Question

Find the temperature function u(x,t)u(x,t) (where xx is the position along the rod in cm and tt is the time) of a 1818 cm rod with conducting constant 0.10.1 whose endpoint are insulated such that no heat is lost, and whose initial temperature distribution is given by:
u(x,0)={5 if 6≤x≤12

{0 otherwise

To start, we have L=18 0.1 Because the rods are insulated, we will use the cosine Fourier expansion. 22 Ac + =1 A cos(" )e| A cos( u(x, t) = 340 e 18 n=1 Solving for Ao gives: r, 0)d = 2 J^ u(x,0)da =60/18 Ag = For the remaining A values, we use the Fourier expansion formula: An=18 Jo cos( u(x,0)dx 18 =S5cos(npix/18) for B 6 da and C 12 Integrating gives: 12 A (90sin(pinx/18))/(pin) =10/(pin)(sin(2pin/3) - sin(pin/3)) We can write out a few terms of the series: u(r, t) A, cos()e 2400 30/18 +...

Answer 1

d (o)n 2 12 J 4(0) d 2 18 u06)d + 12 [9-] 60 18 Nou Cosnnx 18 Sin nA Sin 3 Sin Now TT S2 C Sin (43) Sin A 2 2TT 2n 2.757 -10 3 2 n

A3 L Sin -Sin n 3R 3n A A C AND Sin - Sin añ3 3/2 AN 103 1378 A A n(5 Sin A5 all odd OSte Zeslu ae eocisト So RLeOT enm DO An CosnT A /2 e 32400 -4n2t 3 2400 Cos e 2.753 53 32400 (2.75A e 32400 513 32400 +