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GATE 2022 CE (CE2) – Question 11

Engineering Mathematics · PDE: Fourier series, separation of variables, diffusion, wave and Laplace equations · 1 mark · Multiple choice

$f(x,y)$ satisfies $\nabla^2 f=0$ on the unit circular domain centred at (0,0), with boundary value 3. Find f(0,0).

  1. 0
  2. 2
  3. 3
  4. 1

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Show answer and explanation

Correct answer: (C) 3

Explanation

A harmonic function satisfies the mean-value property: its value at the centre of a circle equals the boundary average. Every boundary value is 3, so that average is 3.

Equivalently, uniqueness for the Dirichlet problem makes the entire solution constant. **f(0,0)=3 (C)**.