Plan The Heat Equation The - Michigan State University€¦ · Section 7.1 i The Heat Equation Plan...
Transcript of Plan The Heat Equation The - Michigan State University€¦ · Section 7.1 i The Heat Equation Plan...
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