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

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

  <math>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>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>

数学 f 左(x1，ldots xn; u，frac { partial x1} ， ldots frac { partial x1} ; frac { partial ^ 2 u }部分 x1} ， ldots frac { partial ^ 2 u }部分 x1} ; ldots frac1 partial x1}右)0。 数学

<math>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  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.

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.

如果偏微分方程是线性函数及其导数，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、 Klein-Gordon 方程和 Poisson 方程。

如果偏微分方程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、 Klein-Gordon 方程和 Poisson 方程。

A relatively simple PDE is

A relatively simple PDE is