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October 4, 2013, 06:09 
using time as a variable to classify a pde

#1 
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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 

October 4, 2013, 08:10 

#2 
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Filippo Maria Denaro
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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 
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nargess
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Isn't there anyone who can help me about the socalled problem?


October 4, 2013, 10:43 

#4 
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Filippo Maria Denaro
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October 4, 2013, 12:15 

#5 
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nargess
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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?


October 4, 2013, 12:42 

#6  
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Filippo Maria Denaro
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Quote:
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 
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nargess
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thank you so much, I will read the book.


October 6, 2013, 05:11 
another problem

#8 
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nargess
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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. 

October 6, 2013, 05:25 

#9  
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Filippo Maria Denaro
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Quote:
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 
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nargess
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thank you so much.


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classifying, time 
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