# 偏微分方程

此词条暂由Yuling翻译，未经人工整理和审校，带来阅读不便，请见谅。

A visualisation of a solution to the two-dimensional heat equation with temperature represented by the third dimension

以三维表示温度的二维热方程解的可视化

In mathematics, a **partial differential equation** (**PDE**) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

在数学中，** 偏微分方程函数 Partial Differential Equation**是包含未知多元函数及其偏导数的微分方程。偏微分方程用于描述涉及多元函数的问题，可以通过人为求解，也可以通过建立计算机模型来求解。常微分方程是偏微分方程一种特殊情况，它处理的是一元函数及其导数。

PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation and quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation and quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.

偏微分方程可以用来描述各种各样的物理现象，如声音，热量，扩散，静电，电动力学，流体力学，弹性力学，重力和量子力学。这些看起来截然不同的物理现象却可以用类似的偏微分方程来描述。正如常微分方程经常对一维动力系统进行建模一样，偏微分方程经常对多维系统进行建模。随机偏微分方程是偏微分方程的一种推广。

## Introduction

引言

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. For example, the position of a rigid body is specified by six parameters,^{[1]} but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, and heat transfer.

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. For example, the position of a rigid body is specified by six parameters, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, and heat transfer.

偏微分方程(简称为PDEs)涉及到方程相对于连续变量的变化率。例如，刚体的位置是由六个参数确定的，而流体的状态是由几个参数的连续分布给出的，如温度、压力等。刚体的动力学过程发生在有限维状态空间中，流体的动力学过程发生在无限维状态空间中。这种区别通常使偏微分方程比常微分方程更难求解，但是线性问题仍然有简单的求解方式。使用偏微分方程的经典领域包括声学、流体力学、电动力学和传热学。

A partial differential equation (PDE) for the function *u*(*x*_{1},… *x _{n}*) is an equation of the form

A partial differential equation (PDE) for the function is an equation of the form

函数 *u*(*x*_{1},… *x _{n}*) 的偏微分方程形式是：

- [math]\displaystyle{ f \left (x_1, \ldots x_n; u, \frac{\partial u}{\partial x_1}, \ldots \frac{\partial u}{\partial x_n}; \frac{\partial^2 u}{\partial x_1 \partial x_1}, \ldots \frac{\partial^2 u}{\partial x_1 \partial x_n}; \ldots \right) = 0. }[/math]

```
[math]\displaystyle{ f \left (x_1, \ldots x_n; u, \frac{\partial u}{\partial x_1}, \ldots \frac{\partial u}{\partial x_n}; \frac{\partial^2 u}{\partial x_1 \partial x_1}, \ldots \frac{\partial^2 u}{\partial x_1 \partial x_n}; \ldots \right) = 0. }[/math]
```

If f is a linear function of u and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

If is a linear function of and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

如果 f 是函数 *u* 及其导数的线性函数，则偏微分方程称为线性函数。线性偏微分方程的常见例子包括热方程、波动方程、拉普拉斯方程、亥姆霍兹方程方程、克莱因-高登方程和泊松方程。

A relatively simple PDE is

A relatively simple PDE is

一个相对简单的偏微分方程：

- [math]\displaystyle{ \frac{\partial u}{\partial x}(x,y) = 0. }[/math]

```
[math]\displaystyle{ \frac{\partial u}{\partial x}(x,y) = 0. }[/math]
```

This relation implies that the function *u*(*x*,*y*) is independent of x. However, the equation gives no information on the function's dependence on the variable y. Hence the general solution of this equation is

This relation implies that the function is independent of . However, the equation gives no information on the function's dependence on the variable . Hence the general solution of this equation is

这意味着函数 *u*(*x*,*y*) 独立于 x 的。然而，这个方程没有给出关于函数和自变量的相关性的信息。因此，这个方程的通解是

- [math]\displaystyle{ u(x,y) = f(y), }[/math]

```
[math]\displaystyle{ u(x,y) = f(y), }[/math]
```

where f is an arbitrary function of y. The analogous ordinary differential equation is

where is an arbitrary function of . The analogous ordinary differential equation is

其中， f 是 y 的任意函数。

类似的常微分方程是：

- [math]\displaystyle{ \frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0, }[/math]

```
[math]\displaystyle{ \frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0, }[/math]
```

which has the solution

which has the solution

它的解为

- [math]\displaystyle{ u(x) = c, }[/math]

```
[math]\displaystyle{ u(x) = c, }[/math]
```

where c is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.

where is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.

这里，c 是一个任意常量。

这两个例子说明常微分方程的一般解包含任意常数，但偏微分方程的解包含任意函数。

A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function *f*(*y*) can be determined if u is specified on the line *x* = 0.

A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function can be determined if is specified on the line 0}}.

偏微分方程的解一般不是唯一的; 一般必须在定义解的区域边界上定义附加条件。 例如，在上面的简单示例中，如果在 *x* = 0 时确定了 u 的值，则可以确定该函数 *f*(*y*)。

## Existence and uniqueness

存在性和唯一性

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf theorem, that is far from the case for partial differential equations. The Cauchy–Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf theorem, that is far from the case for partial differential equations. The Cauchy–Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.

尽管常微分方程解的存在性和唯一性用**弗罗贝尼乌斯定理 Picard–Lindelöf Theorem得到了令人满意的结果，但偏微分方程解的存在性和唯一性却远没有得到解决。**柯西-科瓦列夫斯基定理 Cauchy–Kowalevski theorem指出：对于任意系数在未知函数及其导数中解析的偏微分方程，柯西问题存在一个局部唯一的解析解。虽然这个结果似乎解决了解的存在性和唯一性问题，但是存在一些线性偏微分方程-其系数具有所有级数的导数(尽管这些导数不是解析的) ，但是根本没有解: 见 Lewy (1957)。即使偏微分方程的解存在且唯一，它仍然可能具有不可预料的性质。这些问题的数学研究通常是在更有力的弱解的背景下进行的。

An example of pathological behavior is the sequence (depending upon n) of Cauchy problems for the Laplace equation

An example of pathological behavior is the sequence (depending upon ) of Cauchy problems for the Laplace equation

反常特征的一个例子是拉普拉斯方程的柯西问题的序列(取决于n)

- [math]\displaystyle{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0, }[/math]

```
[math]\displaystyle{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0, }[/math]
```

with boundary conditions

with boundary conditions

具有边界条件

- [math]\displaystyle{ \begin{align} u(x,0) &= 0, \\ \frac{\partial u}{\partial y}(x,0) &= \frac{\sin nx}{n}, \end{align} }[/math]

```
[math]\displaystyle{ \begin{align} u(x,0) &= 0, \\ \frac{\partial u}{\partial y}(x,0) &= \frac{\sin nx}{n}, \end{align} }[/math]
```

where n is an integer. The derivative of u with respect to y approaches zero uniformly in x as n increases, but the solution is

where is an integer. The derivative of with respect to approaches zero uniformly in as increases, but the solution is

其中 n 是整数。u 关于 y 的导数，一致地随着 n 的增加而趋于零，但解是

- [math]\displaystyle{ u(x,y) = \frac{\sinh ny \sin nx}{n^2}. }[/math]

```
[math]\displaystyle{ u(x,y) = \frac{\sinh ny \sin nx}{n^2}. }[/math]
```

This solution approaches infinity if nx is not an integer multiple of 模板:Pi for any non-zero value of y. The Cauchy problem for the Laplace equation is called *ill-posed* or *not well-posed*, since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.

This solution approaches infinity if is not an integer multiple of for any non-zero value of . The Cauchy problem for the Laplace equation is called ill-posed or not well-posed, since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.

对于任何非零的 y，如果 nx 不是模板 模板:Pi 的整数倍，这个解会接近于无穷大。拉普拉斯方程的柯西问题被称为不适定的（可以译为ill-posed或not well-posed），因为解不是连续地依赖于问题的数据。这种不适定问题通常不能满足物理应用。

The existence of solutions for the Navier–Stokes equations, a partial differential equation, is part of one of the Millennium Prize Problems.

The existence of solutions for the Navier–Stokes equations, a partial differential equation, is part of one of the Millennium Prize Problems.

偏微分方程”纳维-斯托克斯方程“的解的存在性就是千禧年大奖难题之一的一部分。

## Notation

符号

In PDEs, it is common to denote partial derivatives using subscripts. That is:

In PDEs, it is common to denote partial derivatives using subscripts. That is:

在偏微分方程中，通常用下标表示偏导数。例如:

- [math]\displaystyle{ u_x = \frac{\partial u}{\partial x} }[/math]

```
[math]\displaystyle{ u_x = \frac{\partial u}{\partial x} }[/math]
```

- [math]\displaystyle{ u_{xx} = \frac{\partial^2 u}{\partial x^2} }[/math]

```
[math]\displaystyle{ u_{xx} = \frac{\partial^2 u}{\partial x^2} }[/math]
```

- [math]\displaystyle{ u_{xy} = \frac{\partial^2 u}{\partial y\, \partial x} = \frac{\partial}{\partial y } \left(\frac{\partial u}{\partial x}\right). }[/math]

```
[math]\displaystyle{ u_{xy} = \frac{\partial^2 u}{\partial y\, \partial x} = \frac{\partial}{\partial y } \left(\frac{\partial u}{\partial x}\right). }[/math]
```

Especially in physics, del or nabla (∇) is often used to denote spatial derivatives, and *̇u*, *ü* for time derivatives. For example, the wave equation (mentioned below) can be written as

Especially in physics, del or nabla () is often used to denote spatial derivatives, and for time derivatives. For example, the wave equation (mentioned below) can be written as

特别是在物理学中，del 或 nabla (∇) 经常用来表示空间导数和时间导数。例如，波动方程(在下文中提到)可以写成：

- [math]\displaystyle{ \ddot u=c^2\nabla^2u }[/math]

[math]\displaystyle{ \ddot u=c^2\nabla^2u }[/math]

or

or

或

- [math]\displaystyle{ \ddot u=c^2\Delta u }[/math]

[math]\displaystyle{ \ddot u=c^2\Delta u }[/math]

where Δ is the Laplace operator.

where is the Laplace operator.

其中，Δ 代表拉普拉斯算子。

## Classification

分类

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，在不同的领域有不同的类型。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

一阶方程

### Linear equations of second order

二阶线性方程

Assuming *u _{xy}* =

*u*, the general linear second-order PDE in two independent variables has the form

_{yx}Assuming u_{yx}}}, the general linear second-order PDE in two independent variables has the form

假设 *u _{xy}* =

*u*，含有两个独立变量的一般线性二阶偏微分方程具有如下形式：

_{yx}

- [math]\displaystyle{ Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0, }[/math]

```
[math]\displaystyle{ Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0, }[/math]
```

where the coefficients A, B, C... may depend upon x and y. If *A*^{2} + *B*^{2} + *C*^{2} > 0 over a region of the xy-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 A, B, C... 一般取决于 x 和 y 。如果在 xy-平面的一个区域上 *A*^{2} + *B*^{2} + *C*^{2} > 0，偏微分方程是二阶的，这种形式类似于圆锥截面的方程:

- [math]\displaystyle{ Ax^2 + 2Bxy + Cy^2 + \cdots = 0. }[/math]

```
[math]\displaystyle{ Ax^2 + 2Bxy + Cy^2 + \cdots = 0. }[/math]
```

More precisely, replacing ∂_{x} by 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.

更准确地说，用 X 替换 ∂_{x}，并同样替换其它变量(通常由傅里叶变换完成)，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)对于偏微分方程的分类最为重要。

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant *B*^{2} − 4*AC*, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by *B*^{2} − *AC* due to the convention of the xy term being 2*B* rather than B; formally, the discriminant (of the associated quadratic form) is (2*B*)^{2} − 4*AC* = 4(*B*^{2} − *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^{2} − AC)}}, with the factor of 4 dropped for simplicity.

正如人们可以根据判别式 *B*^{2} − 4*AC* 将圆锥截面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 *B*^{2} − 4*AC* 是根据交叉项的系数2*B* 而不是 B 给出的，形式上，判别式(关联二次型)是 (2*B*)^{2} − 4*AC* = 4(*B*^{2} − *AC*)，为简单起见，去掉了因子4。

*B*^{2}−*AC*< 0 (*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*x*< 0.

(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 .

*B*^{2} − *AC* < 0 (椭圆型微分方程)：在定义方程和解的区域内部，椭圆型偏微分方程的解在系数允许的程度内光滑。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型偏微分方程近似，欧拉-特里科米方程在 *x* < 0 时是椭圆型偏微分方程。

*B*^{2}−*AC*= 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*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}}.

*B*^{2} − *AC* = 0（抛物型偏微分方程）：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-特里科米方程在特征线 *x* = 0 上是抛物线型的。

*B*^{2}−*AC*> 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*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 .

*B*^{2} − *AC* > 0 (双曲型偏微分方程)：双曲型方程在初始数据中保留了函数或导数的任何不连续性。波动方程就是其中的一个例子。超音速流体的运动可以用双曲型偏微分方程近似，欧拉-特里科米方程在 *x* > 0 时是双曲型的。

If there are n independent variables *x*_{1}, *x*_{2 },… *x*_{n}, 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 n 个自变量 *x*_{1}, *x*_{2 },… *x*_{n}，一般二阶线性偏微分方程的形式是：

- [math]\displaystyle{ 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]\displaystyle{ 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 *a*_{i,j}.

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: 特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.

抛物线形方程:除了一个为零值，特征值全部为正或全部为负。

- 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).

超双形方程: 存在多于一个正特征值和多于一个的负特征值，且不存在零特征值。对于超双曲方程，只存在有限的理论(Courant 和 Hilbert，1962)。

### 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 u is now a vector with m components, and the coefficient matrices A_{ν} are m by m matrices for *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

偏微分方程的分类可以推广到一阶方程组，其中未知量 u 是有 m 个分量的向量。对于 *ν* = 1, 2,… *n*，系数矩阵 A_{ν} 是 m × m 的矩阵。偏微分方程形式如下：

- [math]\displaystyle{ Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0, }[/math]

```
[math]\displaystyle{ Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0, }[/math]
```

where the coefficient matrices A_{ν} and the vector B may depend upon x and u. If a hypersurface S 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 A_{ν} 和向量 B 可能依赖于 x 和 u。如果超曲面是以以下隐式形式给出的，

- [math]\displaystyle{ \varphi(x_1, x_2, \ldots x_n)=0, }[/math]

```
[math]\displaystyle{ \varphi(x_1, x_2, \ldots x_n)=0, }[/math]
```

where φ has a non-zero gradient, then S is a **characteristic surface** for the operator L 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:

其中，存在一个非零的梯度 φ，对于在给定点上特征形式消失的算子，特征曲面 S 形式如下:

- [math]\displaystyle{ 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]\displaystyle{ 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]
```

The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation *do not* determine the normal derivative of u on S, then the surface is **characteristic**, and the differential equation restricts the data on S: the differential equation is *internal* to S.

The geometric interpretation of this condition is as follows: if data for are prescribed on the surface , then it may be possible to determine the normal derivative of on from the differential equation. If the data on and the differential equation determine the normal derivative of on , then is non-characteristic. If the data on and the differential equation do not determine the normal derivative of on , then the surface is characteristic, and the differential equation restricts the data on : the differential equation is internal to .

这个条件的几何解释如下: 如果关于 u 的数据是在曲面 S 上规定的，那么就有可能依据微分方程确定曲面 S 上 u 的法向导数。如果曲面 S 上的数据和上面的微分方程能确定曲面 S 上 u 的法向导数，那么它就是非特征的。如果曲面 S 上的数据和上面的微分方程不能确定曲面 S 上 u 的法向导数，那么曲面是特征的，并且微分方程将数据限制在曲面 S 上：微分方程是在曲面 S 内部。

- A first-order system
*Lu*= 0 is*elliptic*if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.

A first-order system 0}} is elliptic if no surface is characteristic for : the values of on and the differential equation always determine the normal derivative of on .

如果对于 L 没有曲面是特征的，则一阶系统 *Lu* = 0 是椭圆形的：u在 S 的值和微分方程总能够决定 S 上 u 的法向导数。

- A first-order system is
*hyperbolic*at a point if there is a**spacelike**surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation*Q*(*λξ*+*η*) = 0 has m real roots*λ*_{1},*λ*_{2},…*λ*_{m}. The system is**strictly hyperbolic**if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form*Q*(*ζ*) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has m sheets, and the axis*ζ*=*λξ*runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.

A first-order system is hyperbolic at a point if there is a spacelike surface with normal at that point. This means that, given any non-trivial vector orthogonal to , and a scalar multiplier , the equation 0}} has real roots . The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has sheets, and the axis λξ}} runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.

如果在该点存在一个法向量为 ξ的 **类空曲面 Spacclike Surface S ，则一阶系统在那一点是双曲的。这意味着，给定任意正交于 ξ 的非平凡向量 η 和一个标量乘子 λ，方程 Q(λξ + η) = 0 有 m 个实根 λ_{1}, λ_{2},… λ_{m}。如果这些根始终不同，则该系统是严格双曲形的。这个条件的几何解释如下: 特征形式 Q(ζ) = 0 定义了一个具有齐次坐标 ζ的圆锥(法线圆锥)。在双曲形的情况下，这个圆锥体有 m 层，并且轴 ζ = λξ 在这些层中运动: 它不与任何一层相交。但是当从原点偏离η时，这条轴线与每一层都相交。在椭圆形的情况下，法向圆锥没有实层。**

==Yuling（讨论） sheet 这个单词也不很理解，我直译为了“层”

### Equations of mixed type

混合型方程

If a PDE has coefficients that are not constant, it is possible that it will not belong to any of these categories but rather be of **mixed type**. A simple but important example is the Euler–Tricomi equation

If a PDE has coefficients that are not constant, it is possible that it will not belong to any of these categories but rather be of mixed type. A simple but important example is the Euler–Tricomi equation

如果偏微分方程有非常数的系数，那么它可能不属于这些类别中的任何一个，而是属于混合型。一个简单但重要的例子是欧拉-特里科米方程：

- [math]\displaystyle{ u_{xx} = xu_{yy}, }[/math]

```
[math]\displaystyle{ u_{xx} = xu_{yy}, }[/math]
```

which is called **elliptic-hyperbolic** because it is elliptic in the region *x* < 0, hyperbolic in the region *x* > 0, and degenerate parabolic on the line *x* = 0.

which is called elliptic-hyperbolic because it is elliptic in the region , hyperbolic in the region , and degenerate parabolic on the line 0}}.

它在 *x* < 0 的区域上是椭圆形，在 *x* > 0 区域上是双曲形，在 *x* = 0这条线上是退化为抛物线形，因此称之为椭圆-双曲型。

fill in: Dirichlet and Neumann boundaries, hyperbolic/parabolic/elliptic separation of variables, Fourier analysis, Green's functions ...-->

填写: Dirichlet 和 Neumann 边界，双曲 / 抛物线 / 椭圆分离变量法，傅立叶分析，Green 函数... --

### Infinite-order PDEs in quantum mechanics

量子力学中的无限阶偏微分方程

In the phase space formulation of quantum mechanics, one may consider the quantum Hamilton's equations for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of ħ. The evolution equation of the c is also an infinite-order PDE. The quantum trajectories are quantum characteristics, with the use of which one could calculate the evolution of the Wigner function.

In the phase space formulation of quantum mechanics, one may consider the quantum Hamilton's equations for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of Dirac constant|. The evolution equation of the Wigner function is also an infinite-order PDE. The quantum trajectories are quantum characteristics, with the use of which one could calculate the evolution of the Wigner function.

在量子力学中的相空间表述中，我们可以考虑求解量子粒子的轨迹的量子哈密顿的方程。这些方程是无限阶偏微分方程。然而，在半经典展开中，我们在给定ħ阶数下有一个有限的常微分方程组。**维格纳函数 Wigner Function的演化方程也是一个无限阶偏微分方程。量子轨道具有量子特性，通常可以用来计算维格纳函数的演化。**

## Analytical solutions

解析解

### Separation of variables

分离变量法

Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a characteristic of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is *the* solution (this also applies to ODEs). We assume as an ansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.^{[2]}

Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a characteristic of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is the solution (this also applies to ODEs). We assume as an ansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.

线性偏微分方程组可以通过分离变量法简化为常微分方程组。这种方法依赖于微分方程解的一个特性: 如果能找到任何一个满足方程和边界条件的解，那么这个解就是方程的解(这也适用于常微分方程)。我们假设解对空间和时间的依赖可以写成对它们每一项的依赖的乘积，然后再看否可以用来解决问题。

In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.

In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.

在分离变量法方法中，可以将偏微分方程简化为含有更少变量的偏微分方程，如果只有一个变量，那么就变成了一个**常微分方程 Ordinary Differential Equation**--，这些方程也更容易求解。

This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.

This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed " as a coordinate, each coordinate can be understood separately.

对于简单的偏微分方程(称为可分离偏微分方程)来说，这是可能的，而且方程通常定义在一个矩形区域(区间的积)上。可分离偏微分方程对应于对角线矩阵——以“固定值”为坐标，每个坐标可分开理解。

This generalizes to the method of characteristics, and is also used in integral transforms.

This generalizes to the method of characteristics, and is also used in integral transforms.

这种方法可以推广到特征曲线法，也可以用于积分变换。

### Method of characteristics

特征曲线法

In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.

In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.

在特殊情况下，可以找到一些特征曲线，在这些曲线上方程可以变成一个常微分方程——意味着改变坐标从而使这些曲线变直从而达到分离变量的目的，这就是所谓的特征曲线法。

More generally, one may find characteristic surfaces.

More generally, one may find characteristic surfaces.

更一般地说，人们可能会找到特征表面。

### Integral transform

积分变换

An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.

An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.

积分变换可以将偏微分方程转换为更简单的偏微分方程，特别是可分离的偏微分方程。这对应于对角化算符。

An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.

An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.

这方面的一个重要例子是**傅里叶分析 Fourier Analysis**，它使用正弦波的特征基来对角化热方程。

If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.

If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.

如果区域是有限的或周期性的，那么解为无限和的形式是恰当的，例如傅里叶级数，但是一个解的积分，例如傅立叶积分，一般是在无限区域上的。上面给出的热传导方程的点源解法就是使用傅里叶积分的一个例子。

### Change of variables

变量代换

Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes PDE

Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes PDE

偏微分方程通常可以通过变量的适当变化，用已知的解简化为更简单的形式。例如，布莱克-舒尔斯偏微分方程

- [math]\displaystyle{ \frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 }[/math]

[math]\displaystyle{ \frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 }[/math]

is reducible to the heat equation

is reducible to the heat equation

可以简化为热传导方程

- [math]\displaystyle{ \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2} }[/math]

[math]\displaystyle{ \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2} }[/math]

by the change of variables (for complete details see Solution of the Black Scholes Equation，存档于-{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-（存檔日期 April 11, 2008）)

by the change of variables (for complete details see )

通过变量替换(完整的细节见)

- [math]\displaystyle{ \begin{align} V(S,t) &= K v(x,\tau),\\[5px] x &= \ln\left(\tfrac{S}{K} \right),\\[5px] \tau &= \tfrac{1}{2} \sigma^2 (T - t),\\[5px] v(x,\tau)&=e^{-\alpha x-\beta\tau} u(x,\tau). \end{align} }[/math]

### Fundamental solution

基本解

Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source), then taking the convolution with the boundary conditions to get the solution.

Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source), then taking the convolution with the boundary conditions to get the solution.

非齐次方程(常系数偏微分方程)一般是通过先求出基本解(点源的解) ，然后利用带边界条件的卷积来求解。

This is analogous in signal processing to understanding a filter by its impulse response.

This is analogous in signal processing to understanding a filter by its impulse response.

这类似于在信号处理中通过滤波器的脉冲响应来理解滤波器。

### Superposition principle

叠加原理

The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example sin *x* + sin *x* = 2 sin *x*. The same principle can be observed in PDEs where the solutions may be real or complex and additive. superposition

The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example 2 sin x}}. The same principle can be observed in PDEs where the solutions may be real or complex and additive. superposition

**叠加原理 Superposition Principle适用于任何线性系统，包括偏微分方程的线性系统。这个概念的一个常见的可视化是两个同相位的波相互作用结合在一起会产生更大的振幅，例如 sin x + sin x = 2 sin x。在偏微分方程中也可以观察到同样的原理，其中的解可能是真实的或复杂的和可加的。叠加**

If *u*_{1} and *u*_{2} are solutions of linear PDE in some function space R, then *u* = *c*_{1}*u*_{1} + *c*_{2}*u*_{2} with any constants *c*_{1} and *c*_{2} are also a solution of that PDE in the same function space.

If and are solutions of linear PDE in some function space , then c_{1}u_{1} + c_{2}u_{2}}} with any constants and are also a solution of that PDE in the same function space.

若线性偏微分方程在某个函数空间R中有解*u*_{1} 和 *u*_{2}，则*u* = *c*_{1}*u*_{1} + *c*_{2}*u*_{2}也是该偏微分方程在同一函数空间中的解，其中*c*_{1} 和 *c*_{2} 是任意常数。

### Methods for non-linear equations

非线性方程的求解方法

There are no generally applicable methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation.

There are no generally applicable methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation.

对于非线性偏微分方程，目前还没有普遍适用的求解方法。然而，通常是可能知道解的存在性和唯一性(如柯西-科瓦列夫斯基定理)，也是可能得到解的重要定性和定量性质的证明(得到这些结果是分析的主要部分)。非线性偏微分方程的计算解，即分步法，对一些特定的方程适用，比如非线性**薛定谔方程 Schrödinger equation**。

Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

Nevertheless, some techniques can be used for several types of equations. The -principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

然而，一些技巧可以用于几种类型的方程。h-原理是求解欠定方程最有效的方法。里基尔-珍妮特理论是一种可以获得许多解析超定系统信息的有效方法。

The method of characteristics can be used in some very special cases to solve partial differential equations.

The method of characteristics can be used in some very special cases to solve partial differential equations.

特征线法可用于求解某些特殊情况下的偏微分方程。

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

在某些情况下，偏微分方程可以通过扰动分析来求解。在扰动分析中，通常是求解将具有已知解的方程修正后的新得到方程。可供选择的数值分析技术从简单的差分格式到更成熟的多重网格和有限元方法。许多有趣的科学和工程问题都是在计算机上用这种方法解决的，有时是高性能超级计算机。

### Lie group method

李群方法

From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

从1870年起，索弗斯·李的工作为微分方程理论奠定了一个较为令人满意的基础。他指出，通过引入现在所谓的李群，老一辈数学家的积分理论可以引用到一个共同的来源; 承认相同的无穷小变换的常微分方程在积分方面存在相当的困难。他还强调了接触的转变这一主题。

==Yuling（讨论）这句话“He also emphasized the subject of transformations of contact.”中的transformations of contact翻译的不太好。

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.

求解偏微分方程的一般方法是利用微分方程的对称性，即解到解的连续无穷小变换(李理论)。连续群论、李代数和微分几何理论被用来理解生成可积方程的线性和非线性偏微分方程的结构，找到它的Lax对、递归算子、 贝克伦德变换，最后找到偏微分方程的精确解析解。

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

对称方法已被公认可以用于研究出现在数学，物理，工程和许多其他学科的微分方程。

### Semianalytical methods

半解析方法

The Adomian decomposition method, the Lyapunov artificial small parameter method, and He's homotopy perturbation method are all special cases of the more general homotopy analysis method. These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory, thus giving these methods greater flexibility and solution generality.

The Adomian decomposition method, the Lyapunov artificial small parameter method, and He's homotopy perturbation method are all special cases of the more general homotopy analysis method. These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory, thus giving these methods greater flexibility and solution generality.

阿多米安分解法、李雅普诺夫人工小参数方法和何同伦摄动方法都是更一般的同伦分析方法的特殊情况。除了李雅普诺夫方法之外，这些都是级数展开方法，与为人熟知的摄动理论方法相比，它们与小的物理参数无关，因此这些方法具有更大的灵活性和解的通用性。

## Numerical solutions

数值解

The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called Meshfree methods, which were made to solve problems where the before mentioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), Element-Free Galerkin Method (EFGM), Interpolating Element-Free Galerkin Method (IEFGM), etc.

The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called Meshfree methods, which were made to solve problems where the before mentioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), Element-Free Galerkin Method (EFGM), Interpolating Element-Free Galerkin Method (IEFGM), etc.

求解偏微分方程最常用的三种数值方法是有限元分析法、有限体积法和有限差分法，以及其他一些称为无网格法的方法，用于解决前面用提到的方法求解受到限制的问题。在这些方法中，有限元方法，尤其是高效的高阶有限元方法（hp-FEM），占有重要地位。其他有限元法和无网格法的混合形式包括广义有限元分析法(GFEM)、扩展有限元分析法(XFEM)、谱有限元分析法(SFEM)、无网格有限元分析法(DGFEM)、间断伽辽金有限元分析法(DGFEM)、无单元伽辽金法(EFGM)、插值无单元伽辽金法(IEFGM)等。

### Finite element method

有限元分析法

The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.

The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.

**有限元分析法 Finite Element Method** ((其实际应用通常被称为有限元分析(FEA))是一种寻找偏微分方程(PDE)和积分方程近似解的数值技术。这种求解方法要么基于完全消除微分方程(稳态问题) ，要么将偏微分方程转化为常微分方程的近似系统，然后使用标准技术进行数值积分，如欧拉方法、 Runge-Kutta 等。

### Finite difference method

有限差分法

Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.

Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.

有限差分法是一种数值方法，用差分方程近似倒数来近似微分方程的解。

### Finite volume method

有限体积法

Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.

Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.

类似于有限差分法或有限元分析，函数值是在网格化的几何体上的离散位置进行计算的。“有限体积”是指网格上每个节点周围的小体积。在有限体积法中，偏微分方程中含有散度项的面积分用散度定理积分转换成体积分。然后将这些项用于估算每个有限体积表面上的通量。由于进入给定体积元的通量与转移出相邻体积元的通量相同，所以这些方法通过设计保证了质量守恒。

## See also

请参阅

狄利克雷边界条件

射流丛

微分方程中的拉普拉斯变换

动力学系统列表和微分方程主题

矩阵微分方程

诺伊曼边界条件

数值偏微分方程

偏微分代数方程

递推关系

罗宾边界条件

随机过程和边值问题

## The energy method

能量法

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.<ref>Wazwaz, Abdul-Majid (2009). *High Order Difference Methods for Time Dependent PDE*

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.<ref>Adomian, G. (1994). [https://books.google.com/books?id=UKPqCAAAQBAJ&printsec=frontcover#v=onepage&q=%22partial%20differential%22&f=false *High Order Difference Methods for Time Dependent PDE*

能量法是一种数学过程，可用于验证初始边界值问题的适定性。

In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by

In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by

在下面的示例中，将使用能量法决定应在何处施加哪些边界条件，以使得到的IBVP处于适当位置。考虑下式给出的一维双曲PDE

- [math]\displaystyle{ \frac{\partial u}{\partial t} + \alpha \frac{\partial u}{\partial x} = 0, \quad x \in [a,b], \operatorname t \gt 0, }[/math]

```
[math]\displaystyle{ \frac{\partial u}{\partial t} + \alpha \frac{\partial u}{\partial x} = 0, \quad x \in [a,b], \operatorname t \gt 0, }[/math]
```

where [math]\displaystyle{ \alpha \neq 0 }[/math] is a constant and [math]\displaystyle{ u(x,t) }[/math] is an unknown function with initial condition [math]\displaystyle{ u(x,0) = f(x) }[/math]. Multiplying with [math]\displaystyle{ u }[/math] and integrating over the domain gives

where [math]\displaystyle{ \alpha \neq 0 }[/math] is a constant and [math]\displaystyle{ u(x,t) }[/math] is an unknown function with initial condition [math]\displaystyle{ u(x,0) = f(x) }[/math]. Multiplying with [math]\displaystyle{ u }[/math] and integrating over the domain gives

其中 [math]\displaystyle{ \alpha \neq 0 }[/math] 是常数，并且 [math]\displaystyle{ u(x,t) }[/math] 是初始条件是 [math]\displaystyle{ u(x,0) = f(x) }[/math]的未知函数，乘以 [math]\displaystyle{ u }[/math] 并在域上进行积分。

- [math]\displaystyle{ \int_a^b u \frac{\partial u}{\partial t} \operatorname dx + \alpha \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = 0. }[/math]

```
[math]\displaystyle{ \int_a^b u \frac{\partial u}{\partial t} \operatorname dx + \alpha \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = 0. }[/math]
```

Using that

Using that

利用这一点

- [math]\displaystyle{ \int _a ^b u \frac{\partial u}{\partial t} \operatorname dx = \frac{1}{2} \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \quad \text{and} \quad \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = \frac{1}{2} u(b,t)^2 - \frac{1}{2} u(a,t)^2,. }[/math]

```
[math]\displaystyle{ \int _a ^b u \frac{\partial u}{\partial t} \operatorname dx = \frac{1}{2} \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \quad \text{and} \quad \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = \frac{1}{2} u(b,t)^2 - \frac{1}{2} u(a,t)^2,. }[/math]
```

where integration by parts has been used for the second relationship, we get

where integration by parts has been used for the second relationship, we get

第二个关系中采用了分部积分法，我们可以得到

- [math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 + \alpha u(b,t)^2 - \alpha u(a,t)^2 = 0. }[/math]

```
[math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 + \alpha u(b,t)^2 - \alpha u(a,t)^2 = 0. }[/math]
```

Here [math]\displaystyle{ \vert \vert \cdot \vert \vert }[/math] denotes the standard L2-norm.

Here [math]\displaystyle{ \vert \vert \cdot \vert \vert }[/math] denotes the standard L2-norm.

在这里，[math]\displaystyle{ \vert \vert \cdot \vert \vert }[/math] 表示标准的L2-正则。

For well-posedness we require that the energy of the solution is non-increasing, i.e. that [math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0 }[/math], which is achieved by specifying [math]\displaystyle{ u }[/math] at [math]\displaystyle{ x = a }[/math] if [math]\displaystyle{ \alpha \gt 0 }[/math] and at [math]\displaystyle{ x = b }[/math] if [math]\displaystyle{ \alpha \lt 0 }[/math]. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that [math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0 }[/math] holds when all data is set to zero.

For well-posedness we require that the energy of the solution is non-increasing, i.e. that [math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0 }[/math], which is achieved by specifying [math]\displaystyle{ u }[/math] at [math]\displaystyle{ x = a }[/math] if [math]\displaystyle{ \alpha \gt 0 }[/math] and at [math]\displaystyle{ x = b }[/math] if [math]\displaystyle{ \alpha \lt 0 }[/math]. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that [math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0 }[/math] holds when all data is set to zero.

对于适定性，我们要求解的能量是不增加的，即 [math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0 }[/math] ，这种关系可以通过在[math]\displaystyle{ x = a }[/math]处 （如果 [math]\displaystyle{ \alpha \gt 0 }[/math]） 以及 [math]\displaystyle{ x = b }[/math]处 （如果 [math]\displaystyle{ \alpha \lt 0 }[/math]）指定[math]\displaystyle{ u }[/math]的值来实现。这相当于只在流入处附加边界条件。注意，适定性允许在数据项(初始和边界)上的增长，因此它足以表明当所有数据设置为零时应有 [math]\displaystyle{ \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0 }[/math]。

## Notes

- ↑ Sciavicco, Lorenzo; Siciliano, Bruno (2001-02-19) (in en).
*Modelling and Control of Robot Manipulators*. Springer Science & Business Media. ISBN 9781852332211. https://books.google.com/books?id=v9PLbcYd9aUC&pg=PA32. - ↑ Gershenfeld, Neil (2000).
*The nature of mathematical modeling*(Reprinted (with corr.) ed.). Cambridge: Cambridge Univ. Press. p. 27. ISBN 0521570956.

## References

- Adomian, G. (1994).
*Solving Frontier problems of Physics: The decomposition method*. Kluwer Academic Publishers. https://books.google.com/books?id=UKPqCAAAQBAJ&printsec=frontcover#v=onepage&q=%22partial%20differential%22&f=false.*]. Kluwer Academic Publishers. https://books.google.com/books?id=UKPqCAAAQBAJ&printsec=frontcover#v=onepage&q=%22partial%20differential%22&f=false.*

1994年*. Higher Education Press. ISBN 978-3-642-00251-9.*

| first Abdul-Majid | last Wazwaz | publisher Higher Education Press | year 2009 | isbn 978-3-642-00251-9 | page }

- Wazwaz, Abdul-Majid (2002).
*Partial Differential Equations Methods and Applications*. A.A. Balkema. ISBN 90-5809-369-7.

|first=Abdul-Majid|last=Wazwaz|publisher=A.A. Balkema|year=2002|isbn=90-5809-369-7|page=}}

| first Abdul-Majid | last Wazwaz | publisher a.a.Balkema | year 2002 | isbn 90-5809-369-7 | page }

- Zwillinger, D. (1997),
*Handbook of Differential Equations*(3rd ed.), Boston: Academic Press, ISBN 0-12-784395-7.

- Gershenfeld, N. (1999),
*The Nature of Mathematical Modeling*(1st ed.), New York: Cambridge University Press, New York, NY, USA, ISBN 0-521-57095-6.

- Krasil'shchik, I.S. & Vinogradov, A.M., Eds. (1999),
*Symmetries and Conserwation Laws for Differential Equations of Mathematical Physics*, American Mathematical Society, Providence, Rhode Island, USA, ISBN 0-8218-0958-X.

- Krasil'shchik, I.S.; Lychagin, V.V. & Vinogradov, A.M. (1986),
*Geometry of Jet Spaces and Nonlinear Partial Differential Equations*, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, ISBN 2-88124-051-8.

- Vinogradov, A.M. (2001),
*Cohomological Analysis of Partial Differential Equations and Secondary Calculus*, American Mathematical Society, Providence, Rhode Island, USA, ISBN 0-8218-2922-X.

- Gustafsson, Bertil (2008).
*High Order Difference Methods for Time Dependent PDE*. Springer. doi:10.1007/978-3-540-74993-6. ISBN 978-3-540-74992-9.

|first=Bertil|last=Gustafsson|publisher=Springer|year=2008|isbn=978-3-540-74992-9|doi=10.1007/978-3-540-74993-6}}

| first Bertil | last Gustafsson | Springer | year 2008 | isbn 978-3-540-74992-9 | doi 10.1007 / 978-3-540-74993-6}

## Further reading

- Cajori, Florian (1928). "The Early History of Partial Differential Equations and of Partial Differentiation and Integration" (PDF).
*The American Mathematical Monthly*.**35**(9): 459–467. doi:10.2307/2298771. Archived from the original (PDF) on 2018-11-23. Retrieved 2016-05-15.

## External links

- Partial Differential Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.

- Partial Differential Equations: Index at EqWorld: The World of Mathematical Equations.

- Partial Differential Equations: Methods at EqWorld: The World of Mathematical Equations.

- Example problems with solutions at exampleproblems.com

- Partial Differential Equations at mathworld.wolfram.com

- Partial Differential Equations with Mathematica

- Partial Differential Equations in Cleve Moler: Numerical Computing with MATLAB

- Partial Differential Equations at nag.com

- Sanderson, Grant (April 21, 2019). "But what is a partial differential equation?".
*3Blue1Brown*– via YouTube.

模板:Differential equations topics

Category:Multivariable calculus

类别: 多元微积分

Category:Differential equations

类别: 微分方程

Category:Concepts in physics

分类: 物理概念

This page was moved from wikipedia:en:Partial differential equation. Its edit history can be viewed at 偏微分方程/edithistory