UPSC 2024 Maths Optional Paper 2 Q6a — Step-by-Step Solution

20 marks · Section B

Laplace equation: Dirichlet/Neumann, separation of variables · PDEs · asked 4× in 13 yrs · Read the full method →

Question

Show that the solution of the 2D Laplace equation in the upper half-plane ( $y\ge 0$ , $\phi\to 0$ at infinity) with boundary condition $\phi(x,0)=f(x)$ is

\phi(x,y)=\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{f(\xi)\,d\xi}{y^2+(x-\xi)^2}.

Technique

Fourier transform in $x$ ; solve the resulting ODE in $y$ ; apply boundary conditions; invert via the convolution theorem to identify the Poisson kernel.

Solution

Step 1 — Fourier transform.

Let $\Phi(k,y)=\hat\phi(k,y)$ , $F(k)=\hat f(k)$ . Transforming the PDE:

\Phi_{yy}-k^2\Phi=0.

Step 2 — Solve the ODE in $y$ .

General solution: $A(k)e^{-|k|y}+B(k)e^{|k|y}$ . The decay condition $\phi\to 0$ as $y\to\infty$ forces $B(k)=0$ :

\Phi(k,y)=A(k)e^{-|k|y}.

Step 3 — Boundary condition.

At $y=0$ : $\Phi(k,0)=A(k)=F(k)$ , so $\Phi(k,y)=F(k)e^{-|k|y}$ .

Step 4 — Convolution theorem.

$\Phi=F\cdot e^{-|k|y}$ in Fourier space corresponds to $\phi=f*P_y$ in physical space, where $P_y$ is the inverse transform of $e^{-|k|y}$ .

Step 5 — Compute the Poisson kernel.

P_y(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-|k|y}e^{ikx}\,dk=\frac{1}{2\pi}\left[\frac{1}{y-ix}+\frac{1}{y+ix}\right]=\frac{1}{2\pi}\cdot\frac{2y}{y^2+x^2}=\frac{y}{\pi(y^2+x^2)}.

Step 6 — Combine.

\phi(x,y)=\int_{-\infty}^{\infty}f(\xi)\cdot\frac{y}{\pi(y^2+(x-\xi)^2)}\,d\xi=\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{f(\xi)\,d\xi}{y^2+(x-\xi)^2}.

Answer

\boxed{\;\phi(x,y)=\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{f(\xi)\,d\xi}{y^2+(x-\xi)^2}.\;}