Section 7.1 i The Heat Equation

Plan The Heat Equation L space occur

The IBVP i Dirichlet B.c

The IBVP i Neumann B c 4

The Mixed Problemso

the IBV Pi Nairn 13

R k Dirichlet B C i UCI O 0

IgTemperatures

ult L to

Neumann B c i 2 60,0 o Heat Flux

UCE L O

If U is temp is heat flux

R O

YA yo U L U2

Ht'shot constant temperaturescoldDX

Mixed B C Uct o o TempUct L o Heat Flux

B C on d U can be achievedwith insulators

Them The B UP

BWP JEU k 2 U i UCE O O RUct L to L 0

Lhas infinitely many SoCs

act x get 2 on ek T cos

MelG

Them For every piecewise continuous fCx on 0,6I 1

IUP with fCo fCc O there is only one

act x Soc of L given by 2 where

Cn Z J fox cos Cruze dxn o 1,2

proof Buf Separation of Variables

Simple solutionsUct X VCE WCX 2 1403 0

I Uct D o

7 U e R 8 6

Wcx Oct k vets w ex is off w zw

iz VIGIL w a constWCX

f st X

Ect Ravel

w cx tawcx1

maxi vets was fVCE W'co o

Ct WIG oa t 3 W'co o

lw.ch

ircts kavetI w ta w to

f W'co o W'CL o

u i BUR Non Zero Sols

T

Teo W cx 0 Who to W'CL o

EVVCa Co t do X W'Cx doW'co o WH o 06

01Wo 1CkooseWoCx3 1

2 0 W ca t a wax O who to W'CL o

Characteristic polyn 021 2 0 It I KI i

Gen SocIVVcrdec.cos.ci Ix tczSirsHI IW'Cx VIC Sirs VIC cos x

B c O W lo O t VIS L fc2towcx c VIx

W'ex HTC Sin x

B c o W'CL VIC sirs 2

Sirs VIL O KIL UIT n 4,2

an T Wncx an cosC choose cn

tvvncx7cos.fr

Vnctl anrant

Unctix cuKCET't Cos E n 1,2

UoCt X Co I

SuperpositionproportIUCt.H coztE.cu.comtJtcos nzr

ME

Proof i IVP

we fix initial Temperatureact o X f CX tko O f'c4 o

f CX i continuous on IO L

f Cx Uco x coz t 2nF on cos I

JcneZ_JotfcxIcoseIJol.n 0,42

14

Exaurple i Solve 2g Us U 20 XE 0,3

B C 2 46,07 0 IC Uco 2710

XE 0,322 444 0 7 XE CZ 3

Sofb L

Separation of variables T e

7 gfad 1 3 1

Ult X VCE NCD i

th o 7 Iz ex

VI W X constW

X

v Ct duct W ex a WW o

lwicoi ow.esat CBVP Eigenfunction ProbH

VCtl2 0 a 70

2 0 W O WYO 0 Wk O

WH cotdox wie

do D WA Ceo i choose co

1o 0 Wo t

w t a W O

charact polyn At 2 0 re I

E i

Gen Soc WCX C cosCVIX t C sirs HID1 1W'Cx c sirs It VIC cos t

B e O W'co 0 t VI L s

wcxt Ocosli.mxw1cx VIC Seis 44TH

BE O W 3 VI g Seis 6 73

Sirs 1 53 0 VAT 3 not

n l 2

n lwncxi cos.CI

lvnctl cn.ci IUctix coz EI e e

74

fascistsF C TI

ffcx Uco a

l Tg Iz i

f Cx Uco a Cq t NEE Cn cos 4 y 2 3

an 23 Jo fad cos DX n z c

cos ok

Zz sin

ing sinful sin E i sifuT If

cnn sinC I

co 3 Jos f Cx DX

J F DX

3 232

lcor71uct.xt

Et E.ch sisCzIje4cosez

DEE