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GATE 2022 ME (ME1) – Question 12

Engineering Mathematics · Differential Equations: Heat, wave and Laplace equations · 1 mark · Multiple choice

Solution of $\nabla^2T = 0$ in a square domain ($0 < x < 1$ and $0 < y < 1$) with boundary conditions: $T(x, 0) = x$; $T(0, y) = y$; $T(x, 1) = 1 + x$; $T(1, y) = 1 + y$ is

  1. $T(x, y) = x - xy + y$
  2. $T(x, y) = x + y$
  3. $T(x, y) = -x + y$
  4. $T(x, y) = x + xy + y$

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Show answer and explanation

Correct answer: (B) $T(x, y) = x + y$

Explanation

The function $T = x + y$ satisfies Laplace's equation (its second derivatives are zero) and it matches all four boundary conditions: $T(x, 0) = x$, $T(0, y) = y$, $T(x, 1) = x + 1$ and $T(1, y) = 1 + y$. The solution is unique.