Question

Please detail

PDE Utt = Uzx + 2a sin(at) sin(1x) 0 < x <1 0<t< oo BCS S u(0,t) = 0 | u(1,t) = 0 0<t< oo ICs u(x,0) = 0 | u4(,0) = sin(nx) 0 < x <1 u (0,t) = f (t) u (L,t) = g(t) S Use sine transform Uz (0,t) = f(t) uz (L,t) = g(t)) Use cosine transform 2 L S [u (x,t)] = Sn (t) = 1 | u(x, t) sin (ntx/L) dx u (8,8) = È s. (') sin (07.2/L) L C[u (,1)] = Cm(t) = Ě S "u (8,4) cos (19x/L) dx u (.t) = CoC + (t)cas (175) 1) n=1

Please detail

Answer 1

• Pelę U+-4an = 99 Sincort). Since) (0,+): 0 - UCLA) ; Octo U 42,0) - Sin 371 j05x (2,3) (4,5) ni Let Uc»= 2. an Ct). Sin (na) and $277 Sin Cod) = Ï bnct) sin(nou) (**) ? Sino where bnchS 201510011); nel in 1 Note: Fouine device sin(x) = 1. Sn(514) + Io. Sincnau) • Now pult Founver-series of urst) grun by (k) s F.S. of 29 5145+2 Sinoax groen by (**) in (); we get - a { DC4) Sincanx) y anet) Sin conxs} = Ï boc+) Sin coax) = ¿ án" Ct). Sinchana) .- ¿ anco).facnom sin consis by dits anct) twice by dit twice Sin (m ) = 3 bn(t) sincase => 3 [ar"(t) + (19)?anct) - boc177 Sincnou) = {0.Sıncari - D = Qn" (+) + coolanCt) – bn(t) = 0 Chy comparing Caefferrent) = [D +(332 Jaolt) = boct.) T * Qnct) = An Cos(n=1+7+ Bn Sincrot) 12+CON=0 + { bacts = ting D²+(noor Sotins) ₃ which is suluhun of and evelen linean one with Constant 1 . Coefficient of the form (ef)+(P.I.) type • Where I bachi = t {2M SINGA)! = 20 Sinbit) - D²+nner Denul -> Note: Sin cat) ; prouidiel fl-a20 Thus (***)) An (t) = An Custoint) +Bn Sincant) + Meist 2 Sincont) da For Pony Bolith eve giren Inchal Comelihens CamScanner (***) -2+12522 Sincai) = O2)

. Sing U13,0) = 0 = O Sin (now) = anco) - Sininos) - On (0=0 · Also uiry,0) = S(X) - 2 enco) Sun(01029 = Žar'co). Sın tnx) - An'(o)= m(o) whew Castinal LONI here there Condihuns al ancus=o, an'(o)= (n (U) in (4*) anco= 0 0 An CSO -Bn Sino A An' con= color = Coco)= Bn:(092 Gb (0) = Br= Yna .Com Thus (4) become =) B.ES;n=1 to nel anc!) in 1-517) + 23in () 72) SXnEhl (12-1) 7. so required soluhon ut grun by Ucx, t) = Ï [ An Ct). Sincnama) J+91C17 Sin(301) UCX, 4) = -($(54) + È {conf.sin(at)- Sinan ota → Required Iduhun. -2 (na Note: o anch=pa C inal Qolt) ; nei + (-665014) Śnal 1 2 Sin(524) nza 02171 (2) L (241 Sincol) R?+312 ay D + 12 = 0 at D=-7² 90 = 1 (2015116113) =rf sin Git) at -- Coslot) = - Collon) Scanned with CamScanner