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Homework Help: PDE math homework help

  1. Jan 17, 2010 #1
    1. The problem statement, all variables and given/known data

    We are given f [tex]\epsilon[/tex] C(T) [set of continuous and 2pi periodic functions] and PS(T) [set of piecewise smooth and 2pi periodic functions]

    SOlve the BVP

    ut(x,t) = uxx(x,t) ; (x,t) belongs to R x (0,inf)
    u(x,0) = f(x) ; x belongs to R

    Find a solution of this PDE

    2. Relevant equations

    The only relevant equation I can think of is a Fourier Inversion which states that if is continuous and piecewise smooth then
    [tex] f(x) = \sum f^h(n) e^i^n^x ; where f^h = 1/2\pi \int f(x)e^-^i^n^x dx [/tex]

    3. The attempt at a solution

    I have tried solving the first equation ODE till I get by seperation of varaibles
    S''(x) - AS(x) = 0 and T'(t) - AT(t) = 0

    A is real. Three cases: A>0 in which case the solution is S(x) = C(sin(Lx)) + B(cos(Lx))
    where C,B are constants and L = (-L)^(1/2)
    Then I tried using power series expansion of sin and cos to be able to relate it to the fourier series of f (seeing as f(x) = S(x)) but to only get stuck.

    When A = 0 S(x) = Cx + B
    Then I can do this by finding the fourier coefficients([tex]f^h[/tex]) and so on

    When A<0 things become complicated because we get and exponential.

    Anyways, first off, how do we know which A to use!? Then I can try to reach some sort of conclusion.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jan 17, 2010 #2
    Re: Pde

    first, the best thing is to solve it using separation of variable i.e. as an ODE as u did

    u get X''(x)-CX(x)=0 and T'(t)-CT=0

    i think ur missing something in the given such as u(0,t)=u(2\pi,t)
    if so the solution would be X''(x)+L^2X(x)=0
    with L=-n^2*(pi)^2/(2pi)^2

    the solution would be the sum of X_n(x)*T_n(t)
  4. Jan 17, 2010 #3
    Re: Pde

    No I'm not missing anything in the solution. That's the problem. If the constraints on the extremities of x were there it would be a piece of cake. And if we had them then there would be no need for f to be belong to C or PS. Even in the question it tells us to solve the PDE using fourier series.
  5. Jan 18, 2010 #4
    Re: Pde

    Thanks but I found the answer to this. I think this is done.
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