Question

6. Solve the following boundary value problem: 1 U = 34xx, 0 < x < 1,t> 0; u(0,t) = u(1,t) = 0; u(x,0) = 7 sin nx - sin 31x

Answer 1

Ito Ut = 3 Uax Ocull ulot)=ullt=0 U(20) = 7 sinu - SingTTx To 00000 using sepaution of variable U= New Tt) Ut = X(a) IT at Una = T (t) 2 x dx² = 3T(H) ²x Хя) от Jaz at 2x - 72 JT at 3 T(t) daz X (x) 2X cas + 2X1x) 20 duz a +372TH) = 0 at - 372€ TH) = Ge charactestic eqn 52 +02=0 saz di Xea = b, cospathsinda XCos= b=0 X(1) = b, sint=0 sincubato sint=0 e-323724 7=NT Ulx, t) = E Cn (sinntx)) nal DO U (0,0) = = 7 sinta -1 Sin3t = { ln Sinnta g n=1 19 go

(,=7 , C3=-11g and rest Cm - O -37724 - 2772€ ulut) = (7 sinttal - 1/ (sinama) Ano' 9 20