# using time as a variable to classify a pde

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 October 4, 2013, 06:09 using time as a variable to classify a pde #1 New Member   nargess Join Date: Oct 2013 Posts: 7 Rep Power: 3 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

October 4, 2013, 08:10
#2
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Filippo Maria Denaro
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 Originally Posted by nargess_md 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.

 October 4, 2013, 10:24 is there anyone? #3 New Member   nargess Join Date: Oct 2013 Posts: 7 Rep Power: 3 Isn't there anyone who can help me about the so-called problem?

October 4, 2013, 10:43
#4
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Filippo Maria Denaro
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 Originally Posted by nargess_md 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...

 October 4, 2013, 12:15 #5 New Member   nargess Join Date: Oct 2013 Posts: 7 Rep Power: 3 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?

October 4, 2013, 12:42
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Filippo Maria Denaro
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Quote:
 Originally Posted by nargess_md 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

 October 4, 2013, 14:11 #7 New Member   nargess Join Date: Oct 2013 Posts: 7 Rep Power: 3 thank you so much, I will read the book.

 October 6, 2013, 05:11 another problem #8 New Member   nargess Join Date: Oct 2013 Posts: 7 Rep Power: 3 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.

October 6, 2013, 05:25
#9
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Filippo Maria Denaro
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Quote:
 Originally Posted by nargess_md 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

 October 6, 2013, 05:38 #10 New Member   nargess Join Date: Oct 2013 Posts: 7 Rep Power: 3 thank you so much.

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