Consider diffusion inside an enclosed circular tube. let its length (circumference) be 2l. let x denote the arc length parameter where −l ≤ x ≤ l. then the concentration of the diffusing substance satisfies ut = kuxx for − l ≤ x ≤ l u(−l, t) = u(l, t) and ux (−l, t) = ux (l, t). these are called periodic boundary conditions. (a) show that the eigenvalues are λ = (nπ/l) 2 for n = 0, 1, 2, . (b) show that the concentration is u(x, t) = 1 2 a0 +[infinity] n=1 an cos nπx l + bn sin nπx l e−n2π2kt/l 2 .

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Step-by-step explanation:

Consider diffusion inside an enclosed circular tube. Let its length (circumference) be 2l. Let x denote the arc length parameter where −l ≤ x ≤ l. Then the concentration of the diffusing substance satisfies ut = kuxx for − l ≤ x ≤ l u(−l, t) = u(l, t) and ux (−l, t) = ux (l, t). These are called periodic boundary conditions. (a) Show that the eigenvalues are λ = (nπ/l) 2 for n = 0, 1, 2, 3,... . (b) Show that the concentration is u(x, t) = 1 2 A0 +[infinity] n=1 An cos nπx l + Bn sin nπx l e−n2π2kt/l 2 .

-1/2 +2

step-by-step explanation: