## Partial Differential Equation Question

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 $w$

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
$

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
$

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 $w(x,H) = 0$