偏微分方程

此词条暂由彩云小译翻译，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

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

二维热传导方程解的可视化，图中第三个维度表示温度的大小。

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，也可以通过创建计算机模型来求解。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

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.

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，而流体的形状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位形空间中，流体的动力学过程发生在无限维位形空间中。这种区别通常使偏微分方程比常微分方程更难求解，但是在这里，线性问题也有简单的解。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(x₁,… x_n) is an equation of the form

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

对于函数u(x₁,… x_n)，它的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]

[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是函数及其导数的线性方程，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是：

[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

这个关系意味着函数是独立于x的。然而，这个方程没有给出关于函数和变量相关性的信息。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.

这里的c是常量。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; 附加条件一般必须在定义解的区域边界上定义。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

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.

虽然常微分方程解的存在唯一性问题用 Picard-lindel f 定理得到了令人满意的结果，但这与偏微分方程的情形相去甚远。柯西-科瓦列夫斯基定理指出，对于任何系数在未知函数及其导数中是解析的偏微分方程，柯西问题有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，但是有一些线性偏微分方程的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，它仍然可能具有不希望的性质。这些问题的数学研究通常是在更强大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

病态行为的一个例子是拉普拉斯方程的 Cauchy 问题的序列(取决于)

[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]

部分 x ^ 2} + 部分 y ^ 2}0，/ math

with boundary conditions

with boundary conditions

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]

Math begin u (x，0) & 0， frac partial u }(x，0) & frac { sin 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

哪里是整数。关于零的导数一致地随着增加而趋于零，但是解是

[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]

Math u (x，y) frac sinh ny sinn ^ 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.

如果不是任何非零值的整数倍，这个解决方案接近无穷大。拉普拉斯方程的 Cauchy 问题被称为不适定或不适定问题，因为该问题的解并不连续地依赖于该问题的数据。这种不适定问题在物理应用中通常不能令人满意。

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.

偏微分方程方程的解的存在性是千禧年大奖难题方程的一部分。

Notation

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。这就是:

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

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

数学，部分数学，部分数学

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

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

数学 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]

数学 u (xy) frac (frac) partial y ，部分 x frac (frac) partial y 左(frac)部分 x 右)。数学

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(下面提到的)可以写成

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

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

数学，数学，数学，数学

or

or

或

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

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

数学 ddot u ^ 2 Delta u / math

where Δ is the Laplace operator.

where is the Laplace operator.

拉普拉斯算子在哪里。

Classification

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。

Equations of first order

Main article: First-order partial differential equation

Linear equations of second order

Assuming u_xy = u_yx, the general linear second-order PDE in two independent variables has the form

Assuming u_yx}}, 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

[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]

数学 Au { xx } + 2 bu { xy } + Cu {广州欢聚时代} + cdots mbox {(低阶项)}0，/ math

where the coefficients A, B, C... may depend upon x and y. If A² + B² + C² > 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，，... 可能取决于和。如果在-平面的一个区域上，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等式:

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

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

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，替换，同样对于其他变量(从形式上来说，这是由傅里叶变换来完成的) ，将一个常系数 PDE 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)对于分类是最重要的。

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B² − 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B² − AC due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)² − 4AC = 4(B² − 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(B² − AC)}}, with the factor of 4 dropped for simplicity.

正如人们可以根据判别式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是根据项为而不是的约定给出的，形式上，判别式(关联二次型)是4(b sup 2 / sup-AC)} ，为简单起见，去掉了4因子。

1. B² − 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 .

(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。

1. B² − 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}}.

在每一点上都是抛物线型的方程可以通过改变自变量而转化成类似于热方程的形式。抛物偏微分方程。随着转换后的时间变量的增加，解决方案变得平滑。欧拉-特里科米方程在0}处具有抛物型。

1. B² − 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 .

(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。

If there are n independent variables x₁, x₂ ,… 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

如果存在自变量，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]

数学 l u sum { i } ^ n sum { j 1} ^ n a { i，j } frac { partial ^ 2 u } quad { 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 .

这种分类依赖于系数矩阵特征值的签名。

1. Elliptic: the eigenvalues are all positive or all negative.

Elliptic: the eigenvalues are all positive or all negative.

椭圆: 本征值全部为正或全部为负。

1. 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.

抛物线型: 本征值全部为正或全部为负，除了一个为零。

1. 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.

双曲型: 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。

1. 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).

超双曲: 存在多于一个正本征值和多于一个负本征值，且不存在零本征值。超双曲方程只有一个有限的理论(Courant 和 Hilbert，1962)。

Systems of first-order equations and characteristic surfaces

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

偏微分方程组的分类可以推广到一阶方程组，其中未知量是分量向量，系数矩阵是1,2，... n }的矩阵。美国偏微分方程协会采取的形式

[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]

数学 Lu nu 1 ^ n } a nu frac (部分 u) + b0，/ 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

其中系数矩阵和向量可能依赖于。如果超曲面是以隐式形式给出的

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

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

Math varphi (x1，x2，ldots xn)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:

如果有一个非零的梯度，那么如果特征形式消失，则在给定点上算子的特征曲面:

[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]

数学 q 左(部分 x 1) ，左(部分 x 1) ，左(部分 x 1) ，/ 数学

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: 如果数据是在表面上规定的，那么就有可能确定法向导数的微分方程。如果上面的数据和上面的微分方程确定了 on 的正常导数，那么它就是非特征的。如果上面的数据和微分方程的数据不能确定 on 的法向导数，那么表面是特征的，微分方程的数据限制在: 微分方程是内部的。

1. 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 .

如果没有表面具有以下特征，则一阶系统0}是椭圆的: on 和微分方程的值总是决定 on 的法向导数。

1. 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 λ₁, λ₂,… 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.

一阶系统在某一点是双曲的，如果在该点有一个法线的类空曲面的话。这意味着，给定任意正交于的非平凡向量和一个标量乘子，方程0}有实根。如果这些根始终是不同的，则该系统是严格双曲型的。这个条件的几何解释如下: 特征形式0}定义了一个具有齐次坐标的圆锥体(法线圆锥体)。在双曲线的情况下，这个圆锥体有工作表，并且轴}在这些工作表中运行: 它不与工作表中的任何一个相交。但是当从原点偏离时，这条轴线与每一片都相交。在椭圆情况下，法锥没有实片。

Equations of mixed type

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 PDE 的系数不是常数，那么它可能不属于这些类别中的任何一个，而是属于混合类型。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]

数学广州欢聚时代 / 数学

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}}.

它在区域上是椭圆型，在区域上是双曲型，在线上是退化抛物型，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，傅立叶变换家族中的关系，Green 函数... --

Infinite-order PDEs in quantum mechanics

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 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.

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《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

Separation of variables

Main article: Separable partial differential equation

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.

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的重要技术来简化为常微分方程组。这种技巧依赖于微分方程解的一个特征: 如果一个人能找到任何解决方程并满足边界条件的解，那么它就是解(这也适用于常微分方程)。作为一个假设，我们假设解对参数空间和时间的依赖可以写成每个项依赖于一个参数的乘积，然后看看这是否可以用来解决这个问题。

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，一个变量可以将偏微分方程简化为偏微分方程，如果只有一个变量，那么这个变量就是一个常微分方程-- 反过来，这些变量也更容易求解。

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.

对于简单偏微分方程(称为可分离偏微分方程)来说，这是可能的，而且域通常是一个矩形(区间的乘积)。可分离偏微分方程对应于对角线矩阵——以“固定值”为坐标，每个坐标可以分别理解。

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.

这种方法推广到特征值方法，也用于积分变换。

Method of characteristics

Main article: 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.

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的坐标系，从而使这些曲线变直，这就是所谓的特征分离变量法法。

More generally, one may find characteristic surfaces.

More generally, one may find characteristic surfaces.

更一般地说，人们可能会发现特征表面。

Integral transform

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。这对应于对角化操作符。

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立叶变换家族中的关系，它使用正弦波的特征基对角化热方程。

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，那么一个无限和的解是适当的，例如傅立叶级数，但是对于无限区域，一般需要一个解的积分，例如傅立叶积分。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

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，Black-Scholes 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]

数学部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的部分的

is reducible to the heat equation

is reducible to the heat equation

可以归结为热量方程

[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]

数学部分数学部分数学部分数学部分数学部分数学

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 )

通过变量的变化(完整的细节见)

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

\end{align}</math>

End { align } / math

Fundamental solution

Main article: Fundamental 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.

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，通常求解)通常是先求出基本解(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

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。这个概念的一个共同的可视化是两个相位波的相互作用结合在一起产生更大的振幅，例如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 u₁ and u₂ are solutions of linear PDE in some function space R, then u = c₁u₁ + c₂u₂ with any constants c₁ and c₂ are also a solution of that PDE in the same function space.

If and are solutions of linear PDE in some function space , then c₁u₁ + c₂u₂}} 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中有解，则任意常数的 c 子1 / sub u sub 1 / sub + c 子2 / sub u sub 2 / sub }也是该偏微分方程在同一函数空间中的解。

Methods for non-linear equations

See also: nonlinear partial differential 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.

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，存在性和唯一性结果(如柯西-科瓦列夫斯基定理)通常是可能的，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)也是可能的。非线性偏微分方程的计算解，即分步法，存在于一些特定的方程，比如非线性薛丁格方程式。

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。- 原理是求解欠定方程最有效的方法。Riquier-Janet 理论是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，解被认为是对具有已知解的方程的修正。可供选择的方法有数值分析技术，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用这种方法解决的，使用计算机，有时是高性能超级计算机。

Lie group method

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可比的困难。他还强调了接触的转变这一主题。

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，即解的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、 Bäcklund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.

对称方法已被公认为研究微分方程出现在数学，物理，工程和许多其他学科。

Semianalytical methods

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。这些是级数展开方法，除了李雅普诺夫方法之外，与众所周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

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、有限体积法和差分法，以及其他一些被称为无网格法的方法，这些方法都是为了解决前面提到的方法受到限制的问题。在这些方法中，有限元方法尤其是高效的高阶有限元方法占有重要地位。有限元法和无网格法的其他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)、间断有限有限元分析法(DGFEM)、无网格伽辽金法(EFGM)、插值无网格伽辽金法(IEFGM)等。

Finite element method

Main article: Finite element method

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(FEM)(其实际应用通常被称为有限元分析法(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

Main article: Finite difference method

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，用差分方程近似微分方程的解。

Finite volume method

Main article: Finite volume method

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，数值是在网状几何体的离散位置计算的。“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，含有散度项的偏微分方程中的表面积分用高斯散度定理积分转换成体积分。然后将这些项计算为每个有限体积表面的通量。由于进入给定体积的磁通量与离开相邻体积的磁通量相同，这些方法在设计上保持了质量。

See also

• Dirichlet boundary condition

• Jet bundle

• Laplace transform applied to differential equations

• List of dynamical systems and differential equations topics

• Matrix differential equation

• Neumann boundary condition

• Numerical partial differential equations

• Partial differential algebraic equation

• Recurrence relation

• Robin boundary condition

• Stochastic processes and boundary value problems

The energy method

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems. 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

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

| first Bertil | last Gustafsson | publisher | Springer | year 2008 | isbn 978-3-540-74992-9 | doi 10.1007 / 978-3-540-74993-6} / ref 在下面的例子中，使用能量方法来决定应该施加哪些边界条件，使得得到的 IBVP 适定。考虑一维双曲偏微分方程

[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]

在[ a，b ]中，操作者名为 t 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 alpha neq 0 / math 是常数，math u (x，t) / math 是未知函数，初始条件是 math u (x，0) f (x) / math。使用 math u / math 进行乘法，并在域上进行集成

[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]

部分操作名称 dx + alpha int a ^ b u frac 部分操作名称 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]

数学内部一个 ^ b u frac {1}和一个 ^ b u frac {1}和一个 ^ b u frac {1}(b，t)2-frac {1}{2} u (a，t)2,。 数学

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

第二种关系采用了部分整合，明白了吗

[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]

部分 t-vert u-vert 2 + alpha u (b，t) ^ 2-alpha u (a，t) ^ 20. / 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 vert vert cdot vert math 表示标准的 L2-norm。

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。部分 t-vert u-vert 2-leq 0 / math 通过在 math x a / math if math alpha 0 / math 和 math x b / math if math alpha 0 / math 中指定数学 u / math 来实现。这只相当于在进水口附加边界条件。注意，适定性允许在数据(初始和边界)方面的增长，因此它足以证明当所有数据设置为零时 math frac { partial t } vert u vert 2 leq 0 / math 持有。

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

• 模板:Springer

• 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.

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