Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
The one-dimensional heat conduction equation in a homogeneous rod of length $L = \pi$ with thermal diffusivity $\alpha = 1$ is:
$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < \pi, \quad t > 0$
subject to boundary conditions $u(0, t) = u(\pi, t) = 0$ and initial condition $u(x, 0) = 4 \sin(2x)$. The temperature $u\left(\frac{\pi}{4}, 0.25\right)$ is ________ (round off to two decimal places).
Answer Key:
1.47
Step-by-Step Solution & Explanation
**Method 1: Separation of Variables**
The general Fourier series solution for homogeneous Dirichlet boundary conditions on $[0, \pi]$ is:
$u(x, t) = \sum_{n=1}^\infty B_n \sin(nx) e^{-n^2 \alpha t}$
With initial condition $u(x, 0) = 4 \sin(2x)$, all coefficients $B_n = 0$ except for $n = 2$, where $B_2 = 4$.
Given $\alpha = 1$:
$u(x, t) = 4 \sin(2x) e^{-2^2 (1) t} = 4 \sin(2x) e^{-4t}$
Evaluating at $x = \frac{\pi}{4}$ and $t = 0.25$:
$\sin\left(2 \cdot \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{2}\right) = 1$
$e^{-4(0.25)} = e^{-1} \approx \frac{1}{2.71828} \approx 0.36788$
$u\left(\frac{\pi}{4}, 0.25\right) = 4(1) e^{-1} = 4 \times 0.36788 \approx 1.4715 \approx 1.47$