[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

[FMDenaro]

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.

[nargess_md]

Isn't there anyone who can help me about the so-called problem?

[FMDenaro]

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...

[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?

[FMDenaro]

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

[nargess_md]

thank you so much, I will read the book.

[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.

[FMDenaro]

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

[nargess_md]

thank you so much.