Laplace's equation applied to a rectangular plate, where the vertical edges of the plate are held at the constant temperature $0$ and the temperature along the top and bottom edges is given by fixed but not necessarily constant functions $f(x)$ and $g(x)$, is $$\frac{\partial^2 u_1}{\partial x^2} +\frac{\partial^2 u_1}{\partial y^2}=0, \quad \text{ where } 0< x < a, \ 0 < y < b$$ with boundary conditions $$\begin{gather*} u_1(0,y)=0, \quad u_1(a,y)=0,\quad 0 < y < b \\ u_1(x,0) = f(x), \quad u_1(x,b) = g(x), \quad 0<x<a \end{gather*}.$$ It has the solution $$u_1(x,y) = \displaystyle\sum\limits_{n=1}^\infty \left( A_n \cosh\left(\frac{n\pi}{a}y\right)+B_n \sinh\left(\frac{n\pi}{a}y\right) \right) \sin\left( \frac{n\pi}{a}x\right)$$ where

$$A_n = \frac{2}{a}\int_0^a f(x)\sin\left(\frac{n\pi}{a} x\right) \; dx$$

$$B_n = \frac{1}{\sinh\left(\frac{n\pi}{a}b\right)} \left(\frac{2}{a}\int_0^a g(x)\sin\left(\frac{n\pi}{a} x\right) \; dx - A_n\cosh\left(\frac{n\pi}{a}b\right)\right)$$

If, instead of the vertical edges, the horizontal edges of plate are held at constant temperature 0 and the temperature along the vertical edges is given by fixed functions $F(y)$ and $G(y)$, then everything is nearly the same as before except the roles of $x$ and $y$ are swapped. The PDE becomes

$$\frac{\partial^2 u_2}{\partial x^2} +\frac{\partial^2 u_2}{\partial y^2}=0, \quad \text{ where }0< x < a,\ 0 < y < b$$

with boundary conditions

$$\begin{gather*}u_2(0,y)=F(y), \quad u_2(a,y)=G(y),\quad 0 < y < b\\ u_2(x,0) = 0, \quad u_2(x,b) = 0, \quad 0<x<a \end{gather*}.$$ Find the solution to this equation by swapping the roles of $x$ and $y$ in the previous solution

where

The superposition $$u = u_1+u_2$$ of these two solutions solves the general PDE with Dirichlet boundary conditions on a rectangle: $$\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0, \quad\text{ with } 0< x < a,\ 0 < y < b$$ with boundary conditions $$\begin{gather*} u(x,0) = f(x), \quad u(x,b) = g(x), \ 0<x<a\\ u(0,y)=F(y), \quad u(a,y)=G(y), \ 0 < y < b \end{gather*}.$$