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Solution to a PDE (heat equation) with one initial condition

  1. Sep 24, 2012 #1
    1. The problem statement, all variables and given/known data

    By trial and error, find a solution of the diffusion equation du/dt = d^2u / dx^2 with
    the initial condition u(x, 0) = x^2.

    2. Relevant equations



    3. The attempt at a solution

    Given the initial condition, I tried finding a solution at the steady state (du/dt=0) which gives me d^2u / dx^2 = 0, from I found u(x,0)=c_1 + c_2 x = x^2 and now I am stuck. Thanks in advance!
     
  2. jcsd
  3. Sep 30, 2012 #2
    [itex]u_t = u_{xx}[/itex] with [itex]u(x,0) = x^2[/itex].

    The basic solution is of the form [itex]u(x,t) = \varphi(x)\psi(t)[/itex].
    You would use the method of separation of variables.
    $$
    \begin{cases}
    \varphi = A\cos\lambda x + B\frac{\sin\lambda x}{\lambda}\\
    \psi = C\exp(-\lambda^2t)
    \end{cases}
    $$
    The above is basic and nothing ground breaking.
    Right now your solution is of the form
    $$
    a_0 + \sum_{n = 1}^{\infty}\left[\left(A_n\cos\lambda_n x + B_n\frac{\sin\lambda_n x}{\lambda}\right)\exp(-\lambda^2t)\right]
    $$
    When you have non-homogeneous boundary conditions, you would solve the steady state solution. What are your BC?
    I would then use the orthogonality and Fourier series to finish the problem solving for the Fourier coefficients.
     
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