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Math Help - exact solution to heat equation

  1. #1
    Junior Member
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    Sep 2010
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    exact solution to heat equation

    Hey everyone, I need a little help with this question.

    Relevant equations:

    T(x,t)= \sum\limits_{i=1}^{\infty} B_i e^{-(i^2 \pi^2kt)} sin(i \pi x)

    T(0,t) = T(1,t) = 0

    T(x,0) = sin(\pi x)

    \alpha = \frac{k \Delta t}{(\Delta x)^2}


    The question:

    Let N=20 and \alpha = \frac{1}{4}, given the initial condition T(x,0) = sin(\pi x), what is the exact solution at points on the grid?

    So I have worked out \Delta x = \frac{1}{N} = \frac{1}{20} and k \Delta t = \frac{(\Delta x)^2}{4}=0.000625, and then worked out B_i = 1.

    I am just unsure what the exact solution is, since I don't know k.

    Can someone help me with the exact solution.

    Thanks
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  2. #2
    Junior Member
    Joined
    Sep 2010
    Posts
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    Am I supposed to find T_j^n which gives the exact solution at points on the grid, and that j \Delta x = x and n \Delta t = t. I can't find what T_j^n is.

    All I have is:

    T_j^{n+1} = \frac{k \Delta t}{(\Delta x)^2} T_{j+1}^n + (1-2\frac{k \Delta t}{(\Delta x)^2}) T_j^n +\frac{k \Delta t}{(\Delta x)^2}T_{j-1}^n, j=1,2,...,N-1 where N=20.

    I still need some help.

    Thanks
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  3. #3
    Junior Member
    Joined
    Sep 2010
    Posts
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    how would I evaluate:

    T_j^{n} , j=1,2,...,N-1 where N=20.

    I know I need to use:

    T_0^n = T_N^n = 0 for all n

    and

    T_j^{n+1} = \frac{k \Delta t}{(\Delta x)^2} T_{j+1}^n + (1-2\frac{k \Delta t}{(\Delta x)^2}) T_j^n +\frac{k \Delta t}{(\Delta x)^2}T_{j-1}^n

    but I can't seem to get much.
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