Finding Temperature in a Rectangular Plate: $\nabla^2u=0$

[Dustinsfl]

The steady state temperature $u(x,y)$ in a rectangular plate $0\leq x\leq L$, $0\leq y\leq M$, is sought, under the condition that the edge $x = 0$ is maintained at zero degrees, $x = L$ is kept at $u(L,y) = y$ degrees, and the edges $y = 0$ and $y = M$ are insulated. The appropriate differential equation $\nabla^2u = 0$.

Since the vertical boundary conditions are insulated, wouldn't this be the same as just dealing with $u_t=u_{xx}$ since Fourier coefficients for those boundaries will be 0?

 

[HallsofIvy]

Saying that the boundaries are insulated does NOT mean that the Fourier coefficients are 0. It means that the derivatives there are 0 and so the coefficients of the sine terms are 0.

 

[Dustinsfl]

Wasn't thinking figured out this post.