help with PDE

**PvtBillPilgrim** · February 6th 2009, 04:00 PM

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.

**ThePerfectHacker** · February 6th 2009, 08:15 PM

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.

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