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第836行： − 动力学系统和微分方程的列表+ 第859行：
第859行： − 能量法是验证初边值问题适定性的一种数学方法。 时变偏微分方程的高阶差分方法+ − 能量法是一种数学过程，可用于验证初始边界值问题的适定性。在下面的示例中，将使用能量法决定应在何处施加哪些边界条件，以使得到的IBVP处于适当位置。考虑下式给出的一维双曲PDE+ − |first=Bertil|last=Gustafsson|publisher=Springer|year=2008|isbn=978-3-540-74992-9|doi=10.1007/978-3-540-74993-6}}</ref> In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by+ − |first=Bertil|last=Gustafsson|publisher=Springer|year=2008|isbn=978-3-540-74992-9|doi=10.1007/978-3-540-74993-6}}</ref> In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by+ − − | first Bertil | last Gustafsson | publisher | Springer | year 2008 | isbn 978-3-540-74992-9 | doi 10.1007 / 978-3-540-74993-6} / ref 在下面的例子中，使用能量方法来决定应该施加哪些边界条件，使得得到的 IBVP适定。考虑一维双曲偏微分方程 第884行：
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无编辑摘要
微分方程中的拉普拉斯变换
微分方程中的拉普拉斯变换
* [[List of dynamical systems and differential equations topics]]
* [[List of dynamical systems and differential equations topics]]
动力学系统列表和微分方程主题
* [[Matrix differential equation]]
* [[Matrix differential equation]]
矩阵微分方程
矩阵微分方程
The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.<ref>{{cite book |title=High Order Difference Methods for Time Dependent PDE
The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.<ref>{{cite book |title=High Order Difference Methods for Time Dependent PDE
能量法是一种数学过程，可用于验证初始边界值问题的适定性。
In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by
In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by
在下面的示例中，将使用能量法决定应在何处施加哪些边界条件，以使得到的IBVP处于适当位置。考虑下式给出的一维双曲PDE
where <math>\alpha \neq 0</math> is a constant and <math>u(x,t)</math> is an unknown function with initial condition <math>u(x,0) = f(x)</math>. Multiplying with <math>u</math> and integrating over the domain gives
where <math>\alpha \neq 0</math> is a constant and <math>u(x,t)</math> is an unknown function with initial condition <math>u(x,0) = f(x)</math>. Multiplying with <math>u</math> and integrating over the domain gives
其中 <math>\alpha \neq 0</math> 是常数，并且 <math>u(x,t)</math> 是未知函数，初始条件是 <math>u(x,0) = f(x)</math>。乘以 <math>u</math> 并在域上进行积分。
其中 <math>\alpha \neq 0</math> 是常数，并且 <math>u(x,t)</math> 是初始条件是 <math>u(x,0) = f(x)</math>的未知函数，乘以 <math>u</math> 并在域上进行积分。
Here <math>\vert \vert \cdot \vert \vert</math> denotes the standard L2-norm.
Here <math>\vert \vert \cdot \vert \vert</math> denotes the standard L2-norm.
在这里，<math>\vert \vert \cdot \vert \vert</math> 表示标准的 L2-正则。
在这里，<math>\vert \vert \cdot \vert \vert</math> 表示标准的L2-正则。
For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.
For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.