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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{{mvar|f}}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。
如果{{mvar|f}}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、克莱因-高登方程和泊松方程。
<math>\frac{\partial u}{\partial x}(x,y) = 0.</math>
<math>\frac{\partial u}{\partial x}(x,y) = 0.</math>
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
<math>u(x,y) = f(y),</math>
<math>u(x,y) = f(y),</math>
where is an arbitrary function of . The analogous ordinary differential equation is
where is an arbitrary function of . The analogous ordinary differential equation is
其中{{mvar|f}}是{{mvar|y}}的任意函数。类似的常微分方程是
<math>\frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0,</math>
<math>\frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0,</math>
which has the solution
which has the solution
它的解为
<math>u(x) = c,</math>
<math>u(x) = c,</math>
where is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.
where is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.
这里的{{mvar|c}}是常量。这两个例子说明常微分方程的一般解包含任意常数，但偏微分方程的解包含任意函数。
A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function can be determined if is specified on the line 0}}.
A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function can be determined if is specified on the line 0}}.
偏微分方程的解一般不是唯一的; 附加条件一般必须在定义解的区域边界上指定。例如，在上面的简单示例中，如果在行0}中指定了函数，则可以确定该函数。
偏微分方程的解一般不是唯一的; 附加条件一般必须在定义解的区域边界上定义。例如，在上面的简单示例中，如果在{{math|''x'' {{=}} 0}}时确定了{{mvar|u}}的值，则可以确定该函数{{math|''f''(''y'')}}。
== Existence and uniqueness ==
== Existence and uniqueness ==