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General approach to solve separation of variables problems
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Providing a general approach to solve problems involving partial differential equations using the separation of variables method
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Backetosparationof VariablesAll ofthe ideas from previouschapterscanbeappliedto S.o.v problems
EXAMPLE
Solve P'u in the unitsphere o ere s of
o Lit Of Oct
with boundary conditions acres fro and u bounded in the domain
Look for axisymmetricsolutions independentof e
Look for separable solutions u Rcr 010
O'u Sino
f r
Ey Eno 8g e
Separate variables to
find R R D check
ER
Sino Q t Xsino 0 0
Or 1 Ma H 2Mt thH O with Maso and Hfe DCaso
B C on H are that It remainsbounded at O e it or M I s
Recognizethat HCM solves Legendre S L eigenvalue problemwith In nents no integer
and HG PnM O Palaso
Back to radial DDE lookfor Rare
some k
R ER nineR O Euler type equation
K N K nts Exercise
Can alternativelylook for Frobenius solution R
Find I E C a tats o Can C Cnts
Ear't a to
RR an D tics Exercise
int
Given N R arntbr
a bounded when r o R also bounded b o R ar n
General separable solution is ur O Anr PnCaso
B.C at res will determine An
Because ur s AnPicaso
fro E An CLI
CpnPn
An 2n
1Palmfcoano
Recall M cos0
cos gu
, SPECIFIC EXAMPLE
fco 9 981 upperhalfsphere
lowerhalfsphere
IEE
An
any Pacman Inn I fansFIE247771
ants Pn Pats Pn's
Recall An I Pats Pn
12 Pnslo Pats o net
and Ao calculate directly
3 fromintegral
zero
Also recall Pacts is odd pa.fi fundjmsf ground
Andyou can show using Rodriguez formula that Ants Exercise
1 521
4411
EXAMPLE
The displacement ofthe membrane of a drum thatoccupiestheunit circleobeys thewaveequation
242 TU O Eres
Oe OeLit
Supposemembrane is
fixed at r s u o at res and ult o fir 2 t o
gr
Find axisymmetric solutions for air E
Look for RIOTED X
a
I 7
121
T XT D We know
fries o
r s 0
tr r o u bounded R
bounded
Erray becausemembrane can't bedivergent
Bessel'sequationofindexo
R J jor withXkjot joeis the KthrootofJo
Andfor anyK
T j T o T Akcostat Basinjoke
Ur E EÉArcosjeet Basin just Jo jor
Initial conditions fir EgAkJocjar
gir EsBajoJo jor
so a rycr
Effigies go.at
JoCiand Bk
jotgcjpa rgfr
JoCjonrJdr
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