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D^2T/dx^2 + d^2T/dy^2 + d^2T/dz^2 = C

  1. Jan 30, 2013 #1
    I'm trying to solve the heat conduction formula in 3 dimensions when there is constant generation from electrical resistance q'''. This creates a constant C on the right hand side that is equal to q'''/k.

    T=T(x,y,z)
    d^2T/dx^2 + d^2T/dy^2 + d^2T/dz^2 = C

    I found a solution using separation of variables for when the right hand side equals 0, but it doesn't work with a non-zero constant on the right, because you end up with:

    X'''/x + Y'''/y + Z'''/z = C/XYZ
     
  2. jcsd
  3. Jan 30, 2013 #2
    I think I got it, maybe. I can solve the homogeneous equation:

    d^2τ/dx^2 + d^2τ/dy^2 + d^2τ/dz^2 = 0

    and then assume the particular solution to have the form:

    T = τ + Ax^2 + Bx^2 + Dx^2

    That makes

    d^2T/dx^2 + d^2T/dy^2 + d^2T/dz^2 = 0 + 2A + 2B + 2D,

    So 2A + 2B + 2D = -C

    and I can use boundary conditions to find A, B, and D

    ?
     
  4. Feb 1, 2013 #3

    pasmith

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    Homework Helper

    You need to start with a particular solution ([itex]Cx^2/2[/itex] will suffice, but if you expect your solution to have certain symmetry properties then it might be worth looking for a particular solution which shares those properties) and then add complementary functions to satisfy the boundary conditions.
     
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