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GATE 2024 CE (CE2) – Question 11

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

A partial differential equation $\frac{\partial^2T}{\partial x^2} + \frac{\partial^2T}{\partial y^2} = 0$ is defined for the two-dimensional field $T: T(x, y)$, inside a planar square domain of size 2 m × 2 m. Three boundary edges of the square domain are maintained at value $T = 50$, whereas the fourth boundary edge is maintained at $T = 100$.
The value of $T$ at the center of the domain is

  1. 50.0
  2. 62.5
  3. 75.0
  4. 87.5

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Correct answer: (B) 62.5

Explanation

The solution of Laplace's equation is linear in the boundary values. Write $T = 50 + \theta$ where $\theta$ is 0 on three edges and 50 on the fourth. By symmetry, if each of the four edges in turn carried 50 and the other three 0, the centre values would add up to 50 (the constant solution), so each gives $\frac{50}{4} = 12.5$. So $T_{centre} = 50 + 12.5 = 62.5$.