Question

Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find the Sturm-Liouville problem related to the PDE (A) and the bound- ary conditions (BC). c. (10 points) Find all eigenvalues and eigenfunctions of the obtained Sturm-Liouville problem. Show your calculations. d. (9 points) Find a solution u(x, t) of the PDE (A) which satisfies the boundary value and initial conditions (BC) – (IC). e. (1 point) Is such solution classical?

Answer 1

sol":- Giver differential equation is Ut - 9 Uns -o, ocnc 72, tro. with conditions Un (to) = ult, 2) = 0,67,0 u Co, u = Uf0u) = 4 cos²x ot x L kl 2. Let the complete solution of wave equation is Tit. X (e) zu ETX" Now der 324 842 = TX Now from T"X =9TX" CT = K (contout) X a Then. get two ordinary functions Tot it, and x = x (0) ordinary differential equations for т" екст año X" = kx y Here there are three cases anises for the rahve of k. case- 1 : When Kao then we have From T1 = T = arth are constants a, a fromcy) ) and x"=0 X = Betly Honee complete solution is u-crit = (htt) (extly)

so we पा casella when k is positione integer ka (e is any no.) from T" - e c T = 0 on (or apver) T and tp x = 0 (-) x = 2 du buxiliary equation for Auxiliary equation for 7 mr-per e & pc Ł3P m - tp (C23) Pa X C e + C4 eln T = a ces set it + C, е T = Geet Herce complete solution of ① the ret] [ C3 ex tape u(at) [a pet case-14 9 i when к (where pis and is negative le.kanp2 any number) Ther from T + pct T = (C-3) T" +9o T = 0 This is the stuem - Lionville problem on (2 tg pr.)

+ pl y se This is Then also sturm-Liouville problem form, (6 + p) 10 A.E. for 6 Ar Er for 0 ant 9 p = m te 20 ma api = 13pi { x = ca caspat simpa t = Goszett a sin et Hence complete solution of o u (ait) = (a as 3pt the sin 36t) (la cas px + q simpre) are 1) so only sirce periodic functions present in ② ore it shows the physical nature of ware is solution which become the complete solution boundary conditions ; put u (oth = ₂ ( G cos 3pt thong tinapt) so (2= Now use into be comes Hence complete solution ucrit) = Ca simpa ( 4 cosa pt + Ca sinapt) put Tr=&=572 into 11A MA I we get,

u(41, t) = Cq simpa (a csapt + 2 sim 3pt) = 0 [Ca to - since a ) 20,41, 12, ---] This is rakes of the offerind Sturm-Lionville problem Then complete solution of ware equation eigen This ulxit = ca sin (ana) { 'ç ces (font) tosin (6nt) A sin(aand cosent + B sin(nn) sinfurt, required eigen functions, Then general solution of the pde is given by An sin (22x) Ces (ant) + Bn sin (2na). sin (ont) is the u(xit) h21 where An= { }, "f (as sim (ar) dan Z , he gray sin (anas dx 2 c = 3 L = Atz aw (= finy - 1693x Upon a gaya ya here

Thon An= Stay 4 cy² x sim (2mm) du 농 S [Les 3x43 com] sim m (rang du zsin(arn) 630 3x dak 2 - 3 1 1 1 2sim (un). Ceir an som (2n+3) " tsin 2 (2-3) x] dn the It isim (2+1) + sim (2n-1) due CB (ent 3) a [cs (cos com uma [cs cos(2na) a > Jata. [mon emaz] ++ / [mmenti 20x3 42 cos (2001) in 22x1 2nri 2n- = 2 2 2n 3 + 2m +3 (20) ² 32 2n-1 tnt1 (n-1 sn 2 K(4-9) Then Bu 을 An S nxo) .than a cos²x Sir (2nn) da H2 gemein aangaa 6n un LA + sh (4-9) 3kn finn 2 (arn)

Hence the required required solution is u(mit). El or het ons + a camminar sim (nn) cos (6mt) hr 16 + I Bar (96–9) may being sirlarn) sirlant)