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GATE 2022 CH – Question 36

Engineering Mathematics · Differential Equations: Heat, wave and Laplace equations · 2 marks · Multiple choice

The partial differential equation $\frac{\partial u}{\partial t} = \frac{1}{\pi^2}\frac{\partial^2u}{\partial x^2}$ where, $t \ge 0$ and $x \in [0, 1]$, is subjected to the following initial and boundary conditions: $u(x, 0) = \sin(\pi x)$, $u(0, t) = 0$, $u(1, t) = 0$. The value of $t$ at which $\frac{u(0.5, t)}{u(0.5, 0)} = \frac{1}{e}$ is

  1. 1
  2. $e$
  3. $\pi$
  4. $1/e$

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Correct answer: (A) 1

Explanation

The solution is $u = e^{-t}\sin\pi x$, which satisfies the equation ($u_t = -u$ and $\frac{1}{\pi^2}u_{xx} = -u$) and the conditions. At $x = 0.5$ the ratio is $e^{-t} = \frac{1}{e}$, so $t = 1$.