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

[timsea81]

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

 

[timsea81]

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

?

 

[pasmith]

You need to start with a particular solution ($Cx^2/2$ 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.