Math PDEs for Engineering

MATH 3150 PDE’s for Engineers

Homework 2

This homework is mainly designed ”to clean the rust”. That is for the practice of the concepts you have learned in Calculus III (or equivalent) which we already reviewed in the class and also a little to remind you some concepts you have learned in the ODE course.

1. Consider the following Dirichlet heat transfer problem  ut −kuxx = f(x, t) x ∈ (a, b), t > 0 u(x, 0) = g(x) x ∈ (a, b) u(a, t) = ha(t) t > 0

u(b, t) = hb(t) t > 0

Use the Method of Energy Integral to show that if there is a solution to this problem is a unique solution.

2. Consider the following Neumann heat transfer problem  ut −kuxx = f(x, t) x ∈ (a, b), t > 0 u(x, 0) = g(x) x ∈ (a, b) ux(a, t) = ha(t) t > 0

ux(b, t) = hb(t) t > 0

If there is a solution to this problem is a unique solution.

Use the Maximum Principle to show that if there is a solution to this problem is a unique solution.

3. Consider the following mixed heat transfer problem  ut −kuxx = f(x, t) x ∈ (a, b), t > 0 u(x, 0) = g(x) x ∈ (a, b) u(a, t) = ha(t) t > 0

ux(b, t) = hb(t) t > 0

If there is a solution to this problem is a unique solution.

Use any method (Maximum Principle or Method of Energy Integral) to show that if there is a solution to this problem is a unique solution.

4. Solve Questions 1.a, 1.b, 1.c and 1.d in section 12.4. (If you don’t have the book - see next page)

732 Chapter 12 Heat Equation

Problems 1. Deternilne the cqui)ihrunt icnlpcnh)uic ditrihuiion Lie

ii one-dimensional rod i)h constant thermal properlies with tire )‘ol Ion ng sources and boundary iiitdi lions:

(a) Q )). iOU — 0. gi(Li )b) Q . 0, till)) 7’. oL) U

Di, )e) Q 0. —(It) — 0. 11(L) —- I’

Id) (1 = l. nil)) 7’.

ii) - I. iiW( = 7’s. ii) L( = 7’

= .1. uw = 7.

)g) Q 0. till)) I. —(L) ÷ 11(L) )) Dii —

‘(li) Q=)). —(0) hIt)) /l=ll—(I.) —11 u Iliese you univ issii,iit, (tat lii I)) ft_i).

2. 1 niis,dr tflc ei)UI)liirIiIlIi lelilpeilIuo.’di5iiihi,iiii or u:i:iiiriii (itl’—diiiiCilsuihii;l( lIlt) Ivitli s1i1Ilfll.ti. = I ii) ilueriiti) L’flel’CY. siih) to die h,liliid;il’ I,ii(litIi,ii’. ,i)U) I) tuid :0!.) = ‘(ti) Deteiiiiiit,. i)ie )1,.sit elierny CCIICIiiiCil p. I’ 111111 hOC il

-ide the entire -111).

(h) Deieriiuiiie Liii- tea) enemy tihiWilig 1111) if hi’ id per Will III11C ii .1 (1 Bid 11) 1 = 1.,

Ic) VIiaL iCl;iliiiiis(iips Sllilliii) e\ls! liC)5d’Celi liii.’ lliSlICr’ ii par1 (a) alit) I

3. l)crertitinc lie ei4Uili(tli lii (c-iltjlcluituiv’ dIsIlihillIlill (hr 1 iiile—ttiiiieli’,illlfli) lit) eiinitti’i.’tl II itt,, )il)i,’i,,’i,i iii:uieri:I)s in Ocrieci t)tcrliial L,itil;ici ii .1 I - ‘u,: (I . t.t1ieie IS ilnC ifla(Crlil) 1,Ii = ( - l: = I I with cilli’t,i,It source I I) = ) I. svliereis or the inther ) a . 2. lucre ire ill) sources )fj = U, i = 2. K, = 2) )seC Escicise ).5,2I (h ii)0) )) Bid 0)2) =

4. If ho) Ii ends o) ti riid lie 11511)t! el), hens C Ji,,n (Ii,’ /xli’f in! iIul,imBiiluil rqnuliiin lIeU I lie iota) l)il lint) eni’ixy iii the iuti iS utinsitihi I.

5. C illSldl’ It iliie—t)lZiiCiiSiOllH( rod 1) ‘ 1, ii) kitiion

etujith and k non-it ci,iisL:inl I herititi I properties n thom sources. SUppo5e dliii ole wiiipc:iture is in hlliCIllhluil Ctlfl shin) 7’ it ,i L, Deieriuine 1’ ss e know (iii I he shady sti both (ic exipcr.iiuic and lie Itcat (tin at_I hS I),

6. ‘c tSo CIlds or a ii hinul rod It)’ letuth 1. ire insula)d. Thrc is a cox’oanl s. tame of hi unit eitcrey Q, 0 and l:etcnip—CtuiLlrC is itiitai)y i).i 0) fl_iL

Show ii:itheii ,:it catty (li_il I here doi. ioi CS si ally cquilibrtuin icinper,liLirC distribution. I3rii)y epIuin pliysiciilly.

7. I-tr the ftdiowinix p1 iihlems. dcieriniite an equt icniperuturc dintnhution ii one exists). For wha U) are ihrc so)u)iciiis? Explain physically.

(ill 5(a)

— — [I. 111.1.))) flit.

Dii -10(1 I.

iii D’u (LI)

II Di

dii (it —11). I) = I, —IL. I) = /5(a dii lt(i

(c) — = —, ‘) /1. n(i,U) = flit.

——0). ii = I). —)L. i) =

H. l:press tie iliiecnl) L’iiiisert:ttiiin law ill die e’ iii cotistaiti dieiiiiii) prilpeilies. Assu:t,e he I’

is kiii,wii iii h dilLducni ciltisi;itit’. It both cods, .tI,iiiiilt with [especi 0 1111w, deterilliilc (he 011th) diem’, iii lie roil, (I hiii: i’sii die ituiuti) ci,uidiiion Ii) Assuilie (ill_IC ic tilt SiIUli,’C’,,

(Ii) ,\s’,uille Liii’ source’, til i)Ieriiitil eileie%’ ire cc 9. Dense tire inienil L’i,nset Itililli law (iii lie cliiire eotts(tilii iiiertiiai properties h) ii(eui iIiiiC lie hi

(II)) )ii’-SU(tlltiit 1111 StliticL’%( .5(ills, lilt’ result tdent 0 (4),

Dig i( ii hiJO. Supptise ‘j— = t 4. ti(t.1() fit).

5. IL, () Ii. (‘:t(ctt);ttC tire lola) tlirnittal cite i’iii’ tiutiietisiiiiittl ‘lii) lisa )iIllu)u,,li II)(illlCI.

- Do lii DiII. Suppose I- i . iui.ll) = /11). —

f). —-IL.,) = 7. I) (‘tt)etdit)c die (0(a) i)teaiitii eneig ti diiiiiisioiia) mud it’, a )tltictiiiil ii) hind).

(Ii) ‘tiltil pail (a). I)CtCIilililL’ ii 5 tilts.’ uI /4 br eiiuiiihriuiti i’5Itis. lull lids ttitie o( /4.

itt t.,(.

12. Suppose I Ite cttiieeniration lilt . I oil ii iteiiiic luck’s law 1)3), and dIe 51(W) coilci’iuir,ituon I

fha I. (‘oitsudt’r a rcgiiit I) u L I lie flitw is sped red at hol Ii ends (I.

— IL, i) /4 - Assu,iic (5 lIlt) /4 iic coilst,IIti (a) Express tile consers 111(111 toiw Ilir (tie elitire (LI) Determine the total un,oiiiit oF dlidflildii) in

is it ruidlicin of time (using the lii)itil ctmd

dit —(Li) =j)

OCt.01 = flit.

(LI) Calculate tue Odli) llienmtil CflL’iuy in die elliot roil.

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