Find the temperature in Prob. 11 with L = Ï, c = 1, and f (x) = cos 2x Data from Prob. 11 Adiabatic means

Find the temperature in Prob. 11 with L = π, c = 1, and

f (x) = cos 2x

Data from Prob. 11

€œAdiabatic€ means no heat exchange with the neighborhood, because the bar is completely insulated, also at the ends. Physical Information: The heat flux at the ends is proportional to the value of ˆ‚u/ˆ‚x there.

Show that for the completely insulated bar u_x(0, t) = 0, u_x(L, t) = 0, u(x, t) = f(x) and separation of variables gives the following solution, with A_n given by (2) in Sec. 11.3.

,- (cn /L?t u(x, ) - + A cos n=1