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November 3, 2003, 11:56 |
Terminology
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#1 |
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Please who can say: why the equation <br clear="all" /><table border="0" width="100%"><tr><td> <table align="center" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center"> </td><td nowrap="nowrap" align="center"> *<font face="symbol">¶</font>u <div class="hrcomp"><hr noshade="noshade" size="1"/></div><font face="symbol">¶</font>t
</td><td nowrap="nowrap" align="center"> + a</td><td nowrap="nowrap" align="center"> *<font face="symbol">¶</font>u <div class="hrcomp"><hr noshade="noshade" size="1"/></div><font face="symbol">¶</font>x </td><td nowrap="nowrap" align="center"> =0</td></tr></table> </td></tr></table> is named hyperbolic? |
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November 4, 2003, 04:01 |
Re: Terminology
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#2 |
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Please consult any book on partial differential equations. The answer will be on the first ten pages.
Yours, Philipp |
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November 4, 2003, 07:20 |
Re: Terminology
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#3 |
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<title>Books consult: an eqn is hyperbolic</title>
<div class="p"> </div> Books consult: an eqn is hyperbolic if a<sub>11</sub>a<sub>22</sub><font face="symbol">-</font>a<sub>12</sub><sup>2</sup> < 0 but here we have a<sub>11</sub>=a<sub>22</sub>=a<sub>12</sub> <font face="symbol">=</font> 0. Equations with a<sub>11</sub>a<sub>22</sub><font face="symbol">-</font>a<sub>12</sub><sup>2</sup>=0 are called parabolic. |
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November 4, 2003, 08:17 |
Re: Terminology
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#4 |
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The definition you are refering to applies to second order equations. For first order equations you can use the following definition:
PDE u_t + A u_x = 0 where A could be a matrix, is said to be hyperbolic if A has only real eigenvalues and a complete set of linearly independent eigenvectors. The scalar equation u_t + au_x=0 is trivially hyperbolic by this definition. The distinct characteristic of hyperbolic pde is that they admit wave-like solutions. You can write the second order pde as a system of first order pde and try to check the above definition which will lead to the same inequality. |
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November 4, 2003, 08:33 |
Re: Terminology
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#5 |
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Thank you. I forgot that there is a separate classification for the first-order equations.
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