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   (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''² − ''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。

{{math|''B''² − ''AC'' < 0}} (椭圆型微分方程)：在定义方程和解的区域内部，椭圆型偏微分方程的解在系数允许的程度内光滑。例如，拉普拉斯方程的解在它们被定义的区域内是解析的，但可能假设边界值是不光滑的。亚音速流体的运动可以用椭圆型偏微分方程近似，欧拉-特里科米方程在 {{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''² − ''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。

{{math|''B''² − ''AC'' {{=}} 0}}（抛物型偏微分方程）：在每一点上都是抛物线型的方程可以通过改变自变量从而转化成类似于热方程的形式。随着转换后的时间变量的增加，方程的解变得平滑。欧拉-特里科米方程在特征线 {{math|''x'' {{=}} 0}} 上是抛物线型的。

   (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''² − ''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。

{{math|''B''² − ''AC'' > 0}} (双曲型偏微分方程)：双曲型方程在初始数据中保留了函数或导数的任何不连续性。波动方程就是其中的一个例子。超音速流体的运动可以用双曲型偏微分方程近似，欧拉-特里科米方程在 {{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''₁, ''x''₂,… ''x''_''n''}}，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''₁, ''x''₂,… ''x''_''n''}}，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 .

  Elliptic: the eigenvalues are all positive or all negative.

  Elliptic: the eigenvalues are all positive or all negative.

椭圆形方程: 本征值全部为正或全部为负。

椭圆形方程: 特征值全部为正或全部为负。

# Parabolic: the eigenvalues are all positive or all negative, save one that is zero.

# Parabolic: the eigenvalues are all positive or all negative, save one that is zero.

  Parabolic: the eigenvalues are all positive or all negative, save one that is zero.

  Parabolic: the eigenvalues are all positive or all negative, save one that is zero.

抛物线形方程: 本征值全部为正或全部为负，除了一个为零。

抛物线形方程:除了一个为零值，特征值全部为正或全部为负。

# Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.

# Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.

  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，只存在一个有限理论(Courant 和 Hilbert，1962)。

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