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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，'''<font color = "#ff8000">偏微分方程函数 Partial Differential Equation</font>'''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。
在数学中，'''<font color = "#ff8000">偏微分方程函数 Partial Differential Equation</font>'''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。
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。
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。
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
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{{mvar|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
这个关系意味着函数是独立于{{mvar|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
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'''<font color="#ff8000">弗罗贝尼乌斯定理 Picard–Lindelöf Theorem</font>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。'''<font color="#ff8000">柯西-科瓦列夫斯基定理 Cauchy–Kowalevski theorem</font>指出，对于任何在未知函数及其导数中系数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有力的弱解的背景下进行的。
虽然常微分方程解的存在唯一性问题用'''<font color="#ff8000">弗罗贝尼乌斯定理 Picard–Lindelöf Theorem</font>得到了令人满意的结果，但应用到偏微分方程上却不尽如意。'''<font color="#ff8000">柯西-科瓦列夫斯基定理 Cauchy–Kowalevski theorem</font>指出，对于任何在未知函数及其导数中系数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，它仍然可能具有人们所不希望的性质。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。
where is an integer. The derivative of with respect to approaches zero uniformly in as 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{{mvar|n}}是整数。{{mvar|u}}关于{{mvar|y}}的导数一致地随着{{mvar|n}}的增加而趋于零，但是解是
其中{{mvar|n}}是整数。{{mvar|u}}是关于{{mvar|y}}的导数，一致地随着{{mvar|n}}的增加而趋于零，但是解是
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.
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{{mvar|y}}如果{{mvar|nx}}不是{{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问题（可以译为ill-posed或not well-posed），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。
对于任何非零的{{mvar|y}}，如果{{mvar|nx}}不是{{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问题（可以译为ill-posed或not well-posed），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.
一种偏微分方程——纳维-斯托克斯方程——的解的存在性就是千禧年大奖难题的一部分。
== Notation ==
== 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。例如:
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
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 ({{math|∇}})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
特别是在物理学中，[[del]]或nabla ({{math|∇}})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
where the coefficients , , ... may depend upon and . If over a region of the -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
where the coefficients , , ... may depend upon and . If over a region of the -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
其中的系数{{mvar|A}}, {{mvar|B}}, {{mvar|C}}... 一般取决于{{mvar|x}}和{{mvar|y}}。如果在{{mvar|xy}}-平面的一个区域上{{math|''A''<sup>2</sup> + ''B''<sup>2</sup> + ''C''<sup>2</sup> > 0}}，偏微分方程在该区域是二阶的。这种形式类似于圆锥曲线的方程:
其中的系数 {{mvar|A}}, {{mvar|B}}, {{mvar|C}}... 一般取决于{{mvar|x}}和{{mvar|y}}。如果在{{mvar|xy}}-平面的一个区域上 {{math|''A''<sup>2</sup> + ''B''<sup>2</sup> + ''C''<sup>2</sup> > 0}}，偏微分方程在该区域是二阶的。这种形式类似于圆锥曲线的方程:
More precisely, replacing by , and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
More precisely, replacing by , and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
更准确地说，用{{mvar|X}}替换{{math|∂<sub>''x''</sub>}}，对于其他变量做同样的操作(从形式上来说，这是由傅里叶变换来完成的)，将一个常系数偏微分方程转换成一个相同次数的多项式，最高次数的项(齐次多项式，这里是一个二次形式)一般会用于偏微分方程的分类。
更准确地说，用 {{mvar|X}} 替换 {{math|∂<sub>''x''</sub>}}，对于其他变量做同样的操作(从形式上来说，这是由傅里叶变换来完成的)，将一个常系数偏微分方程转换成一个相同次数的多项式，最高次数的项(齐次多项式，这里是一个二次形式)一般会用于偏微分方程的分类。
===~~ most significant for the classification 意译为用于偏微分方程的分类。 significiant直译过来感觉不太合适
===~~ most significant for the classification 意译为用于偏微分方程的分类。 significiant直译过来感觉不太合适
Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention of the term being rather than ; formally, the discriminant (of the associated quadratic form) is 4(B<sup>2</sup> − AC)}}, with the factor of 4 dropped for simplicity.
Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention of the term being rather than ; formally, the discriminant (of the associated quadratic form) is 4(B<sup>2</sup> − AC)}}, with the factor of 4 dropped for simplicity.
正如人们可以根据判别式{{math|''B''<sup>2</sup> − 4''AC''}}将圆锥曲线和二次型分为抛物型、双曲型和椭圆型一样，对于给定点的二阶偏微分方程也可以这样做。然而，偏微分方程中的判别式{{math|''B''<sup>2</sup> − 4''AC''}}是根据交叉项的系数{{math|2''B''}}而不是{{mvar|B}}给出的，形式上，判别式(关联二次型)是{{math|(2''B'')<sup>2</sup> − 4''AC'' {{=}} 4(''B''<sup>2</sup> − ''AC'')}}，为简单起见，去掉了因子4。
正如人们可以根据判别式 {{math|''B''<sup>2</sup> − 4''AC''}} 将圆锥曲线和二次型分为抛物型、双曲型和椭圆型一样，对于给定点的二阶偏微分方程也可以这样做。然而，偏微分方程中的判别式 {{math|''B''<sup>2</sup> − 4''AC''}} 是根据交叉项的系数{{math|2''B''}} 而不是 {{mvar|B}} 给出的，形式上，判别式(关联二次型)是 {{math|(2''B'')<sup>2</sup> − 4''AC'' {{=}} 4(''B''<sup>2</sup> − ''AC'')}}，为简单起见，去掉了因子4。
(elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where .
(elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where .
{{math|''B''<sup>2</sup> − ''AC'' < 0}}(椭圆形微分方程)：在定义方程和解的区域内部，椭圆型偏微分方程的解光滑到系数允许的程度。例如，拉普拉斯方程的解在它们被定义的区域内是解析的，但是解可能假设边界值是不光滑的。亚音速流体的运动可以用椭圆偏微分方程近似，其中欧拉-特里科米方程在{{math|''x'' < 0}}是椭圆型偏微分方程。
{{math|''B''<sup>2</sup> − ''AC'' < 0}} (椭圆形微分方程)：在定义方程和解的区域内部，椭圆型偏微分方程的解光滑到系数允许的程度。例如，拉普拉斯方程的解在它们被定义的区域内是解析的，但是解可能假设边界值是不光滑的。亚音速流体的运动可以用椭圆偏微分方程近似，其中欧拉-特里科米方程在 {{math|''x'' < 0}} 时是椭圆型偏微分方程。
0}} (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where 0}}.
0}} (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where 0}}.
{{math|''B''<sup>2</sup> − ''AC'' {{=}} 0}}（抛物线形偏微分方程）：在每一点上都是抛物线型的方程可以通过改变自变量从而转化成类似于热方程的形式。随着转换后的时间变量的增加，方程的解变得平滑。欧拉-特里科米方程在{{math|''x'' {{=}} 0}}特征线上是抛物线形的。
{{math|''B''<sup>2</sup> − ''AC'' {{=}} 0}}（抛物线形偏微分方程）：在每一点上都是抛物线型的方程可以通过改变自变量从而转化成类似于热方程的形式。随着转换后的时间变量的增加，方程的解变得平滑。欧拉-特里科米方程在 {{math|''x'' {{=}} 0}} 这条特征线上是抛物线形的。
(hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where .
(hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where .
{{math|''B''<sup>2</sup> − ''AC'' > 0}}(双曲形偏微分方程)：双曲形方程在初始数据中保留了函数或导数的任何不连续性。波动方程就是一个例子。超音速流体的运动可以用双曲形偏微分方程近似，其中欧拉-特里科米方程在{{math|''x'' > 0}}是双曲型的。
{{math|''B''<sup>2</sup> − ''AC'' > 0}} (双曲形偏微分方程)：双曲形方程在初始数据中保留了函数或导数的任何不连续性，波动方程就是其中的一个例子。超音速流体的运动可以用双曲形偏微分方程近似，其中欧拉-特里科米方程在 {{math|''x'' > 0}} 时是双曲型的。
If there are independent variables , a general linear partial differential equation of second order has the form
If there are independent variables , a general linear partial differential equation of second order has the form
如果存在{{mvar|n}}个自变量{{math|''x''<sub>1</sub>, ''x''<sub>2 </sub>,… ''x''<sub>''n''</sub>}}，一般二阶线性偏微分方程的形式是
如果存在 {{mvar|n}} 个自变量 {{math|''x''<sub>1</sub>, ''x''<sub>2 </sub>,… ''x''<sub>''n''</sub>}}，一般二阶线性偏微分方程的形式是
The classification depends upon the signature of the eigenvalues of the coefficient matrix .
The classification depends upon the signature of the eigenvalues of the coefficient matrix .
这种分类取决于系数矩阵本征值的符号（正负性）。
Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
超双形方程: 存在多于一个正本征值和多于一个的负本征值，且不存在零本征值。对于超双曲方程，只有一个有限理论(Courant 和 Hilbert，1962)。
超双形方程: 存在多于一个正本征值和多于一个的负本征值，且不存在零本征值。对于超双曲方程，只存在一个有限理论(Courant 和 Hilbert，1962)。
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown is now a vector with components, and the coefficient matrices are by matrices for 1, 2,… n}}. The partial differential equation takes the form
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown is now a vector with components, and the coefficient matrices are by matrices for 1, 2,… n}}. The partial differential equation takes the form
偏微分方程组的分类可以推广到一阶方程组，其中未知量{{mvar|u}}是有{{mvar|m}}个分量的向量。对于{{math|''ν'' {{=}} 1, 2,… ''n''}}，系数矩阵{{mvar|A<sub>ν</sub>}}是{{mvar|m}} × {{mvar|m}}的矩阵。偏微分方程形式如下：
偏微分方程组的分类可以推广到一阶方程组，其中未知量 {{mvar|u}} 是有 {{mvar|m}} 个分量的向量。对于 {{math|''ν'' {{=}} 1, 2,… ''n''}}，系数矩阵 {{mvar|A<sub>ν</sub>}} 是 {{mvar|m}} × {{mvar|m}} 的矩阵。偏微分方程形式如下：
where the coefficient matrices and the vector may depend upon and . If a hypersurface is given in the implicit form
where the coefficient matrices and the vector may depend upon and . If a hypersurface is given in the implicit form
其中系数矩阵{{mvar|A<sub>ν</sub>}}和向量{{mvar|B}}可能依赖于{{mvar|x}} 和 {{mvar|u}}。如果超曲面是以隐式形式给出的
其中系数矩阵 {{mvar|A<sub>ν</sub>}} 和向量 {{mvar|B}} 可能依赖于 {{mvar|x}} 和 {{mvar|u}}。如果超曲面是以隐式形式给出的
where has a non-zero gradient, then is a characteristic surface for the operator at a given point if the characteristic form vanishes:
where has a non-zero gradient, then is a characteristic surface for the operator at a given point if the characteristic form vanishes:
其中存在一个非零的梯度，那么如果特征形式消失，则在给定点上算子的特征曲面形式如下:
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 .
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 {{mvar|u}} 的数据是在曲面 {{mvar|S}} 上规定的，那么就有可能依据微分方程确定曲面 {{mvar|S}} 上 {{mvar|u}} 的法向导数。如果曲面 {{mvar|S}} 上的数据和上面的微分方程能确定曲面 {{mvar|S}} 上 {{mvar|u}} 的法向导数，那么它就是非特征的。如果曲面 {{mvar|S}} 上的数据和上面的微分方程能确定曲面 {{mvar|S}} 上 {{mvar|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 {{mvar|S}} 上：微分方程是在曲面 {{mvar|S}} 内部。
这个条件的几何解释如下: 如果关于 {{mvar|u}} 的数据是在曲面 {{mvar|S}} 上规定的，那么就有可能依据微分方程确定曲面 {{mvar|S}} 上 {{mvar|u}} 的法向导数。如果曲面 {{mvar|S}} 上的数据和上面的微分方程能确定曲面 {{mvar|S}} 上 {{mvar|u}} 的法向导数，那么它就是非特征的。如果曲面 {{mvar|S}} 上的数据和上面的微分方程不能确定曲面 {{mvar|S}} 上 {{mvar|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 {{mvar|S}} 上：微分方程是在曲面 {{mvar|S}} 内部。
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 .
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 {{mvar|L}} 没有曲面是特征的，则一阶系统 {{math|''Lu'' {{=}} 0}} 是椭圆形的：{{mvar|u}}在 {{mvar|S}} 的值和微分方程总能够决定 {{mvar|S}} 上 {{mvar|u}} 的法向导数。
# A first-order system is ''hyperbolic'' at a point if there is a '''spacelike''' surface {{mvar|S}} with normal {{mvar|ξ}} at that point. This means that, given any non-trivial vector {{mvar|η}} orthogonal to {{mvar|ξ}}, and a scalar multiplier {{mvar|λ}}, the equation {{math|''Q''(''λξ'' + ''η'') {{=}} 0}} has {{mvar|m}} real roots {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>,… ''λ''<sub>''m''</sub>}}. The system is '''strictly hyperbolic''' if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form {{math|''Q''(''ζ'') {{=}} 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has {{mvar|m}} sheets, and the axis {{math|''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 {{mvar|S}} with normal {{mvar|ξ}} at that point. This means that, given any non-trivial vector {{mvar|η}} orthogonal to {{mvar|ξ}}, and a scalar multiplier {{mvar|λ}}, the equation {{math|''Q''(''λξ'' + ''η'') {{=}} 0}} has {{mvar|m}} real roots {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>,… ''λ''<sub>''m''</sub>}}. The system is '''strictly hyperbolic''' if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form {{math|''Q''(''ζ'') {{=}} 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has {{mvar|m}} sheets, and the axis {{math|''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.
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 {{mvar|ξ}} '''<font color="#ff8000">类空曲面 Spacclike Surface</font> {{mvar|S}} ，则一阶系统在某一点是双曲的。这意味着，给定任意正交于 {{mvar|ξ}} 的非平凡向量 {{mvar|η}} 和一个标量乘子 {{mvar|λ}}，方程 {{math|''Q''(''λξ'' + ''η'') {{=}} 0}} 有 {{mvar|m}} 个实根 {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>,… ''λ''<sub>''m''</sub>}}。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 {{math|''Q''(''ζ'') {{=}} 0}} 定义了一个具有齐次坐标 ζ的圆锥(法线圆锥)。在双曲线的情况下，这个圆锥体有 {{mvar|m}} 层，并且轴 {{math|''ζ'' {{=}} ''λξ''}} 在这些层中运行: 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。在椭圆情况下，法锥没有实层。
如果在该点存在一个法向量为 {{mvar|ξ}}的 '''<font color="#ff8000">类空曲面 Spacclike Surface</font> {{mvar|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 {{mvar|ξ}} 的非平凡向量 {{mvar|η}} 和一个标量乘子 {{mvar|λ}}，方程 {{math|''Q''(''λξ'' + ''η'') {{=}} 0}} 有 {{mvar|m}} 个实根 {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>,… ''λ''<sub>''m''</sub>}}。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 {{math|''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 {{mvar|m}} 层，并且轴 {{math|''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。
which is called elliptic-hyperbolic because it is elliptic in the region , hyperbolic in the region , and degenerate parabolic on the line 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 {{math|''x'' < 0}} 的区域上是椭圆形，在 {{math|''x'' > 0}} 区域上是双曲形，在 {{math|''x'' {{=}} 0}}这条线上是退化抛物线形，因此称之为椭圆-双曲型。
它在 {{math|''x'' < 0}} 的区域上是椭圆形，在 {{math|''x'' > 0}} 区域上是双曲形，在 {{math|''x'' {{=}} 0}}这条线上是退化为抛物线形，因此称之为椭圆-双曲型。
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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.
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，我们可以考虑用于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[[Dirac constant|{{mvar|ħ}}]]阶数下有一个有限的的常微分方程组。'''<font color="#ff8000">维格纳函数 Wigner Function</font>的演化方程也是一个无限阶偏微分方程。由于量子轨道的量子特性，所以它通常可以用来计算维格纳函数的演化。
== Analytical solutions ==
== Analytical solutions ==
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这一重要方法来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征: 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)。我们假设解对参数空间和时间的依赖可以写成对它们每一项的依赖以及一个参数的乘积，然后再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，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'''<font color = "#ff8000">常微分方程 Ordinary Differential Equation'''</font>-- 反过来，这些方程也更容易求解。
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'''<font color = "#ff8000">常微分方程 Ordinary Differential Equation</font>'''-- 反过来，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 " 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(区间的乘积)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===
===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——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。
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'''<font color = "#ff8000">傅里叶分析 Fourier Analysis</font>'''，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===
===Change of variables===
变量代换
is reducible to the heat equation
is reducible to the heat equation
可以简化为热传导方程
If and are solutions of linear PDE in some function space , then c<sub>1</sub>u<sub>1</sub> + c<sub>2</sub>u<sub>2</sub>}} with any constants and 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<sub>1</sub>u<sub>1</sub> + c<sub>2</sub>u<sub>2</sub>}} 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{{mvar|R}}中有解{{math|''u''<sub>1</sub>}} 和 {{math|''u''<sub>2</sub>}}，则{{math|''u'' {{=}} ''c''<sub>1</sub>''u''<sub>1</sub> + ''c''<sub>2</sub>''u''<sub>2</sub>}}，也是该偏微分方程在同一函数空间中的解，其中{{math|''c''<sub>1</sub>}} 和 {{math|''c''<sub>2</sub>}} 是任意常数。
若线性偏微分方程在某个函数空间{{mvar|R}}中有解{{math|''u''<sub>1</sub>}} 和 {{math|''u''<sub>2</sub>}}，则{{math|''u'' {{=}} ''c''<sub>1</sub>''u''<sub>1</sub> + ''c''<sub>2</sub>''u''<sub>2</sub>}}也是该偏微分方程在同一函数空间中的解，其中{{math|''c''<sub>1</sub>}} 和 {{math|''c''<sub>2</sub>}} 是任意常数。
===Methods for non-linear equations===
===Methods for non-linear equations===
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'''<font color="#ff8000">薛定谔方程 Schrödinger equation</font>'''。
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===
===Lie group method===
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。这些方法用于解决前面提到的方法受到的限制。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。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)、无网格伽辽金法(EFGM)、插值无网格伽辽金法(IEFGM)等。
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.
'''<font color = "#ff8000">有限元分析法</font> Finite Element Method(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。
'''<font color = "#ff8000">有限元分析法 Finite Element Method</font>''' ((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===
===Finite difference 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，函数值是在网状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。由于进入给定体积的通量与转移到相邻体积元的通量相同，所以这些方法在设计上保证了质量守恒。
==See also==
==See also==
For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 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>\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>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 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>\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，我们要解的能量是不增加的，即 <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> ，这种关系可以通过在<math>x = a</math>处 （如果 <math>\alpha > 0</math>） 以及 <math>x = b</math>处 （如果 <math>\alpha < 0</math>）指定<math>u</math>的值来实现。这只相当于在入流处附加边界条件。注意，适定性允许在数据(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>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>。