Question

2. Show that the function u(x, 1) = C, exp(-n?n?) sin nax = solves the heat conduction problem uxx = u, with boundary conditions u(x,0) = Cn sinnix u(0, 1) = u(1, 1) = 0

Answer 1

Solutton Chwn thal -ko Cnal ue will touo thi equaton by uing mothed of Sereaton of vanablen. thin, -C ko ime partial difeentiaton of equation Then we got ut) xluTCE) TC) JX) -T) xn) UnXTH) cnd again difeentiale wrt x(n T(t) T(t)xtn) and farkaly dlifeentiale auation wrt t X(x1 dT) JXxIT(A) - xcT XCx) T') in (X) uwe get A ( and 4) x ( ITCH) wITG) T4)

cn Gmulta neous equation, ) ela in Conant an Aight ics A and Condanl x(xT Chay) T) -) x(x) -入X「】» O XCH and T AT1 Boundary Condikom aA. u(ot) (1()-0. and u(vt) X(v)T4) wen. utot)-0 x (o) T (t) (o,t) 0 (TCH1=0 Thon ue qel only Trial hlution shich innot of ay une.) fmilast wler u4t) 0 t10 (1)T () xo1- o

Now thope arne thre Canen Cane: Inl when o th eguaton A 6)+8=0 BMG and L AO Only thuial tian. use will aeiedt thi Cane Cane: 2nd whon Thon. Than arhany epuion in m-2-o mat -l0nu. X (nl A eux ge-ux Nau Apeying we haus ) o A+8 G)

AeBeo-) CA A-8 (fmG) Ci) 8e48e-o BC-ee)o -e e-to Fnu, again tisial o lutton Sa we will Pojed thin Cane again Care: 3 ulon Ao 1e Than xnixtnl Then aurilay epuan ma22-0 m2-y? Heno. A con (ux)8 n(2 x(nl A (OA (o) +8 hnle) o A+B.0 :Ao al ady) O Bnu)

Thn w haue only thwial dolyion, ue munt hau Ginulso V- nT 1 nez. n(nl=Bnin (n ) and TC) Tr: -TA)O T1 4127)-0 CA-9) 1 TCH takin intagsaten heth bido we havu T(H) -u2ttCi log (T) e-bci T(0) e e' TA)ce- haus Cn e- =etnn Tn ) un(xt) Xn(h) Tn () unt,)Bnbn(n) Cn e n(1Bnin in(nnn) e -(nnj

b pintinal of huporfoiHon un (.t) n in(nxe (n 1 ut en n Dn Dn in(ome-(n ma1 E um(Mt) ニ u(x,t) - く Dn fin(nix) e-(nfk4 Now ApMting Tnitial canditon u haus uo) Cn hin (nTx) Cn hnlnm) nl T

Cn Dn (m)'t e-(nf u(nt) Cn hin (nnn) e く = nol hunu, it in olution of with BCA ulot)ut)P Cnd IC u(at Cn hninh) nal