EE IISS'S S'S RR
— The _wave_equakon
@
Thee infin
The infiuile String Praflem +
for ther nolubion & Tha wave 04 a alrall derive pf alembert forme!®
with te Ul ft) amd Moe infinite abring CA0e?
Camider the following, LvP: ,
Poe, 2° s
Bee Be, mca ce, be oO
pes lao) = fia) (inthal dinplacement)
ot Uplaye) = 4) Cinikiol oeloity) jw
ee Comparing (1) wi 7 ie
raping cake ent we Fo
Az),
Gneow
Reo Care
° y
, pt Une F her O°
C1 in of Mppompe lngper belie type POE.
the elecwractoress htm ou" to, yorts of He opm
Pdva Bate * Oo tm,
gre ree te fe te:
He choractorisbe es are
a we Fe
scleiny, Oe qh oak es gud abe oa
Lak un choose, q ynak, We ak
ue Uy ee Mata En
Une (us Un)
(45000) GE» By (Men) ae
2 “ uw
a Og 2G TM
Sinitorty Uy # a (Ugg ~ E4Gy +Uan )
2.
74
Duloak tubing Here enbramiens in (1) , we ¥
Us, =? ed ae Gs
% ,
OM ustiich ta lenown on conenien! fu sty ant ee
joling (B) wre GO
ae i : 2 $e) whore PQ) i om orbitvar
Oa ) , funekion HM twdefendet
5 ink rating, werd Y we at
u(gn) fewer + ¥($)
ufere (4) fom
F §
2 ¥ 4 OT
2. Ulae) = y (atek) ag(a-t) 2 - (4)
Now, qe Ula) = Fa) gee.
we) + eo) = Fa) _. - -—-—)
wend Up nse) |e) WEP
uve) PO) TI (®)
z © we cbfain
B Inteqrotin (6) From ro & m
v t ewe) 2 fort mo (3)
~
Di filemboert polulion be the nve (1-02:
hain formule io CF great interest in itowlf, amd it ouside the-
preblem s eam vergence f infinite purity tm |Be Fouviur
herr obpreoch- I EEIET'ES!C”CS”CSC~CS
9 ;
Re D’Alembert’s relution ho» an inferesting. ink b
ng. inferpretobien =
tere
given im Figs
canider the at- Fl .
ak - plone and a poist Pe?
Pie te) > Draw bere chavackeris bes
Hrough P beckwards, until Th
welt (Pe initial Une , we
Re aeania of A ond B- ‘
Pe quia of Mee Gwe chareekrs
~ hes are aaot = ach:
Equ (3) yaveals at the polubon ula8)
at Pp (tete) s*” be sbloined by averaging He value & f of
A ond B and wmlegretiy 4 obry Re wow
leuk jwitel valu® the
Aun ia depew on the
jet lhe too. Hever Re seqme’
am usta te value Sf unt)
domain f defendence GP
p 4 eudas 34
PAS eolted the characterfsie
in of, deferminemes,
Al facet)
Fags chaxeclerishe trang
Ria aolutin be phow Rat the preblem ia well pored«
are pel Sf He #98:
gu po SH _ wen cm, EXO
at~ aay 7
with = eF 3 uta) 2 FO), Fen
te austin) 2 J) tye Te
ot
gute Bok Agefal ce oot Wr Tel Ce
From @) years (fir) - fo] otf fort) -f.0-4)]
+e [tae 460] de
ak anh
7 ~
sy Vs
&
=
‘
Le tse e ee) ~
Shue fer ony finite Hme juferval He amell change in Leaders ®
es initial condition onl produce preber Benalla, pmall chan
ain Whe ool” obows fat He problem io wellbened: t
Special care 9’ Alembert's formula ®
caseE: (initial velocity 270) ° Supper’
ulas) > §)
uel) 2 O°
Khe ateing ler Ie
SBe 9! Alembert oetetten in
ulate) = a+ Leo) 3 fardt)]
Corer? (Initial dinglacraente 207°) \ Supper the otring les Ber
fellewsing Let ulae) 2?
Uy Ws) qo).
Tm his core, The pelution ir,
: ard
sf awdt.
ult) = Ze
ae
ee.
Problem &
yeonte the Linglacemet of He phring and He ends J
eo mekon ofthe atring in dercribed
of ut) te : i
the Kee te onlay yalue problem (reve):
- ** pet ahs - = 4, 2haths eckeo -&)
ee: es 20; UCL) = 0, OCECE ee
ype: alae) = fa, Ue) = 4%, oeaeh --()
the wend f condition mG) reflec’ he foct the obring fo held
fined ot fre be end pointe are and mele
Wwe phell ery fhe methed t arbaration ef variables te
pave Vile [prefalew-
okt? Ler uat)= XOTID = - - @
wate © oe Oe get
Z
Subobluting (y)
xt i(e) =e” XO) v(t) m; ar" le) x" (2) :
ert) Xe) ok i usBure Kiso eanaten=
Wato que, pile) = MR TIE)A Oo - sy)
xia) - Wx) =? -- -@
Uhree froasible cape Gccur ,
coneh? Lek ko: Setting k= AM. $Re aoluliom ore " bq.
jet - det
Th) = AC + BSE
An -4
XM) sce + Be *
= joer
Now, Be let) = ° ent uwe)=e g”
aL -AL
aap eo ome Se ape 72
Tromerjeneou
dD.
Site fe oo oyptem fa Linen asus in coe
Were, He coeffierent determinant = | to \
ae
a ,
é
2 ohh ae.
re a unique polnvien fur e wd D, ourhrek ia
Pen one eck
» ylat) 2°, which i « briviel ost
cont, Uk kee “them astuing, (5) and (6), we hava
Tis AETB, ¥Oe ea td.
( Ak) = A
Applying wear -U(@,e) = ° cand Up Cha) © 0 You
' ua,t) 2, iPieh ix @ tewal all.
ee Lek Keo. st yeah for come jem and Ate.
She aolubow Of T(E) eeu Te)= 2 ie
rey 2 A DINCAHD * % eon (EAE) ,
he nolubiorw Sf x") a Ara) eo ia,
Ka) = ¢ pin (Ar) +p co Ya) ushere A,B, ¢, 0 ore,
, covttonk
stem, last) = XO) TH
~ Crain (eat) gem (0)] [eninGr)+ Dem Qa] = — ©
doen vai =
4
‘he courhy prcblom,
Let A Uagt Belay + CU yy + F (2,4,4,42,%4) 20 ~
ushers Al), BA) and <(x,f) ore uma F a omd d
and fF be o eurve in the au4- plone « Be preblem
Es finding. He polation utay) of Re PPE () tn the nbd
ger paliaf ying the fallowing. condi kions
ue fe) a gee 2!
Boe GP) = @)
on Tw eollel a Cauchy problem She condi bows (2)
ond a) are called fRe Couches pene Paes
Riemann’r Methed:
Consider
Ly) = Vay cmc
be and F axe combi nuourlg differentioble
et ke KH aomdy4-. Shin toa Unear, and
tien udueh to in the canonical farm.
nt buy eeu > flay) -- -
ume o,
for oth Yop’
ex ly perbolle eque
Whe charockerishes are
AL = content and 4 = constant
tek vii) be o function having. ceninuows and order
ord
partial derivative. Shen
a9 ony ~U Vay = (vung = (avy)
avin = (avu)y - 4 vr)
lov ty = Covt)y — ub»),
rgL{u) 2 VYxp roVUa Thvuy 4 cud
= Way + Woody (ery 2 ou), ular )n
ste Fa, * (oeey) - u (oy) 4 coe
(omy - ad ely ree] + Gee seeile
un hela a Le) = ull Cv) + Ma tNy
nctext LEW) = Vay — (a), 7 (UY) pa cv
‘4 eo
M = avn Uy
and No = VUx thou
vel) - ole (v) = MytNy = = = (2)
Re cherater Bb) Lig called He adjeinr eferaler
LU: Q) ie called elon lin identty.
oF L= LC, Ren the cherator L ia nord to be
pag offen:
T Green's Meorem:
&F imbegrabion D ond U ond V be differen -
Re region
- Goble tore wD and conlinucws on. hen Green's
eorem pteter Yat
i (a7 Vy) 4894 $ (ay-vex) - -
D> We phall now discuss Me Souchy. brablem for hy perbolre
Lek € be ow cloned curve beundin
eq LAW) © Ways ata a aug eA =Fay) --o
Lek FP be a pmect® mitial eurve - We opsume WRat
Re Jengent be Tin nowfere porallel bo He a ex or yn,
(or ae cave, 4 ceonstant are characteris he of)
We pubbere Rak aw amd Uy (er Uy) are prereribed
oleng Ts Let P,0) be o pein of wich ue peek
f the couchy problem. Let fhe characherishes
Broagh ee) intersect the inibial curve P ob ond Rr
Let D be Re region Jpeanded by We Pu,
ho cloned contour PARP (94,2) Fi
We pholl now apply Green! &
kReerem im Hin region.
ce polulion Re ently ©), we sblain
Qu aoing |
u) - — ut?) dad 4 ne
ij (Gam ‘4 M May - Nan) + fiasy Jen
on RPG nat 4 dune
re Deis e- ee]
fv o-
Now, { Naan = { (Vy t bou)da
p e
w
=f bvada aval -
\e0 Javad
e
o
: tle *{ (oo = vada
. Wf (ee - ov (oT) dua = [ Smayente) * J. mag
dD
- cork + “Cos, +fa(oe- cva)da
e
(l= {J [rue oir OF} andy - f * andy -ne)
D P a
— Pmdy + Corde. + Puloo- vada
R p
Lekw, it penible, choose Re function (A440) &
loa te polution af Re adjoint aque LF (v) 29,
pabralyny Ro andibions :
ze bv 4 = e
ay 2 avon A=%
ok xeaord yee.
Lik entato,
~v 2h
Such O funckion 3 OS ae) ,
co Riernonn Fonction J Reimann Green'n fan «
(adadat © ase)
ia called
TW, - [wre > s ay (ody baa) i
eg Sofon ey ---@)
Eau? (2) give 4 of e ufen gy Bd Un ore lprencribed oleng
Re curve [- Pe if
However, if wand “4 are freroribed | the Honk.
/ Tard, - Cul, = ik COW), da + (av) i) qe,
Fo Ge}, fee coy) = Nonna
+f F opt andy. ~~)
Further on adding, (yond (u), ee ge
C= [eosdes con - [ovted be)
-+ FPa(onts -ryh) 2% mania)
4+ [j ofoneety
He pelulien at He point (6A) debends only on He Couch
doke along Yee ave OR anf.
oR ee
+G8 ma
ex Frese Rat fer We eau LW= Uns * ?
Re Riemer four in V4 6) » J. ( (A),
seu J, de a unl Bf bof oreo.
Be! \pvity thal He Green'a be ee a
=o
a ay eas ay)
pobjat be wae, Bios aah om yore fiver 4
wus Sn) = GSN ton) eo) ee tn
(ter)? Uniquen or
aolubion : Cheever) we
BR polution Ff fe [ellewing- frstlem, if if tkeniote, tn unique.
de - = Van = EE) | © ench | Ere
alas) = FM > cansh
poss) = YO > oaneh
ule, &) = UCbe) =o bro
D> Let Rere be cu rolvbens vy avd UL F ae above
proflem: Ren y= UUs will be @ al” F He flowing
pretlem :
Comp i der
yer Vanes eneh, be
0 (n°) =°, oeneh
Pelee) oo oemet
eee) SP Ohk) 72, bye
d
ei eS (ove aed
4
@inerve Rat Ele) in o di fferentiolle Ga EE ata
(ae) 3a feirce Aifferertiole:
sel). S (& wanda Verte) a®
ab
L
2 S Vere a [em wifi f < “gan
lek) 8 2 VMeGED AD VEE
waht) 20° a Usk) #2 , WEY
ak
as ee
won Whar ole) =o
Hence 0, 20, PERE
Yyoin in pemible onl if
Sines v0)
ve (Me ” oan) dn =0-
eu) = combovk.
A nimer Vie) eo Taplr
Aa aye) 2 o>
Ewer
EWs
oe te) ou Gt) 2b -