Question

Use the solution of the vibrating string with fixed ends obtained with separation of variables to solve the following initial boundary value problem on the interval [0,1], and sketch the solution for t 0, t= 1/2 and t 1. ial diffe au u ot2 x x<1, t>0, u(x, 0) f(x)-sin zx, (x, 0) 0, u(0,)=u(1,t)= 0

Answer 1

Let u1X, H)= X(X) TCH) be Bolulmon of Then for 2 x'T 4 $ 1 2 X" from CD, X"=x's - X" = "= k1823) =) xxx=047_KT=0 UCO,H=0 =7XO) TCH)=0 =) Xo)=0 UCIH)=0 =) XCI) TCH 3o =) XCI)=0 -- X(O)=0 4X(1)=0 Easelt) - 130 then X504 mto The solutions are X(X) = x+ - TH=gtta (0)=0 =) C116)+G=0 =) GO Xonzo =) CCI)30 =) (i=0 f1 (20) X 0 U4+1=0 is toimial solution

casella).- 130, Bay dap then x'+p8x70 t_pt=0 The solutions are X(() = cep q qe pe 1. TCH = Gefa geft Xo) 20 ) Geotheo 20 =) CHG20 A X(i)=0 =) Gelreco B &® holds only if 9704G70 X0=0 ucat-o is trivial solution Caselii) :-xco say ta-p² then x4 BX=0 & T'4PTO The solutions Je X ha Gelaterte X(X)=468P><tc sinpx ¢ 7CH)=G COPE+ Casinpl X(0)20 ) C18080+Casino =0 » 9=0 Xci) =0 = 968P+Simpso - Gusinpro fagad - Simpso (+670) 5) Sinp=sint е Релст

X60) = C2 Sinpx 1896)= Sinntis (Chade Cz=1) TheCH = G G3 P7+Guinpt Inch = onesnazt din sinnat I choose Gant Let Up Art) = M 60) Pinct) Qadin & UX,Hamil Ande) Thct) Then UCX, Hana Con Ghat+onsinmou) Sinne is solution of GF 466,0) = R2) = - Smuz pom@, uexpo (An@do+onslino) Stomax PODER Cn Stones which is fourier sine series in (0,1) where 69 = $ ( 'Pc) Simizdx sa 1-2 Sintro sintux de

Co= 1-Zsinde sino doc=0 1-2 Sin Sintix de 3-2 / Sinox dx =% | '- coemt das --[x-smonty --[(i - spytan) -(0-0)] G=-661-0-0) = -1 TG=-) Bornola CM = sincnta) Simwox dy 2 Sinntix Sinny dx = { [ CoS (– exame]abc

= SM A+D 112C Sino 137' . (HADI Coon to ESMERADT - Sinn-))-(0-0)) T RADT Trial . Cnco fonol Brom UcXH) - 42,) Cao@natton Smit) Simona pa do Simont-Simonpix HCBT+d, Sinna sinta