108
个编辑
更改
→Analytical solutions
== Analytical solutions ==
== Analytical solutions ==
解析解
===Separation of variables===
===Separation of variables===
分离变量法
{{main|Separable partial differential equation}}
{{main|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，那么这就是方程解(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。
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。
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，也用于积分变换。
===Method of characteristics===
===Method of characteristics===
特征曲线法
{{main|Method of characteristics}}
{{main|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，这就是所谓的特征分离变量法。
===Integral transform===
===Integral transform===
积分变换
An [[integral transform]] may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An [[integral transform]] may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
积分变换可以将偏微分方程转换为更简单的偏微分方程，特别是可分离的偏微分方程。这对应于对角化算符。
An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.
An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.
这方面的一个重要例子是傅立分析，它使用正弦波的特征基来对角化热方程。
===Change of variables===
===Change of variables===
变量替换法
Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes PDE
Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes PDE
偏微分方程通常可以通过变量的适当变化，用已知的解简化为更简单的形式。例如，布莱克-舒尔斯偏微分方程
<math> \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> \frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 </math>
is reducible to the heat equation
is reducible to the heat equation
可以归结为热传导方程
<math> \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}</math>
<math> \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}</math>
by the change of variables (for complete details see )
by the change of variables (for complete details see )
通过变量替换(完整的细节见)
:<math>\begin{align}
:<math>\begin{align}
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]
\tau &= \tfrac{1}{2} \sigma^2 (T - t),\\[5px]
\tau &= \tfrac{1}{2} \sigma^2 (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).
\end{align}</math>
\end{align}</math>
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。
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 ==
== Numerical solutions ==