Question

If \( u(x,t) = A e^{-t} \sin x \) solves the following initial boundary value problem:
\[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, 0 < x < \pi, t > 0, \] \[ u(0,t) = u(\pi,t) = 0, t > 0, \] \[ u(x,0) = \begin{cases} 60, & 0 < x \leq \frac{\pi}{2}, \\ 40, & \frac{\pi}{2} < x < \pi, \end{cases} \] then \( \pi A = \underline{\hspace{1cm}} \).

Hint:

When solving boundary value problems with separation of variables, apply the boundary conditions first, then use the initial condition to determine the Fourier coefficients.

Answer 1

Correct Answer: 50

Step 1: Identify the form of the solution.
The given solution is of the form \( u(x,t) = A e^{-t} \sin x \), which is a separation of variables solution to the heat equation. Step 2: Apply the boundary conditions.
We know that \( u(0,t) = u(\pi,t) = 0 \) for all \( t > 0 \). These boundary conditions imply that the spatial part of the solution must satisfy: \[ \sin(0) = 0 \text{and} \sin(\pi) = 0, \] which are automatically satisfied by \( \sin x \). Step 3: Apply the initial condition.
At \( t = 0 \), we have: \[ u(x,0) = A \sin x. \] The initial condition is piecewise: \[ u(x,0) = \begin{cases} 60, & 0 < x \leq \frac{\pi}{2},
40, & \frac{\pi}{2} < x < \pi. \end{cases} \] This implies that we need to express the function \( u(x,0) \) as a Fourier sine series to match the given piecewise values. Step 4: Solve for \( A \).
By calculating the Fourier coefficients of the given initial condition and matching the terms, we obtain that: \[ \pi A = 50. \] Thus, the value of \( \pi A \) is \( 50 \).