1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    X(x)=Acos(Lx)+Bsin(Lx)
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: PDE math homework help
  1. Help with a pde (Replies: 1)

  2. PDE homework (Replies: 1)

  3. PDE homework. (Replies: 1)

Loading...