← 2016 Paper 2

UPSC 2016 Maths Optional Paper 2 Q8a — Step-by-Step Solution

20 marks · Section B

Heat equation · PDEs · asked 3× in 13 yrs · Read the full method →

Question

Find the temperature $u(x,t)$ in a bar of silver of length 10 cm and constant cross-section of area 1 cm². Let density $\rho=10.6$ g/cm³, thermal conductivity $K=1.04$ cal/(cm sec °C) and specific heat $\sigma=0.056$ cal/g °C. The bar is perfectly isolated laterally, with ends kept at 0°C and initial temperature $f(x)=\sin(0.1\pi x)$ °C. Note that $u(x,t)$ follows the heat equation $u_t=c^2 u_{xx}$, where $c^2=K/(\rho\sigma)$.

Technique

Separation of variables for the heat equation with homogeneous Dirichlet ends; eigenfunctions $\sin(n\pi x/10)$ with decay $e^{-c^2(n\pi/10)^2 t}$; the initial condition is the $n=1$ eigenfunction, so only that single mode survives — no Fourier coefficients to integrate.

Solution

Setup. One-dimensional heat equation on $0\le x\le 10$:

with $c^2=\dfrac{K}{\rho\sigma}$.

Step 1 — Diffusivity

Step 2 — Separation of variables (general)

Seeking $u=X(x)T(t)$ with $u(0)=u(L)=0$, $L=10$, gives the eigenfunctions and modes

so the general solution is the Fourier sine series

Step 3 — Match the initial condition

The initial profile is $f(x)=\sin(0.1\pi x)=\sin\dfrac{\pi x}{10}$ — exactly the first eigenfunction ($n=1$). Hence $B_1=1$ and all other $B_n=0$; no series summation is needed.

Step 4 — Temperature

With $n=1$, the decay rate is

Answer