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15) (10.5) A rod of length coincides with the interval [0, π] on the x-axis. The rod has initial temperature f(x) = x for 0 < x < with the ends at temperature zero for all t > 0. Assume the rod satisfies all the assumptions of the heat equation with a constant a of the square root of two. Set up a boundary-value problem for the temperature u(x, t), then find the temperature u(x, t).
15) For u(x, t), 0≤x≤n, t≥0 𐐀u at = 2 a²u 𐐀х2 0<x<π, t> 0 u(x,0) = x, 0<x< π u(0,t) = 0, u(π, t) = 0, t> 0 An = bn = (-1)n+1² n 00 u(x, t) = (-1)n+1²e-2n²t sin(nx) n=1