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删除49字节 、 2020年11月13日 (五) 00:14
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Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic and elliptic. Others, such as the Euler–Tricomi equation, have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions.
 
Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic and elliptic. Others, such as the Euler–Tricomi equation, have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions.
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一些线性二阶偏微分方程可分为抛物型方程、双曲型方程和椭圆型方程。其他的方程,如欧拉-特里科米方程,在不同的区域有不同的类型。这种分类有助于选择适当的初始和边界条件以及提高解的平滑性。
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一些线性二阶偏微分方程可分为抛物型方程、双曲型方程和椭圆型方程。其他的方程,如欧拉-特里科米方程,在不同的领域有不同的类型。这种分类有助于选择适当的初始和边界条件以及提高解的平滑性。
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Assuming  u<sub>yx</sub>}}, the general linear second-order PDE in two independent variables has the form
 
Assuming  u<sub>yx</sub>}}, the general linear second-order PDE in two independent variables has the form
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假设 {{math|''u<sub>xy</sub>'' {{=}} ''u<sub>yx</sub>''}},含有两个独立变量的一般的线性二阶偏微分方程具有如下这样的形式
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假设 {{math|''u<sub>xy</sub>'' {{=}} ''u<sub>yx</sub>''}},含有两个独立变量的一般线性二阶偏微分方程具有如下形式:
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where the coefficients , , ... may depend upon  and . If  over a region of the -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
 
where the coefficients , , ... may depend upon  and . If  over a region of the -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
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其中的系数 {{mvar|A}}, {{mvar|B}}, {{mvar|C}}... 一般取决于{{mvar|x}}和{{mvar|y}}。如果在{{mvar|xy}}-平面的一个区域上 {{math|''A''<sup>2</sup> + ''B''<sup>2</sup> + ''C''<sup>2</sup> > 0}},偏微分方程在该区域是二阶的。这种形式类似于圆锥曲线的方程:
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其中的系数 {{mvar|A}}, {{mvar|B}}, {{mvar|C}}... 一般取决于 {{mvar|x}} 和 {{mvar|y}} 。如果在 {{mvar|xy}}-平面的一个区域上 {{math|''A''<sup>2</sup> + ''B''<sup>2</sup> + ''C''<sup>2</sup> > 0}},偏微分方程是二阶的,这种形式类似于圆锥截面的方程:
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More precisely, replacing  by , and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
 
More precisely, replacing  by , and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
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更准确地说,用 {{mvar|X}} 替换 {{math|∂<sub>''x''</sub>}},对于其他变量做同样的操作(从形式上来说,这是由傅里叶变换来完成的),将一个常系数偏微分方程转换成一个相同次数的多项式,最高次数的项(齐次多项式,这里是一个二次形式)一般会用于偏微分方程的分类。
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更准确地说,用 {{mvar|X}} 替换 {{math|∂<sub>''x''</sub>}},并同样替换其它变量(通常由傅里叶变换完成),将一个常系数偏微分方程转换成一个相同次数的多项式,最高次数的项(齐次多项式,这里是一个二次形式)对于偏微分方程的分类最为重要。
 
===~~ most significant for the classification 意译为用于偏微分方程的分类。      significiant直译过来感觉不太合适
 
===~~ most significant for the classification 意译为用于偏微分方程的分类。      significiant直译过来感觉不太合适
  
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