Question

The partial differential equation
\[ \frac{\partial u}{\partial t} = \frac{1}{\pi^2} \frac{\partial^2 u}{\partial x^2} \]
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

• A. 1
• B. $e$
• C. $\pi$
• D. $1/e$

Hint:

For heat-equation problems with sinusoidal initial conditions, the solution always decays exponentially in time as $e^{-t}$ when the eigenvalue and diffusion constant cancel neatly.

Answer 1

The Correct Option is A

This is a 1-D heat equation with homogeneous boundary conditions and an initial condition $\sin(\pi x)$.
The standard solution for such a PDE using separation of variables is:
\[ u(x,t) = \sin(\pi x)\, e^{-t} \]
since the eigenvalue is $\lambda_1 = \pi^2$ and the diffusion coefficient is $\frac{1}{\pi^2}$, giving decay factor:
\[ e^{-\lambda_1 \cdot \frac{1}{\pi^2} t} = e^{-t} \]
Now evaluate at $x = 0.5$:
\[ u(0.5,t) = \sin\left(\frac{\pi}{2}\right) e^{-t} = 1 \cdot e^{-t} \]
and
\[ u(0.5,0) = 1 \]
The given condition is:
\[ \frac{u(0.5,t)}{u(0.5,0)} = \frac{1}{e} \]
Substitute the expressions:
\[ \frac{e^{-t}}{1} = e^{-1} \]
Thus:
\[ e^{-t} = e^{-1} \]
\[ t = 1 \]
Therefore, the correct value of $t$ is 1.
Final Answer: 1