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

The Mixed Problemso

The Mixed Problem

Solve Heat Ef R O

yaIz U CtX R Exact2insulation

ICE 0 O E L O on t c 6,00 X C Co Li a1 i with B Cy o X L X

1 a Act O 0 Uct 4 02 a insulation

or

act X b KUCE01 0 UCI L o

Them The B Pi

IBUP JEU k 2 U UCE O O Ra UCE L o L 0

Lhas infinitely many SoCs

k can tact x 2 one sincerely 2

Mel

Furthermore if fCX UCI o X satisfies CL

then Cn Z fix Seis n i DX

Them The B UP

IB UP JEU k 2 U 2x UCE O O Rb U Ct L o L 0

Lhas infinitely many SoCs

knILIJ2tuct.xE en e Costin EM 7

Furthermore if fCX lecto X satisfies CL

then one to fix cos n iz oct

Proof a Separation of Variables

UCE X VCE WCx UCE 05 0

sauce.is ode U e k 22 U

to Ct Wco Ovee who o

w a consoWCX Il

f is VX

Neo Oicts ka w c tTWlwiakit

vcts.cc Eigenfunction Problem

KoA 0 i r't 2 0 LEI IVI e IVI if

Gen Soli Wcx c CosdIx t ca Seis XIXB C o Wco g L t Ca co

wcxi cssiu.ITB c W'Cx Ca VI cos C x

W'Cc carat cos Lok

cos L o KIL kn c LIM l 2

an ntYI wncxi sisGntaEVn Ct c e Kant1

unit x an ek

Sin flat x1ult x on Ek It sin z x1 1

I C U Cto X f Cx

f case on Sin can 422 x

Lemma i Any piecewise continuous f on co L

Iff awogshine series expansion as an

Cne Z J fox sin n 4242J ok

ThrmCa T

Example Solve 2 Us 2 U t o XE Co 3

I Ic ucoLo

xe 0,337 XE CZ 3

soul 7 aseparation of Variables 7

T 2 3UCE X Vct WCX on tz

a ex3

2 U IU Uct 03 0 Vet Wco e O

y 2x act 23 0vet W 4 0

44th Kw a constµ

totuco to

lie 1W tow o wu L

w a AEigenfunction Problem

vets ce 0

2202Calculation in Proof of Them 2n 2nt2I

fwncxh s.us C2n iIEL 3

Gen Socn

DgIJ2t1uct.xEI en e sin n i x

T eI C Alt o X f Cx z

q113 Iz i

f ex cu Seis Clan 11ft xLemma i

Cn 3 J f Cx Sin n c II x ok

23 J 7 Sin n it x ok

131 ED coffin 4 Ex

22 ayn DE concern is Sz

0

2 El C cos n is

cnn.EE cosGn oEI

uct.xi 2 I cosLanish e Sinseny1 1YES

Series Expansions of on co L

Thrun If f is piecewise continuous or co 2Then f can be expanded as i

f fox bn Sirs DX

bn 3 f fCX Seis 7 DX

2 fix doz t an cosf.nu oCx

an J fCx cos DX

00Lemma 3 fCx 2 In sin n 1 Lgin Turin a 4 1

In Z J fCt Sin Girt DX

Needed 4 fCx AAI t ZI In cos cant xTIED

ay to fax cohen KE

o

Remark i 1 was proven by i odd extension of fto C L L

2 was proven by i even extension of fto C L L

3 is proven by Symmetric extension of fto CO 22Odd extension To 22,22

4 Exercise

Proof Given f on O L gLemmain f First extension to 2L iTuinal

fee fax XE cost

f X f 2L X X c L 2LL

ya gaf f te

i Il isL 2L X L 2L X

B Second Extension to C 2L 2L as odd

y t y af te f te

Al e2 I L is

L 2L X 2 L 2L X

we know that fEz is odd so it can beexpanded as a Siue Series i

00

FEET obn Seis Iz

2Lbn IT 12 fezcx sirs g ok

0061 odde

even

J fezcx Sin If ok

f Jj fCx Sin dx t f J FCK x Sin oh

Y LL X dy ok

J fck x sin f x dx J fcy Seis nzI 2L gCoty

J f sin a NII dy

f f Ca Seis na 1 DX

Sin T NEI sirs cosCz coscut SinzI 11

t

C IT Sif.nogLItfC2L

x Sin ngIxJolx C i5fofcx1Sus nIzxJoHf

bn f Jaffa Sinz

oh CLI f fCx Sin oil

bn Iz l GM J fCx SirsCz DX

Me 2k l C c2k

I I 0 bar O

n 2k I l C c2K

11 1 2

bar Z J f Cx Sin 2K DII x DX

Change name k es n

5n bzn J fcx Sin 2n 4gIX o

fcx 2 bnSin hta emits