A solute diffuses in a cell of length whose ends are held at zero concentration…

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 solute diffuses in a cell of length LL whose ends are held at zero concentration (absorbing, Dirichlet ends), so its profile expands in sines. Its concentration profile tt seconds after an unknown initial profile ϕ(ξ)\phi(\xi) was set up is f(x)=∫0LK(x,ξ) ϕ(ξ) dξ,K(x,ξ)=∑n=1∞2L e−n2π2Dt/L2sin⁡ ⁣(nπxL)sin⁡ ⁣(nπξL),\begin{aligned} f(x) &= \int_0^L K(x,\xi)\,\phi(\xi)\,\mathrm{d}\xi, \\ K(x,\xi) &= \sum_{n=1}^{\infty}\frac{2}{L}\,e^{-n^2\pi^2 Dt/L^2}\sin\!\left(\frac{n\pi x}{L}\right)\sin\!\left(\frac{n\pi \xi}{L}\right), \end{aligned} with DD the diffusion constant. For this cell π2Dt/L2=5.0\pi^2Dt/L^2 = 5.0 (dimensionless), and ff is measured with relative noise 10−410^{-4}. ((a)) Classify the equation relating ff to ϕ\phi. ((b)) Predict how many sine modes of ϕ\phi can be recovered reliably, and explain why the rest cannot.

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