Question

Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut; 0<x< 6; t> 0; B.C.: ux(0,t) = 0; uz(6,t) = 0; t> 0; I. C.: u(x,0) = 12 + 5cos (x) – 4cos(27x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or mixed Dirichlet-Neumann. Then, state the general solution for u(x, t). (Note: you don't need to derive the general solution u(x,t).) (7 marks) (e) Find the particular solution of this initial-boundary value problem making use of your general solution in Q2(b). (10 marks)

Answer 1

IC Ang guxx = Ut B-C Ux 10,7): 0, Ux 16,4)=0 214.0) - 12+5 cos (TTY) - 4 (05/2.17 X) ay U13,0) = 12+5 (05/1X3) -4 (OS (217 x3) 12 to 4 =8 ī by B.L are in diff form so it is newman Boundary condition Let 21/2, t) = X(X) Tlt) is solution Ur (x, t) = x '(x)T(H) Uxloit) = x '(0)T(+)=0 = X '10)=0 Ux 16,t) - X'(6) Tlt)=0 =) X16) 30 9 X"(x) TL+) = XWT' (t) X"(x) T'lt = -L? 1 sepration constant) X(X) gilt) - X'(x) + X(X) = 0 X(X) = A cosaX + B sin 2x X'(x) = - LA Son X + 2B cos2x Xlo)= 0 =) B=0 X'/6)=0 = - 2 A Sinbox = 0 n=1,2,. 6x=0 a=no 6

Xn(X) = Ancos/nix) T'lt) + 9 2² +14) = 0 2 -92²t T (t) = Bn -gnn? t t = Bn e 36 - Bne Un(t) = XnlX) Tn (t) Let An Bn-an 5377 an [ Cosphex 1] 2 21 (8,t) = E An 'cos / h) -nt't e 4 n=1 (C). 414,0)= { an cos (NTY J x 1 = 12 + 5los Intre n=1 - 4 (os 2 TX 6 6 P o nel 0 By multiply Cos (MX) and integrate from oto 6 Eans cos/nITY ) Cos/MTX ) dx = S f(x) Cosmmx y de for min L.H.S of above is O so we will cokulate only for men 6. an Stros magn ] dx = $ 12 (Oshry ) dx + $500s (18) Cos/rory) dx 4105 (2113) COSTATY 2 0 Cos (AT Jdx

6. 6 3 S [Cos homelde s pros It cos2/0TH cos intereJode 0 o anlat) .xt 276 2xИТ 12 - 1 x6 = 3 S 12 Cos/nT ) dx = 0 sky $265(%, cosynt yde sif Ceyhan, vege]3 042-]ose for ntl 6. Sen (n+1) In (n-1) TY I (1) (h-)) O for n=1 = 15 5X 3 5 s 6 6 cos (los) los (os (77) dx Cos(21x) Cost nemo do = 25 cos [a +9 ) >) +60 + 60173-23)]]bre 2 for h 12 0 for n=12 6 = 4x 3 = 12 4 Skos (25 x 17 8 x U(x,t) =

4 ai x3 = ot 15 - 0 a = 5 A 2 X 3 = 0 tot 12 O2 = 4 1124 |u(x, t) = A. Cos/T) e 4 367²2 tar los(2x) e IT't -361727 Ulv,t) = 5 cos De 9 + 4 cos (27%) e Ane
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