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Old   November 3, 2003, 12:56
Default Terminology
  #1
z175
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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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Old   November 4, 2003, 05:01
Default Re: Terminology
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P. Birken
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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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Old   November 4, 2003, 08:20
Default Re: Terminology
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z175
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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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Old   November 4, 2003, 09:17
Default Re: Terminology
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Praveen
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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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Old   November 4, 2003, 09:33
Default Re: Terminology
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z175
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Thank you. I forgot that there is a separate classification for the first-order equations.
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