# help with PDE

• February 6th 2009, 05:00 PM
PvtBillPilgrim
help with PDE
Can someone help me with this, it's slightly different from other separable equation PDE's I've done before:

Consider a thin rectangular plate (0 <= x <= pi, 0 <= y <= pi/2) with the following steady-state temperature distributions on the edges
u(x,0) = 0
u(0,y) = 0
u(pi,y) = 0
u(x, pi/2) = 60sinx + 20sin2x

Find u(x,t) to determine the rate of change.

Thank you very much for any help. It's greatly appreciated.
• February 6th 2009, 09:15 PM
ThePerfectHacker
Quote:

Originally Posted by PvtBillPilgrim
Can someone help me with this, it's slightly different from other separable equation PDE's I've done before:

Consider a thin rectangular plate (0 <= x <= pi, 0 <= y <= pi/2) with the following steady-state temperature distributions on the edges
u(x,0) = 0
u(0,y) = 0
u(pi,y) = 0
u(x, pi/2) = 60sinx + 20sin2x

Find u(x,t) to determine the rate of change.

Thank you very much for any help. It's greatly appreciated.

The solution is given by $u_{xx}+u_{yy}=0$ on $[0,\pi]\times [0,\tfrac{\pi}{2}]$.
Look for a solution $u(x,y) = X(x)Y(y)$.
Notice that, $\frac{X''}{X} = - \frac{Y''}{Y}$.
Therefore, $\frac{X''}{X} = k \text{ and }\frac{Y''}{Y} = -k$.

Case $k=0$: Then $XY = (ax+b)(cy+d)$. Does this satisfy the boundary conditions?

Case $k>0$: Then $XY = \left( a\sinh(\sqrt{k}x) + b\cosh(\sqrt{k}x) \right) \left( c\sin (\sqrt{k}y) + d\cos (\sqrt{k}y) \right)$. Does this satisfy the boundary conditions?

Case $k<0$: Then $XY = \left( a\sin(\sqrt{k}x) + b\cos(\sqrt{k}x) \right) \left( c\sinh (\sqrt{k}y) + d\cosh (\sqrt{k}y) \right)$. Does this satisfy the boundary conditions?

Try to work out the problem from here.