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Math Help - PDE - Trouble starting

  1. #1
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    PDE - Trouble starting

    Hi,
    I have the question

    Use eigenfunction expansion to solve

    u_t = u_{xx} + q(x,t)

    with IC: u(x,0) = f(x) and BC's: u(0,t) = u_0 and u(\pi,t) = u_{\pi} where u_0 and u_{\pi} are given constants.

    With this method, I am supposed to start with a trial solution based on

    u_t = u_{xx} which based on the boundary conditions I would guess to be

    \displaystyle u(x,t) = \sum_1^{\infty} b_n (t) \sin (nx)

    but when checking this against the boundary conditions I find that u(0,t) = u(\pi,t) = 0 and do not meet the boundary conditions. What am I missing here?

    Thanks
    Last edited by mr fantastic; November 7th 2010 at 12:25 PM. Reason: Title.
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  2. #2
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    You will need to transform your PDE to one with BCs fixed to zero. Try u = v + ax + b and choosing a and b such that

    v(0,t) = v(\pi,t) = 0.
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  3. #3
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    Quote Originally Posted by Danny View Post
    You will need to transform your PDE to one with BCs fixed to zero. Try u = v + ax + b and choosing a and b such that

    v(0,t) = v(\pi,t) = 0.
    I'm not quite sure what you mean... You say transform the pde by trying  u = v + ax + b . The only thing I can think of is \frac{\partial^2 v + ax + b}{\partial t^2} =\frac{\partial^2 v + ax + b}{\partial x^2}  So confused
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  4. #4
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    Let me show you a bit more. First off,

    u_t = v_t, and u_{xx} = v_{xx} so the PDE u_t = u_{xx} + q(x,t) becomes v_t = v_{xx} + q(x,t).

    Now for the BC's. You want v(0,t) = 0, v(\pi,t) = 0 so

    u(0,t) = v(0,t) + a \cdot 0 + b\;\; \Rightarrow\;\; u_0 = 0 + b
    u(\pi,t) = v(\pi,t) + a \cdot \pi + b\;\; \Rightarrow\;\; u_\pi = a \cdot \pi + b. These you solve for a and b.

    Next, the IC

    u(x,0) = f(x) \; \text{so}\; v(x,0) + ax + b = f(x) \; \text{so}\; v(x,0) = f(x) - ax - b

    Now you have a new problem

    v_t = v_{xx}
    v(0,t) = 0, v(\pi,t) = 0
    v(x,0) = f(x) -ax - b

    This you can solve by the separation of variables. Once you have the solution, then use

    u = v + ax + b.
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