Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 12

For Laplace's equation, seek a nonzero separated product $f=X(x)Y(y)$. Dividing by $XY$ gives $X^{\prime \prime }/X=-Y^{\prime \prime }/Y$, so both sides are a separation constant. Choosing it as $-k^2$ gives $X^{\prime \prime }+k^{2}X=0$, $Y^{\prime \prime }-k^{2}Y=0$, with trigonometric and hyperbolic functions, respectively. A zero constant gives linear factors; a positive constant interchanges the two types. Boundary conditions determine which modes are allowed, and linearity permits their superposition to match the remaining boundary data.

The zero values at $x=0,a$ select $X_n=\sin (n\pi x/a)$, and the zero value at $y=0$ selects $Y_n=\sinh (n\pi y/a)$. Expanding the constant upper-edge value as a Fourier sine series gives coefficients $b_n=2F[1-(-1{)}^n]/(n\pi )$. The resulting harmonic function in a rectangle with one constant boundary value isEach term satisfies Laplace's equation and the three zero boundary values. At $y=a$, the series equals $F$ for $0<x<a$. On compact subsets of the interior it and its derivatives converge exponentially, which justifies the termwise equation.

For $F\ne 0$, the top-corner boundary prescriptions conflict: a corner cannot be both zero on a vertical edge and $F$ on the upper edge. The bounded interior solution has the specified limits along the open sides; it has no single continuous extension at those two corners. Values assigned only at the isolated corners do not change the interior solution.