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

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

如果偏微分方程的系数不是常数，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

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

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。

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

== Analytical solutions ==

== 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。

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

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

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

在分离变量法方法中，可以将偏微分方程简化为含有更少变量的偏微分方程，如果只有一个变量，那么就变成了一个'''<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>'''--，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。

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