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PDE Separation of variables

  1. Sep 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Solve the problem.
    utt = uxx 0 < x < 1, t > 0
    u(x,0) = x, ut(x,0) = x(1-x), u(0,t) = 0, u(1,t) = 1


    2. Relevant equations



    3. The attempt at a solution
    Here is what I have so far but I'm not sure if I am on the right path or not.

    u(x,t) = X(x)T(t)
    ut(x,t) = X(x)T'(t) ux(x,t) = X'(x)T(t)
    utt(x,t) = X(x)T"(t) uxx(x,t) = X"(x)T(t)
    X(x)T"(t) = X"(x)T(t)
    T"(t)/T(T) = X"(x)/X(x) = λ
    T"(t) = λT(t) X"(x) = λX(x)

    λ = 0 -----> X(x) = Ax + B
    b.c. u(0,t) = A(0) + B = 0 --------> B = 0
    u(1,t) = A(1) + B = 1 --------> A = 1

    λ > 0 --------> λ = ω2
    X(x) = Acosh ωx + Bsinh ωx
    X(0) = Acosh ω(0) + Bsinh ω(0) = 0
    = Bsinh ω(0) = 0 ------> B = 0

    λ < 0 ---------> λ = -ω2
    X"(x) = λX(x) --------> X"(x) = -ω2X(x)
    X(x) = Acosωx + Bsinωx
    X(0) = Acosω(0) + Bsinω(0) = 0 --------> A = 0
    X(1) = Acosω(1) + Bsinω(1) = 1
    X(1) = Bsinω = 1 B ≠ 0
    ω = ∏/2 + 2m∏ for any interger m

    T"(t) = ω2T(t)
    T"(t) = C cosωt + Dsinωt

    u = (C cosωt + Dsinωt)sinux

    Okay this is all I have. Am I on the right path and where do I go from here?
    Thanks!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 20, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Other than the fact that you mean "sin ωx", not "sin ux", that's good. Now you have to try to make that fit the "initial vaue conditions", u(x,0)= 0 and [itex]u_t(x, 0)= x(1- x)[/itex].

    You won't be able to do that with just a single "ω" so since this is a linear equation try, instead, a sum:
    [tex]u(x,t)= \sum_{m=0}^\infty (cos(\pi/2 + 2m\pi)t + Dsin(\pi/2 + 2m\pi)t)sin(\pi/2 + 2\pi)x[/tex]
     
  4. Sep 20, 2009 #3
    Sorry to be so dense, but I get lost at this point.

    I think I am then suppose to do

    ut=X(x)[-C(∏/2 + 2m∏)sin(∏/2 + 2m∏)t + D(∏/2 + 2m∏)cos(∏/2 + 2m∏)t)
    ut(x,0) = D(∏/2 + 2m∏)cos(∏/2 + 2m∏) = x(1-x) ----> D ≠ 0

    t = 0 f(x) = ∑ Dsin (∏/2 + 2m∏)t

    u(x,t) = ∑ D sin ((∏/2 + 2m∏)t sin (∏/2 + 2m∏)x

    Am I on the doing this correctly? Do I then do the integral from 0 to m∏?

    Thanks!
     
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