(Steady state and transient temperatures) Let a laterally insulated rod with initial tem- perature u(2,0) = f(2) have fixed endpoint temperatures: u(0,t) = A. u(L, t) = B (a) It is observed empirically that: lim u(x, t) = U55(2) t+o where the steady state temperature uss (1) corresponds to setting uy = 0 in the boundary value problem. Thus uss (2) is the solution to the endpoint value problem: 22458 0 ar? us(0) = A Us: (L) = B Find us (30) MA 303 Assignment 4, Page 2 of 2 Summer 2021 (b) The transient temperature utr(2,t) is defined to be: Utr(x, t) = u(, t) - ug () Show that ut satisfies the boundary value problem: aut k ar2 aut at Utr(0,t) Utr(2, 0) Utr(L, t) = 0 g(2) = f(x) - Ug(2) (c) Conclude from the formulas (30) and (31) in Section 9.5 in the textbook that: ur, t) = u(2) + Utr(x, t) = mᎢᏆ ' Us (2) + Cn exp '-n²rkt L sin n=1 where C = Ź!" (z) – um(a) sin (PE dc)
