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Exomple — (overdamped Pondubim )
y= St
Jk byl sees =o 4 oat
Y() = Yo
Ly/@) =Me
Ta the onacdamprd Qumit (b> Wo) inectia t& neg Leguble
=> yen becomes Nonishon aly anal
=> xpack singulac partucbatton Cpamston
Roseale variables; y -Ax | T> Qt
Yr by'+ury = OAK 4 bZAK + WEAR RO, k=
> ns a
ae
ops
& A
Satving obletay A= 7%) SL=V/(bA) = Wie ,
& a
= gta e Wh aL fey
BEA = Se (22) cA,
Rescaled equation: gy +x +% =o
Jnitiad Conditions: x)= Lyo)< OY = %
. XO =-@AyYO= buy =X
Sork golukion inthe Loren oh a pethacbadion ox fenston:
KOE EDS KEE EKL) FE Ka(EP te
ECS FE, Ferg) + Kot Et Eat) + Ceo tyr + Bet}
= Got yd) + 2O% +4) EG A at A) Hs =O
. -t
Ab 2°: fot y= O > Yon Ae] .
act oe Yur (a-At\e7*
et Yaw er Kw=
Ct he a= X= (AGB> At\o-t 5 %, a(C- (Razayt+ At?) at
Colle oking everything: ie
x= CA + E(B-AL) + e2(c-(Br2a\t +A te) +o] et
Boundary Conditions: x)= ArEBtE Cte =% ) Ye
=> A=%e ) B= C=O
Selo) = - A- 28 (AtB)- O° ZA4 B+) 4 No
ee?
-Use-2e%e eV |
(oh course te unperturbed problern (exc) is s*ordec ove !)
Somme as in abgrbcore equoktons: Small conte ok highest order
ons att,
Peobeems Cannot sakisty both tnitiakunditong atonce.
Non-uniforrn convergence: Lim & Chey = x( 40) , tro
>o
| g.
eo. (48) A x (9,9)
y Bao, KEY = x (Kod 2) hing % (6D = (9) 5 Ntzo
Boundasy Coyets
First, find the characteristic time safes ;
XN 2 MK) V 0rd) => t=ocd),
Ve -E00N) => t= ord le)
(unless. y+ V = ord(e))
ER+AK+K=0 & |
Genucceally , x fo) +¥ (0) = xo) + X(0) = +0) = ved (1)
ARB
Plecewise approximation ;
, Kean by 2) 5 bkeale)
%KLESE) = ' out Hye), ce > 08)
Con we find a single conbinrous opproxination?
Notice thot Of) terms tn YXoag lt) ©) and Kout Lye) ace
the Samet -
™ Yo be 44) = %a7 40 (8)
Bae
tas Sor Xo
Yntlorm agproximaktans Aapbor Sot
FO ee
hut 2) = Xout Ch ES +X tonlty 2) ~ K mateh hy > =
“ok . 4 aA _
= OPEC Me SE SOT (ERE LE one
‘ At
Ext e4%X=0 5 ke —> EN4R44 =o
We already Solved ChE 2q uaktons Dys -h 4008S
An= —£ +04)
—%,
=> xlyed= Ad + Bo an
slow coho. eae! Cae dean siont
4 boundary t
(hd oun OS Me \poundaly bay .
, aS L N=0 (no ~slip
Sturt 9 > FM bom
Conalition )
Kout (42)
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