ... may depend upon and . If over a region of the -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
where the coefficients , , ... may depend upon and . If over a region of the -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
其中的系数{{mvar|A}}, {{mvar|B}}, {{mvar|C}}... 一般取决于{{mvar|x}}和{{mvar|y}}。如果在{{mvar|xy}}-平面的一个区域上{{math|''A''<sup>2</sup> + ''B''<sup>2</sup> + ''C''<sup>2</sup> > 0}}，偏微分方程在该区域是二阶的。这种形式类似于圆锥曲线的方程:
<math>Ax^2 + 2Bxy + Cy^2 + \cdots = 0.</math>
<math>Ax^2 + 2Bxy + Cy^2 + \cdots = 0.</math>
More precisely, replacing {{math|∂<sub>''x''</sub>}} by {{mvar|X}}, and likewise for other variables (formally this is done by a [[Fourier transform]]), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a [[homogeneous polynomial]], here a [[quadratic form]]) being most significant for the classification.
More precisely, replacing {{math|∂<sub>''x''</sub>}} by {{mvar|X}}, and likewise for other variables (formally this is done by a [[Fourier transform]]), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a [[homogeneous polynomial]], here a [[quadratic form]]) being most significant for the classification.
More precisely, replacing by , and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
More precisely, replacing by , and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
更准确地说，用{{mvar|X}}替换{{math|∂<sub>''x''</sub>}}，对于其他变量做同样的操作(从形式上来说，这是由傅里叶变换来完成的)，将一个常系数偏微分方程转换成一个相同次数的多项式，最高次数的项(齐次多项式，这里是一个二次形式)一般会用于偏微分方程的分类。
===~~ most significant for the classification 意译为用于偏微分方程的分类。 significiant直译过来感觉不太合适
Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention of the term being rather than ; formally, the discriminant (of the associated quadratic form) is 4(B<sup>2</sup> − AC)}}, with the factor of 4 dropped for simplicity.
Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention of the term being rather than ; formally, the discriminant (of the associated quadratic form) is 4(B<sup>2</sup> − AC)}}, with the factor of 4 dropped for simplicity.
正如人们可以根据判别式{{math|''B''<sup>2</sup> − 4''AC''}}将圆锥曲线和二次型分为抛物型、双曲型和椭圆型一样，对于给定点的二阶偏微分方程也可以这样做。然而，偏微分方程中的判别式{{math|''B''<sup>2</sup> − 4''AC''}}是根据交叉项的系数{{math|2''B''}}而不是{{mvar|B}}给出的，形式上，判别式(关联二次型)是{{math|(2''B'')<sup>2</sup> − 4''AC'' {{=}} 4(''B''<sup>2</sup> − ''AC'')}}，为简单起见，去掉了因子4。
(elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where .
(elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where .
(椭圆型微分方程) : 椭圆偏微分方程的解光滑到系数允许的程度，在定义方程和解的区域的内部。例如，拉普拉斯方程的解在它们被定义的区域内是解析的，但是解可能假设边界值是不光滑的。亚音速流体的运动可以用椭圆偏微分方程近似，其中欧拉-特里科米方程是椭圆型的。
{{math|''B''<sup>2</sup> − ''AC'' < 0}}(椭圆形微分方程)：在定义方程和解的区域内部，椭圆型偏微分方程的解光滑到系数允许的程度。例如，拉普拉斯方程的解在它们被定义的区域内是解析的，但是解可能假设边界值是不光滑的。亚音速流体的运动可以用椭圆偏微分方程近似，其中欧拉-特里科米方程在{{math|''x'' < 0}}是椭圆型偏微分方程。
# {{math|''B''<sup>2</sup> − ''AC'' {{=}} 0}} (''[[parabolic partial differential equation]]''): Equations that are [[parabolic partial differential equation|parabolic]] at every point can be transformed into a form analogous to the [[heat equation]] by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where {{math|''x'' {{=}} 0}}.
# {{math|''B''<sup>2</sup> − ''AC'' {{=}} 0}} (''[[parabolic partial differential equation]]''): Equations that are [[parabolic partial differential equation|parabolic]] at every point can be transformed into a form analogous to the [[heat equation]] by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where {{math|''x'' {{=}} 0}}.
0}} (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where 0}}.
0}} (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where 0}}.
{{math|''B''<sup>2</sup> − ''AC'' {{=}} 0}}（抛物线形偏微分方程）：在每一点上都是抛物线型的方程可以通过改变自变量从而转化成类似于热方程的形式。随着转换后的时间变量的增加，方程的解变得平滑。欧拉-特里科米方程在{{math|''x'' {{=}} 0}}特征线上是抛物线形的。
# {{math|''B''<sup>2</sup> − ''AC'' > 0}} (''[[hyperbolic partial differential equation]]''): [[hyperbolic partial differential equation|hyperbolic]] equations retain any discontinuities of functions or derivatives in the initial data. An example is the [[wave equation]]. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where {{math|''x'' > 0}}.
# {{math|''B''<sup>2</sup> − ''AC'' > 0}} (''[[hyperbolic partial differential equation]]''): [[hyperbolic partial differential equation|hyperbolic]] equations retain any discontinuities of functions or derivatives in the initial data. An example is the [[wave equation]]. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where {{math|''x'' > 0}}.
(hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where .
(hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where .
(双曲型偏微分方程) : 双曲方程在初始数据中保留了函数或导数的任何不连续性。波动方程就是一个例子。超音速流体的运动可以用双曲偏微分方程近似，其中欧拉-特里科米方程是双曲型的。
{{math|''B''<sup>2</sup> − ''AC'' > 0}}(双曲形偏微分方程)：双曲形方程在初始数据中保留了函数或导数的任何不连续性。波动方程就是一个例子。超音速流体的运动可以用双曲形偏微分方程近似，其中欧拉-特里科米方程在{{math|''x'' > 0}}是双曲型的。
If there are independent variables , a general linear partial differential equation of second order has the form
If there are independent variables , a general linear partial differential equation of second order has the form
如果存在{{mvar|n}}个自变量{{math|''x''<sub>1</sub>, ''x''<sub>2 </sub>,… ''x''<sub>''n''</sub>}}，一般二阶线性偏微分方程的形式是
<math>L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\partial^2 u}{\partial x_i \partial x_j} \quad \text{ plus lower-order terms} =0.</math>
<math>L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\partial^2 u}{\partial x_i \partial x_j} \quad \text{ plus lower-order terms} =0.</math>
The classification depends upon the signature of the eigenvalues of the coefficient matrix .
The classification depends upon the signature of the eigenvalues of the coefficient matrix .
这种分类取决于系数矩阵特征值的符号（正负性）。
Elliptic: the eigenvalues are all positive or all negative.
Elliptic: the eigenvalues are all positive or all negative.
椭圆形方程: 本征值全部为正或全部为负。
# Parabolic: the eigenvalues are all positive or all negative, save one that is zero.
# Parabolic: the eigenvalues are all positive or all negative, save one that is zero.
Parabolic: the eigenvalues are all positive or all negative, save one that is zero.
Parabolic: the eigenvalues are all positive or all negative, save one that is zero.
抛物线形方程: 本征值全部为正或全部为负，除了一个为零。
# Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
# Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
双曲形方程: 只有一个负特征值，其余的都是正特征值，或者只有一个正特征值，其余的都是负特征值。
# Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
# Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
超双形方程: 存在多于一个正本征值和多于一个的负本征值，且不存在零本征值。对于超双曲方程，只有一个有限理论(Courant 和 Hilbert，1962)。
==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） 我对于limited theory这个单词对应的相关理论不太了解。
=== Systems of first-order equations and characteristic surfaces ===
=== Systems of first-order equations and characteristic surfaces ===
一阶方程组和特征曲面
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown {{mvar|u}} is now a [[Euclidean vector|vector]] with {{mvar|m}} components, and the coefficient matrices {{mvar|A<sub>ν</sub>}} are {{mvar|m}} by {{mvar|m}} matrices for {{math|''ν'' {{=}} 1, 2,… ''n''}}. The partial differential equation takes the form
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown {{mvar|u}} is now a [[Euclidean vector|vector]] with {{mvar|m}} components, and the coefficient matrices {{mvar|A<sub>ν</sub>}} are {{mvar|m}} by {{mvar|m}} matrices for {{math|''ν'' {{=}} 1, 2,… ''n''}}. The partial differential equation takes the form
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown is now a vector with components, and the coefficient matrices are by matrices for 1, 2,… n}}. The partial differential equation takes the form
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown is now a vector with components, and the coefficient matrices are by matrices for 1, 2,… n}}. The partial differential equation takes the form
偏微分方程组的分类可以推广到一阶方程组，其中未知量{{mvar|u}}是有{{mvar|m}}个分量的向量。对于{{math|''ν'' {{=}} 1, 2,… ''n''}}，系数矩阵{{mvar|A<sub>ν</sub>}}是{{mvar|m}} × {{mvar|m}}的矩阵。偏微分方程形式如下：
<math>Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0,</math>
<math>Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0,</math>
where the coefficient matrices and the vector may depend upon and . If a hypersurface is given in the implicit form
where the coefficient matrices and the vector may depend upon and . If a hypersurface is given in the implicit form
其中系数矩阵{{mvar|A<sub>ν</sub>}}和向量{{mvar|B}}可能依赖于{{mvar|x}} 和 {{mvar|u}}。如果超曲面是以隐式形式给出的
<math>\varphi(x_1, x_2, \ldots x_n)=0,</math>
<math>\varphi(x_1, x_2, \ldots x_n)=0,</math>
where has a non-zero gradient, then is a characteristic surface for the operator at a given point if the characteristic form vanishes:
where has a non-zero gradient, then is a characteristic surface for the operator at a given point if the characteristic form vanishes:
若存在一个非零的梯度，那么如果特征形式消失，则在给定点上算子的特征曲面形式如下:
<math>Q\left(\frac{\partial\varphi}{\partial x_1}, \ldots\frac{\partial\varphi}{\partial x_n}\right) =\det\left[\sum_{\nu=1}^nA_\nu \frac{\partial \varphi}{\partial x_\nu}\right]=0.\,</math>
<math>Q\left(\frac{\partial\varphi}{\partial x_1}, \ldots\frac{\partial\varphi}{\partial x_n}\right) =\det\left[\sum_{\nu=1}^nA_\nu \frac{\partial \varphi}{\partial x_\nu}\right]=0.\,</math>
在《相空间表述量子力学，我们可以考虑量子哈密顿的量子粒子轨迹方程。这些方程是无限阶偏微分方程。然而，在半经典展开中，我们有一个有限的狄拉克常数序列 | 的常微分方程组。维格纳函数的发展方程也是一个无限阶偏微分方程。量子轨道具有量子特性，利用量子轨道可以计算维格纳函数的演化。
在《相空间表述量子力学，我们可以考虑量子哈密顿的量子粒子轨迹方程。这些方程是无限阶偏微分方程。然而，在半经典展开中，我们有一个有限的狄拉克常数序列 | 的常微分方程组。维格纳函数的发展方程也是一个无限阶偏微分方程。量子轨道具有量子特性，利用量子轨道可以计算维格纳函数的演化。
== Analytical solutions ==
== Analytical solutions ==