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Math proof formula

  1. Sep 8, 2009 #1
    Consider the diffusion equation:
    ut = uxx, 0 < x < 1, t > 0,
    u(0,t) = p(t), u(1,t) = q(t), t > 0,
    u(x,0) = f(x), 0 < x < 1,
    for t < tfinal

    Suppose that the boundary condition at x = 0 is now replaced by
    [tex]\partial[/tex]u/[tex]\partial[/tex]x(0,t) = 0

    Using the forward difference formula
    [tex]\partial[/tex]u/[tex]\partial[/tex]x(x,t) = ((-u(x+2[tex]\Delta[/tex]x,t) + 4u(x+[tex]\Delta[/tex]x,t) - 3u(x,t))/2[tex]\Delta[/tex]x) + O([tex]\Delta[/tex]x)2

    show that the Euler algorithm needs to be modified to include the formula for U0,k+1:
    Uj,k+1 = rUj-1,k + (1-2r)Uj,k + rUj+1,k, j = 1,2,...,N-1,
    U0,k+1 = (4U1,k+1 - U2,k+1)/3.

    I don't have a clue where to start. Please help.
     
  2. jcsd
  3. Sep 9, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: Proof

    If you are, in fact, taking a course in the numerical solution of differential equations and have no "clue where to start" with all of the problems you have posted, then you have far worse problems that we can help you with! Talk to your teacher about this.
     
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