Question

[points=4] Q4. Solve the heat equation subject to the given conditions: д?u ди О<x<п, to дх? at' ди ди - (0,t) = 0, - (п,t) = 0, t>0 дх дх u(x,0) = п-х Paragraph В І := =

Answer 1

Here the given PDF problem is tyo 22u am o La LT a = 2 x2 at 2 Boundary condition are Sur (0,t) = 0, (7,4) = 0 Initial condition f(n) = M(0,0) = an 3

Assume that the solution of ois of the form X is ucal, t) = x (m) T (t) © a function of n alone and T is a fauction only substituting Xoy T' (t) = xucu) T() of t in G X! T'(+) Х *T(+) clearly the L.His of 5 is fonetion of a only ord R.Hrs is a formation of t only e since n and to independent variables Bean hold good if each side is eamal to o constant i esay) leads to the Dio ODE. are to Ć 2 X-K X Т"- КТ using the Boundary condition 2 we get from @ X') TELO x'(a) T (t) = 0 sinee Tl) co gives 30, gives uso, so suppose that TCL) 70 have from @ x(0) co) Xica) solve runder the Boundary conditions Then we co 6 we wow

case - is Let koo, Then solution of X (ng - An + B 11) xiony = A Using Boe , ① gives , Aco Then 1 reduces to X(X) = B Again corry ponding to kro, G gields d T so that T = corstout a T = constant = ko say dt corresponding to kao solution of the given boun value problem from G is dery given by Eo 2 12 E6 23 ucu,t)= BX

3 Ai en + B, e-da 14 (14 da b Case=1 Let kan2 d&o, Then solution of 6 is xCx ) = x'(W) E Ardern-do,e-da using the boundary conditions I, we get from Aidt. Bino => Ait Bl=0 [170] And Aiden • Bine-da=o Aret_ Biedt salving we get AI=BIFO so that X (2) to, hence Mint) = 0 which does not satisfies ③ so we rejeet Kad2 case=11 Let k=-d² dto Then solution of X (n) = neosda of Bi sinda co 6 is 13

04 using the =) Th(t) = Bu e-n²t X 1(n) = - Ain sinir & Breosaa boundary condition @ 9 gives BAYAN Bid = 0 =) Bi [F] And X'(a) = 0 And sinda) = 0 - Aisinda zo [ id Fo] we take Alfo for non trivial solution we mayt have Sindaco a dan n=1,2,3 Hence non Zero solutions Xn (m) of o are 16 Xn (n) = An ers(nx) using 6 reduces to SO dk - na (15) given by 5), dT dt = - 12 whose general solution is 17

... Un(x, t) = Xu (2) Tu(t) (98 where . - En cosma), e-n't Ena AnBn is another arbitrary constant In order to obtain a solution which also satisfying ③ we consider more general They and (18) constitute a set of infinite solution of are consider a liner combinations of these solutions. Hence complete solution of 0 may be taken in the following form un, t) = + E Un (x, t) ( 2 Eo 2 n=1 nos plos Eo 2 E Eners muy.e-n? + ne! and fo 2 no substituting tao.in 29 d using , we have of (n) = of Z En coston) which is fourier eosine series, so the Eo and En are given by 6. = Ź S. cosunda { [ sinna 19 co ㅈ a

K En= (arn) eosnuda ㅈ 2 2 a so Kersnadn T cosna iter s cosunda ? [Sinma] - [usin ore] + žin/ Sinnude in [- in { i- eosino int[i-Gon] . n ot tot = = Hence the required solution of the given A Heat Eamation o under the given Bee @ initial condition ® is and e-net u(x, t) = Eima [-enn] [.-Ein] posmu.e Ź no1