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Old   October 4, 2013, 06:09
Default using time as a variable to classify a pde
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nargess
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Hello Dear friends
I have a problem with classifying PDEs which is I don't know when to use 'time' as a variable for classifying the equations! I would be grateful if I can find the answer of my question as soon as possible.
thanks alot
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Old   October 4, 2013, 08:10
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Originally Posted by nargess_md View Post
Hello Dear friends
I have a problem with classifying PDEs which is I don't know when to use 'time' as a variable for classifying the equations! I would be grateful if I can find the answer of my question as soon as possible.
thanks alot
the time variable appears for parabolic, hyperbolic equations. Just consider a standard classification method for a PDE in terms of x1, x2, ...xn independent variables and assume x1=t.
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Old   October 4, 2013, 10:24
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Isn't there anyone who can help me about the so-called problem?
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Old   October 4, 2013, 10:43
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Isn't there anyone who can help me about the so-called problem?
you should be more specific in what you need... many textbooks show the classification of time-dependent problems...
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Old   October 4, 2013, 12:15
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Thanks for your reply. I have studied Hoffman's CFD book, sometimes it uses only the coefficient matrix of d/dx to define the type of equation and somewhere else it uses the coefficient of time derivative either. for example in equation du/dt+a du/dx=0 , should i use time coefficient to determine the equation type?
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Old   October 4, 2013, 12:42
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Originally Posted by nargess_md View Post
Thanks for your reply. I have studied Hoffman's CFD book, sometimes it uses only the coefficient matrix of d/dx to define the type of equation and somewhere else it uses the coefficient of time derivative either. for example in equation du/dt+a du/dx=0 , should i use time coefficient to determine the equation type?
Note that du/dt+a du/dx=0 is a first-order PDE and can not be classified since it is always hyperbolic for any value of the coefficient a.

Classification is done for PDE of the type:

a(x1,...xn,f,..)*d2f/dx1^2 + ..... = 0

For example in a 2D case the PDE

a* d2f/dx1^2 + b * d2f/dx1dx2 + c*d2f/dx2^2 =0

can be classified by analysing the characteristic curves. That can be found in many textbook. If am right, you can see the Hirsch book that uses the eigenvectors analysis
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Old   October 4, 2013, 14:11
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thank you so much, I will read the book.
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Old   October 6, 2013, 05:11
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here is what I got :if 'time' derivative is of the highest order in a equation I have to use it's coefficients in chlassifying the equation or sytem of equation. am I right? for example in the following system
du/dt+a du/dx+b dv/dx=0
dv/dt+c du/dx+ d du/dx=0


but there is another question here:
if we have a system like this:.
du/dt+a du/dx+b dv/dy=0
dv/dt+c du/dx+ d du/dy=0

here we have 3 variables but as I studied in different books we just use d/dx and d/dy coefficient matrixes to classify the equation.
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Old   October 6, 2013, 05:25
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Quote:
Originally Posted by nargess_md View Post
here is what I got :if 'time' derivative is of the highest order in a equation I have to use it's coefficients in chlassifying the equation or sytem of equation. am I right? for example in the following system
du/dt+a du/dx+b dv/dx=0
dv/dt+c du/dx+ d du/dx=0


but there is another question here:
if we have a system like this:.
du/dt+a du/dx+b dv/dy=0
dv/dt+c du/dx+ d du/dy=0

here we have 3 variables but as I studied in different books we just use d/dx and d/dy coefficient matrixes to classify the equation.
you have two dependent variables u=[u,v] and three independent variable x,y,t.
You can use a matrix form of the system to write

du/dt + A. du/dx+B. du/dy = 0

use u= uk*exp(i k.x)

and develep an aigenvalue analysis
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Old   October 6, 2013, 05:38
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thank you so much.
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