A metal rod of length

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Conditions of Validity, Ch Integral equations, kernels and the Fredholm alternative for the last time

Problem

A metal rod of length L=0.1 mL=0.1\,\mathrm{m}, held at zero temperature at both ends, has thermal diffusivity D=1×10−5 m2 s−1D=1\times10^{-5}\,\mathrm{m^2\,s^{-1}}. Its temperature at time tt is related to its temperature at t=0t=0 by the first-kind equation f=Kϕf=K\phi with kernel K(x,ξ)=∑n=1∞2L e−n2π2Dt/L2sin⁡ ⁣(nπxL)sin⁡ ⁣(nπξL),K(x,\xi)=\sum_{n=1}^\infty \frac{2}{L}\,\mathrm{e}^{-n^2\pi^2Dt/L^2}\sin\!\left(\frac{n\pi x}{L}\right)\sin\!\left(\frac{n\pi\xi}{L}\right), whose singular values are the damping factors σn=e−n2π2Dt/L2\sigma_n=\mathrm{e}^{-n^2\pi^2Dt/L^2}. A temperature measurement made at t=100 st=100\,\mathrm{s} is used to try to reconstruct the temperature profile at t=0t=0; the measurement carries a relative noise of 10−410^{-4} in each Fourier mode. ((a)) Compute π2Dt/L2\pi^2Dt/L^2. ((b)) Give σ1,σ2,σ3,σ4\sigma_1,\sigma_2,\sigma_3,\sigma_4 to three significant figures. ((c)) State how many Fourier modes of the initial profile can be recovered above the noise floor.

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