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The partial differential equation $$\frac{\partial u}{\partial t} = \frac{1}{\pi^{2}} \frac{\partial ^{2} u}{\partial x^{2}}$$ where, $t \geq 0$ and $x \in [0,1]$, is subjected to the following initial and boundary conditions$$u\left ( x, 0 \right ) = \sin \left ( \pi x \right )$$$$u\left ( 0, t \right ) = 0$$$$u\left ( 1, t \right ) = 0$$

The value of $t$ at which $\dfrac{u\left ( 0.5,t \right )}{u\left ( 0.5,0 \right )} = \frac{1}{e}$ is

  1. $1$
  2. $e$
  3. $\pi$
  4. $\frac{1}{e}$
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