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Laplace's Equation Boundary Problem

  1. Aug 8, 2007 #1
    1. The problem statement, all variables and given/known data

    I have a two part question, the first part involves solving Laplace's equation

    [tex]
    u_{xx} + u_{yy} = 0
    [/tex]

    for the boundary conditions
    [tex]
    u_x(0,y) = u_x(2,y) = 0
    [/tex]

    [tex]
    u(x,0) = 0
    [/tex]

    [tex]
    u(x,1) = \sin(\pi x)
    [/tex]
    for
    [tex]0 < x < 2, 0 < y < 1[/tex].

    The second part now states a new boundary problem for the same equation, involving the square area defined on [tex]0 < x < 1, 0 < y < 1[/tex]. This time we have
    [tex]
    u_x(0,y) = u(1,y) = 0
    [/tex]

    [tex]
    u(x,0) = 0
    [/tex]

    [tex]
    u(x,1) = 2\sin(\pi x)
    [/tex]
    The question asks me to use the solution from the first boundary problem to solve this problem directly (using a theorem/principle).

    2. Relevant equations



    3. The attempt at a solution

    I have solved the first part using the standard method of separation of variables but I'm rather puzzled as to what this mystery theorem/principle is that can allow me to take my solution from the first problem and apply it directly to the second boundary problem :frown:
     
  2. jcsd
  3. Aug 8, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Since the first problem has x between 0 and 2 and the second has x between 0 and 1, what would happen if you replaced the "x" in the first solution by "2x"?
     
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