Question

7. (a) Find the solution of the heat conduction problem: Suxx = ut, 0<x< 5, u(0, 1) = 20, tu(5, 1) = 80, 1>0 u(x,0) = f(x) = 12x + 20 + 13sin(tor) - 5sin(3 tex). (b) Find lim u(2, t). (c) If the initial condition is, instead, u(x,0) = 10x – 20 + 13sin( Tox) - 5sin(3 7ox), will the limit in (b) be different? What would the difference be?

Answer 1

on we've given + Hoat eqncy Bugx = ut & conditions - OLX<5 , U10,t1= 20 , 115,t) - 80, tro Cu(x0) = f(3) = 12x +20 +1351WX)-55'1877) Now, we we separation of variables les U (2, ) = X (3) TCL) then from ci) 8x"(X). T(t) = 804) T'(t). r also ulo,t) = 20 X(0)T(U)=20 ito or & 115,t)=80 – X$)714)=80.2307 Robot - - constant) le tour a"-BX=D functhy Fon - fuhed og x only V : T'- BKT-OU So there are theree comes for k le kyo akzo & kco, u cave when kz 0 : - we get from liv) T=0 & IT = G] [ By integrating it] & foom (lü ) X"=0 7 X'= CZ 71X = CqX+(3] :. Som is 14(mt) - Cilcox+3) - (0) = case ). when k so! Suppose k = 12e it is any no. : from tin & ciu) I = 812 Int= 8 1²t+c, [ by integration] or to post er or F= edit al & X"- *x=0 so we put slo? ;?)x= 0 (2nd-order lineardiff. ean) so auxillary eqnis + m 2=0 mots =) XGent Gen Hence soilshun # fucriti= ceatt be ax co

Cade liti)k<0 Tel Ko-/?. So foom (ij) drivs I -812 integrating it w gel .In T=-01?tte, - - T = 60 -8727 d' C' 091 { x+1?x=0 (linean and order ordinary . diff. egin. (D?++?)X=0 + Auxillary eqnis - n2+d2=0 &m=thi. . X = (C. COSAY + casinux) = Cicostx+cosintx & Son of ci) in this case is → Tule,t? = d'étoit lacus +4+(qsintx) - Vii) ; we've got three coins of ci) 1 eqn (u)(uis & these three eqq only li) is suitable for which if t1 then ut So how we need to find this condit. & ean which Satis his this condith will the son. . 2. In case of eqn (u) this is not involued & That case isn't ponible .th eqncvis ulit=c'edit (Czet+egédk) a t 700,440 le when t increasen u is also - This soll in celso not ponible. increasing. Now we've only egh (vís left + 1 In this eqn as t o uto - tp aut. y cuid in the actual soin of 0) pulizo in cuill we get uloita de ott G1 = 20 cicle dit - 20 - alo yu15, t\ = ceola Cos5t+io Sin51) = 80 also we're given urol 12x + 20 ta sin (7x) -ssins - Ulsiti - 2010 (lios sa taging a -ssing ) 54+5 in 51)= do

from (vii) 100%,0) = (a cos dx +(SindX). i cicleos tact God'sind) = lox -20 +13sin (7x) - 5 Sin BTTX) or Acasty + Brinde =llox-20) +13 Sincax) - 5 sin(37x) from (*) (**) & (ff* 1 we'll get conc&d. A) voweto u conta de 814 [Ciclosu 2+(2Sind 2)) Het 412+ = 0 1 20170) (C) : é "=0 - limit doesn't change here in case we change the initial condits.

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