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Math Help - Finite difference equation - parabolic PDE

  1. #1
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    Finite difference equation - parabolic PDE

    Consider the finite difference scheme
    \frac{3 U^n_p - 4U^{n-1}_p + U^{n-2}_p}{2 \Delta t} - a\frac{U^n_{p+1} - 2 U^n_p + U^n_{p-1}}{\Delta x^2} = F^n_p
    where, with the usual notation, 2 \leq n \leq N$ and $1 \leq p \leq P. Find \rho such that U^n_p = \rho^n \sin (k \pi x_p / L) is a solution of the equation in the case f \equiv 0.

    Let \displaystyle u^0(x) = \phi_k(x) = \sin \frac{k \pi}{L} x, then

    \frac{3 U^n - 4U^{n-1} + U^{n-2}}{2 \Delta t} - a A_P U^n =& 0

    3 U^n - 4U^{n-1} + U^{n-2} =& 2 a \Delta t A_P U^n

    4U^{n-1} - U^{n-2} =& 3 U^n  - 2 a \Delta t A_P U^n

    4U^{n-1} - U^{n-2} =& (3 - 2 a \Delta t A_P) U^n

    U^n =& \frac{4U^{n-1} - U^{n-2}}{3 - 2 a \Delta t A_P}

    where A_P is the tridiagonal coefficient matrix. I am stuck here. How can I derive \rho in the question?
    Last edited by Ackbeet; June 4th 2011 at 11:58 AM. Reason: Cleaned up HTML breaks by eliminating extra lines in LaTeX code.
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