GATE 2022 CE (CE2) – Question 48

Engineering Mathematics · PDE: Fourier series, separation of variables, diffusion, wave and Laplace equations · 2 marks · Numerical answer

$s_t=Ds_{xx}$ on $0\le x\le L$, with $s_x=0$ at both ends. Initially s=s₀ on 0≤x≤0.4L and zero elsewhere (figure). D=0.1 m²/s, s₀=5 μmol/m, L=10 m. Find steady concentration at L/2 (integer).

Initial pollutant concentration s0 from x=0 to x=0.4L and zero from x=0.4L to L. See the attached source image.

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Correct answer: 2

Explanation

Zero spatial derivative at both ends means no pollutant flux, so total mass is conserved. At steady state, $s_{xx}=0$ and the two zero-gradient conditions make concentration uniform.

Initial concentration is 5 over 0.4L and zero elsewhere, giving mean $5(0.4)=\mathbf2$ μmol/m. Diffusivity controls approach time, not this final value.