Question

A. : Suppose that u(x, t) satisfies Ut = Uzr +1, € (0,2) u(x,0) = 0 u(0,t) = u(2,t) = 0 Solve for u(x,t). What is lim u(x,t)? B. Consider the heat equation in the region 0 < x < 1, but supoose that the system is heated with a source. This is represented by: Ut = Uzz + cos(2), 1 € (0,7) u(x,0) = 1+ cos(2x) U (0,t) = U7(TT,t) = 0 Solve for u(x, t).

Answer 1

Criven, U = Unat! ice. 24 - * ine 27 222 + Fs{ nu } 27 2 2 .eu mo lo 22? 2 2 Ces non NR 2 2 nia as ne} NT 2 2 2 n 220 22 dt Taking the finite Fourier sine transform (with l22) of both sides of (1), we lieve = ls { man ti su sinnar dra sin nondits² sin naa da [ame linna lo scos nar da + ( 2 Can om at stucat) sin nan e) sin nga do] {{ccant) Jucart) sin nax an } + min Leij"- ;) dus ų (not) t Cnit) + main levyn.] lihere as enit) = fucat) din nord ] 2222 es mat) a C”- 4 ODE which is a first ordes linear 30; I. F s na at e [EUW-1): e + where A being arbitrary Constans. ow, trz - donat 8 co us = 23 a3 [61?,] put, t=0, U (no) = & [figh- 1] + A or, A-41,") + suca, oj sin nga da os [l-613) [ ; (2,0)20] 2 dus t NA enant 4 s 2 na lg. e 4 at A ënn ne 43

ne) 00 N2L Til & [£1)²-1] sinna ma) censtem. Us enit) = Figh,]{ine en } N3² Taking the inverse finide Fourier sine tomsform, we get, u ca, t) = 2 & us on, 4) sinni ulait) 도 n33 2 When to lim uex,t) to dim to nel na? Σ τη B.) due due 2e+») 201n) 20-n) [.! (tot) s leg (Not) =0 din na 2 we get, i where being arbitrary do ou 07 crivers, le = Uxat coss.et of Taking tinitus Carine transform of both sides of the givers equation we gerzone come Cosmaz az In cas (m2) de + cos2 et cas 22 [ pe Oranj)," +con sen onaj da + eS"craina) isz da [uz Crit). 69 823).*+.n{lucitycomeno*uco,t) &, fajdy} tet. [SinBicy-m)+ ' tin [(4+»] ; where he (not) = 1 unit) as 622) de Cost Coslada = sin(al-»)], sen [2(1-»)), an (3 (1+ » )) Loo, It Jo 20-n) a o, due = n²ue (nit) Ontegrating Me Enit) A e-n? E

pul, t=0, A = ų (no) T s uca,o) asha) da ["@+ Csaa) as ) da f" asowy do + je "casas Kosta a) din (sin(20.-22.7: Sch[2 m)], s sinna tn + 2 (279) . Us cnit) = 0 transform Taking finite inverse founier cosine wie geta T n=1 K e Co) at 2 E uentes (2) 2 ☆s It cos. an) ds to [a lin 22 U Cx, t) =

finede Fourier Cerine hun form =). The finite Fourier Cosine fraus form of a function f(x), where f(x) is sectionally (Piecewise) Continuous over the indi X6 6,1 ) ) defined by Fe (n) = f fex) eos n da where indenval oz2nd n is an integer. Inversion formula for tinite Fourier cosine poussern of Fe (2) is the finite Fourier son tosine freun ferry of f(x) over the interval 6, 2) then the invesse tinide Fourier cesine trous form is given by fla) = I Fe (0) + 2 2 Fe (n) Cos N12 nel