Question

Find the solution u(a, t) to the initial boundary value problem for the heat equation 4urx te (0, 00), a e (0,5), with initial condition e [0,) e ,5 | 3, u(0, ar) f(ar) = 4, and with boundary conditions ug (t, 0) = 0, un (t, 5) = 0.

Answer 1

4-4Uzx, oX<S xe Co, 52) e [52,57 4 42tt,o)= 0, u (t,5)=o The 2equined sel - K ba ulx,t)= Do. Dn f(x) vandx Dn= C fix) dx Here K4 5 5 fxd Do 12 3dxt $2 -C 4 3 X St 22 x

flx) Cao nxx dx IS $72 3 CaonAX dxt 4 Ca 12 aino - 3 in nAx +4 in nxx 15 Sin mn/ Sin nas in nA 20 2 nki 15 in n (= Sin 20 nA u 2y 3in 2 8in n when 0 RNem urhan 2 n us odd. utx,t)= (-h) 2 Sin nA ConTAX + X e