Question

Solve the heat problem Ut = Uzx + 5 sin(4x) - sin(2x), 0 < x <7, u(0,1) = 0, u(,t) = 0 u(x,0) = 0

FInd u(x,t) and lim u(x,t)

Answer 1

Given U - U + 5 Sin(4*) - Sinex) t CX au at = a'u an² +5 sin(4x) - Sin (23) the b.si L7 Take we on L get + (55cn (4 ) scalen]L{1} => su (5) - 4(340) = du (5) sin(49) - senen)-5 - { au at 4 EL? a'u a x2 + da² Given u BM = 0 2 5 sin(4n) -sch (22) co su - =7 + da² s su - ay du s - sin (2x) -5507 (4) da? The differential operator foom is 0²_s) u sin (22) -5 sin (4) s The Auxiliary can m?s=0 =7 SX qe + Ce is s m = + S x CF = and p.I = sin (2) ( si 5.sin(4x) D² s - S (5 son(4) put D² = -16 2 put D=-4

I (5 son(40) -4-5 -16-s +*+ (sin(2-)) son(52) Il The general solution is given by 5 sin(4x) - = sin (2) st4 St16 u C.Ft p. I .ſsa =7 (145) =q + Gel + 15 sin(4x) Son(2x) s+16 st4 Ś s w (1) Ginen ut) = 0 => L { u (e,t)} = L{o} => u (@,s) put iw (1), we get uce,s) qt qto a=0 C: Sino =o) -7 (2) =9 G+2=0 u (Tt) = 0 Also given ey L{ ultit)} =L{0} u (TT, S) = 0 (1), we get en put a = π G Sinnit =o) to ge tge (T/S lu TI o > (3) 9 è iis е t qe => the cans solving the eans (2) & (3), we get G =o q

subo ។ ។ values in G Sin (2x) u(ays) ot 5 sim (4x) s+16 we Take the inverse Lit on bos, { (1), we get con la get l' { u g, 5)% = szon () 416)sambel esting u leyt) = 5.500 (4*) w/4"{$stic). ta] senten) (['{ $-54++] son ero ( 114*} - *{stiel Scm (2x) { ["{}}- t'{573}] syu(tyt) = 5 sen(x) [ 1 +)- Esimew[l-ent] 5 91 4 -167 th (12) leis 91 hane ē otro we .a (x2) 16:5 / 2 - (+4) 635 2 / 2 = (72)n +4117 16 4