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D^2T/dx^2 + d^2T/dy^2 + d^2T/dz^2 = C 
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#1
Jan3013, 05:08 AM

P: 90

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 nonzero constant on the right, because you end up with: X'''/x + Y'''/y + Z'''/z = C/XYZ 


#2
Jan3013, 06:59 AM

P: 90

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 ? 


#3
Feb113, 01:03 PM

HW Helper
Thanks
P: 943

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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