Question:
Consider the problem

 \displaystyle u_{xx} + u_{yy} = y \ 0 \leq x \leq L \ \ 0 \leq y \leq H

with boundary conditions  u(x,0) = 1, u(x, H) = 0, u(0,y) = 0 and  \displaystyle \frac{\partial u}{\partial x} (L,y) = 0

This problem is best solved by making the substitution  u(x,y) = v(y) + w(x,y) and choosing v(y) so that  w(x,y) satisfies a homogenous equation.

Find an explicit expression for v(y) and state the precise boundary conditions and equation which w(x,y) must satisfy. Do not find an explicit expression for w

Attempted Answer:


Let u(x,y) = v(y) + w(x,y)

\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 v}{\partial y^2} = y

Choose \frac{\partial^2 v}{\partial y^2} = y<br />
Then

V' = \frac{1}{2} y^2 + A and V = \frac{1}{6}y^3 + Ay + B

V(0) = 1 so B = 1

V(H) = 0 so \frac{1}{6} H^3 + AH + 1 = 0 and therefore A = - \frac{1}{H} - \frac{1}{6} H^2

V = \frac{1}{6}y^3 - \frac{y}{H} - \frac{1}{6}H^2 y + 1<br />
and then

u(x,0) = v(0) + w(x,0) = 1 so w(x,0) = 0

u(x,H) = v(H) + w(x,H) = 0 so w(x,H) = 0

Can somebody please tell me if this is correct, and if not, could somebody please help me out?