Question

Solve Laplace’s equation inside a rectangle 0 <= x <= L, 0 <= y <= H, with the following boundary conditions

Answer 1

u(0,y) = g(y), u(L,y)=0,
\[\frac{\partial u}{\partial x} (x,0)=0\]
, u(x,H) = 0

Answer 2

I think it helps to draw a picture, although it's unnecessary it can really help to organize your thoughts.

Make sure to use separation of variables to save yourself. You can also use the picture you draw to help you flop the boundary conditions over to the ODEs you get from the separation. \[u(x,y)=X(x)Y(y)\]

Answer 3

\[\nabla^2u(x,y)=0\\[1ex]
0<x<L,\qquad\ \ u(0,y) = g(y), \qquad u(L,y)=0\\
0<y<H,\qquad u_x(x,0)=0,\qquad \ \ \ \ u(x,H) =0\]

\begin{align}
u_{xx}+u_{yy}&=0\\
&& u(x,y) &= X(x)Y(y)\\
&& u_{xx}&=X''Y\\
&& u_{yy}&=XY''\\
X''Y+XY''&=0\\[1ex]
X''/X=-Y''/Y&=\text{const.}=-\lambda
\end{align}