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Math Help - PDE - a few general questions

  1. #1
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    PDE - a few general questions

    Hi, I just had a few questions on PDE's just to make sure I'm doing thigs right.

    Question 1
    Given a function f(x,y) is the integration with respect to one variable (say x) of zero c + g(y) where c is a constant?

    Question 2
    How woul you solve a question like u_y +u_xy = 0? I was thinking maybe you could devide by u_y but that seems way to easy...

    Question 3
    After finding the u(x,t) = F(x+ct) where c is a constant is the solution to a PDE, how would you apply the initial conditions? Am I looking to find a value for c?

    Thank you!
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  2. #2
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    Quote Originally Posted by angels_symphony View Post
    Hi, I just had a few questions on PDE's just to make sure I'm doing thigs right.

    Question 1
    Given a function f(x,y) is the integration with respect to one variable (say x) of zero c + g(y) where c is a constant?

    Mr F says: I can't quite make sense of your question ... I'll answer what I think you're asking: If you integrate f(x, y) wrt x, y is treated as a constant. The arbitrary constant of integration will therefore be a function of y, g(y) say (because the partial derivative of g(y) wrt x is zero). The familiar 'C' is contained in g(y).

    Eg. f(x, y) = xy.  \int f(x, y) \, dx = \int xy \, dx = \frac{yx^2}{2} + g(y). Depending on the boundary conditions, it may be that g(y) = y^3 + y + 1, say.



    Question 2
    How woul you solve a question like u_y +u_xy = 0? I was thinking maybe you could devide by u_y but that seems way to easy...

    Mr F says: I'd re-write it as \frac{\partial}{\partial y} \left( u + \frac{\partial u}{\partial x} \right) = 0. Then, integrating wrt y (after having re-read the above), u + \frac{\partial u}{\partial x} = g(x). So you're now, effectively, solving the ODE \frac{du}{dx} + u = g(x) since u will be totally a function of x. btw You don't know what g(x) is but you might get it from the boundary conditions. (btw the integrating factor method can be used [in principle if g(x) is not known] to solve this ODE).


    Question 3
    After finding the u(x,t) = F(x+ct) where c is a constant is the solution to a PDE, how would you apply the initial conditions? Am I looking to find a value for c?

    Mr F says: No. This is 'half' of the general solution to the wave equation. The wave equation will already have the value of c in it .... The initial conditions are used to get the functional form of F .... I strongly suggest you research D'Alembert's solution of the wave equation.

    Thank you! Mr F says: You're welcome.
    ..
    Last edited by mr fantastic; January 29th 2008 at 03:38 AM. Reason: Added a little bit more about g(x) in what Mr F says to Q2
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  3. #3
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    Quote Originally Posted by angels_symphony View Post
    Hi, I just had a few questions on PDE's just to make sure I'm doing thigs right.

    Question 1
    Given a function f(x,y) is the integration with respect to one variable (say x) of zero c + g(y) where c is a constant?
    If you like, but it realy is the same thing as h(y) .

    Question 2
    How woul you solve a question like u_y +u_xy = 0? I was thinking maybe you could devide by u_y but that seems way to easy...
    \frac{\partial}{\partial y}u + \frac{\partial^2}{\partial x \partial y}u=v+\frac{\partial}{\partial x}v=0

    where v=\frac{\partial}{\partial y}u

    Question 3
    After finding the u(x,t) = F(x+ct) where c is a constant is the solution to a PDE, how would you apply the initial conditions? Am I looking to find a value for c?

    Thank you!
    F(x)=u(x,0)

    RonL
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