MATH 212 Homework 2, Fall 2011

Tlhedh J2 ios. Soltion 2 rv ep. Aur = cH, whert US gin tk ws kt ‘ # \o s : The pe Hel devivediveo cow ome Qa exish Since ! ot - Ke ws kel eos kx a - Kte* sn kel cos kx , exists Din¥ e US = = Kein Ket ers Ca F exists . 2 i j v And c 25 c* (x sin Ket cos Kc) - _ re? sir Ket cos rx = y*u a Then UW ic a solution f die POE oy ams, sores TiquntS ic and c. (2.3 # tl. U(x, t here B= 2 [- where y = 7 J fo Sin an Jn ye 2 ( B. cos AAt + B" sin \,t ) gin ot mt r, = eal ok e ub since the intial velocity Te oF JB ie — 0 -L=1 a cv= 1 =< take e-1 ( we will get Vhe Suwel an sw oY =a rate enim) Ent 7 ¢ @ Sg) ce cost = \ cos Vhus u(x,t ) = a B. cos nlit einnnz«x Colelate B dh OIF OS RS ne % i (xn) = a A ™ f ) a-t rt tc ne t x + YH, if Lex < 2 O 2 = a 4 if Zex e] B = a | fea) Sin nTl 2 ax ° Vis %, B= 2 | (x- L) asin (na )dx4 a | (-x+2) sin (wit 20) dx a % mall _ 2 wv nm = \e sin (nit ) RK ra \@ sin( nw) \ -L Cos (nx) ff + eos (nT) 0 SS all gin (nia) Oo ~S sin(nt7) az nen B - d[-@-b)d5 tos (niV2t) +t simcotix) | %, | ae x) = S\ nv) |" + 2 |- (-2+%) eos WM) — pe nen) " nv i g Lf stag) + ae Z) - 1 sin(ntt) | n> Gnd “2 nv = n'n™ - : i > a2 cise Sa (AE capede mOB) + ADs aw OJ g - 4+ sin(od)\- 4 sinot)- 4s T a ii (oF) oe n(Z) on eal a a -s wher 4" ee he pertedic odd extengton of 4 Oh lat, 04 + f*(a-2) is the Avanslabte x of p" byy 4 to the righ {" (a4 +) 1S fhe trawsdation +f f” by ‘ be +12, eft, both ko, si daa on [o,1]- These (ere cee) Ay 2B 3 C= 0 (oh v= &) mn 7 dy’ 5 A -oO r -2 jhen fle nur ( 7 ') ll vontelles ONL y= 4 Qa Tne 4) = a +C w= (9) > wim D Cay = Fy) ees 47% | sven a= 4 fie. By chain seal win 4 A LOBE Dy Yu 2 By Wy tg = Me 7 Uy = Us ~ Yow +Vou f\ssueina, Prob oo omd ifs deivetives ome wor Vi nwews ae Kiow = Us, then: Unae Ma LUyy 4+ Uno uu =< Ue-Us)y - y vi a ( ry Moe Vy + Ur yo [Us % a U uw y ) Yay = CA. gig — Uw ‘ Uy = Ua” + Uy. es z Uy U = UA , 10) _ By) wy 4 + Uwe ly 2 HU. Subs hing in» ® Uy = 2M ou + Yow + Vy - UX + ow =O =>) Use =O: WN ovmel forwe Tntesnath nj wiv. bk Ay Uy =f ( w) > ms v {tw) + Sow) y pwr deve 5) CAL owl vary cL Purch ions Therefor uC yw) = af{(y-%) + ag 2 es 1L.dy, A= AC_8~ A-4-(£) S ifs Cle Ay 2 2By = -Y4U, - 44, - Cu, -4UL) m - Cay uy ny + 4+ Uyy =< O @) , Ax | / j3= + 7 c-4 ~ 4 te . leet ~ 2 2o 7 4 4 4 Hy pevboli'c PDE 4 Cor oD where y= de ) nn ~4 (Us), aye mv U. = U My 4 Uy, Wa, - = Aby Subsh poling in Ux) ie 3 2 ee a SA ~ 28 Wy 20 Wow + be Mev +} Wve 4 db Wow? eas ~ Uj =o =) [Uo a = o|