Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations

255 Pages·1958·5.66 MB·english

by  B. Noble

OTHER TITLES IN THE SERIES
ON PURE AND APPLIEMDA THEMATICS
IntrodutcotA ilgoenb raTiocp ology
Vol1..
by
A.H .W ALLACE
Circles
Vol2..
by
D.P EDOE
AnalytCiocnailc s
Vol3..
by
B.S PAIN
IntegErqaula tions
Vol. 4.
by
S. MIKHLIN
ProbleimnEs u clidSepaanc e:
Vol. 5.
ApplicatoifCo onn vexity
by
H.G .E GGLESTON
HomoloTghye oroynA lgebraViacr ieties
Vol6..
by
A. HW.A LLACE
METHODSB ASED OTNH E
WIENER-HTOEPCFH NIQUE
fotrh seo luotpfi aornt ial
differeeqnutaitailo ns
by
B. NOBLE
SeniLoerc tuirnMe art hematics
TheR oyaClo lloefgS ec ieanncdTe e chnology
Glasgow
PERGAMON PRESS
LONDON · NEW YORK ,P ARIS, LOS ANGELES
1958
PERGAMON PRESS
LTD.
5F itzrSoqyu arLeo,n don
4 & W.l
PERGAMON PRESS, INC.
Eas5t5 tSht reeNte,w York
122 22,N .Y.
P.O.B ox LosA ngeleCsa,l if.
47715,
PERGAMON PRESS S.A.R.L.
Rued esEc olesP,a ris
24 Ve
Copyright
©
1958
B.N OBLE
LibraorfCy o ngreCsasr dN o.
58-12676
PrintIenNd o rtIhreerlnaa tTn hdUe n ivePrrseiBstesil.ef sa st
CONTENTS
PAGE
Preface
vii
Hombea snioct atainodrn e suflrtoCsmh apIt er
x
I.C OMPLEXV ARIABLAEN D FOURIETRR ANSFORMS
Introduction 1
1.1
Complex variable theory 5
1.2
Analyftuincc tdieofinnsbe ydi ntegrals 11
J.3T heF ouriinetre gral 21
I. 4 Thew aveeq uation 27
1.1)
Contionutre gorfaa c lesr ttayipne 31
l.aT heW ienHeorp-pfr ocedure 36
1.7
Miscelleanxeaomupsal nerdse suIl ts 38
IIB.A SIPCR OCEDUR:E HSALF-PLAPNREO LBEMS
Introduction 48
2.1
22. Jon'emsse thod 52
A duailn teegqruaalt mieotnh od 58
2.3I ntegral feoqrumautliaotni ons 61
2.4
2.1iS olutoifto hnie n teegqruaalt ions 67
26. Discusosfti hoseno lution 72
Comparoifms eotnh ods 76
2.7
Boundcaornyd istpieocnisbfi yge edn efruanlc tions 77
2.HR adiatiobno-utnydcpaoern yd itions 83
2.9M iscelleaxnaemopualsne rds e suIlIt s 86
IIIF.U RTHERW AVEP ROBLEMS
Introduction 98
:1.1
A plawnaev ien cidoentn wtos emi-ipnafirnaipltlleae nle s1 00
:1.2
:Eadiaftriootmnw op araslelmeil- ipnlfiantietes 105
:I.:J
:1.l�4a diaftriooamn cylindrical pipe 110
Sem-niifinistter piaprsa tlolt ehwlea lolfas d uct 181
:1.[; A straicpr oss a duct 122
:I.Ma iscelleaxnaemopualsne drs e suIlItIs 125
lV.E XTENSIOANNSD LIMITATIOOFNT SH EM ETHOD
Introduction 141
·1.T1 heH ilbperrotb lem 141
4·.2
Genecroanls iderations 147
4·,3
SimultaWnieeonuesr -eHqoupaft ions 153
4.4 Approxifmaactteo rization 160
4.5
Laplea'ecsq uatiinpo onl caor- doirnates 146
4. (}
Miscelleaxnaemopualsne drs e suIlVt s
167
v
VI UONTEN'rs
APPHOXIMATEM E'l'HODFl
V. SOMJ<:
5.1I ntroduction 178
5.2S omep roblewmhsic cha nnboets olveexda ctly 180
5.3G enertahle oorfya s peceiqaula tion 184
5.4D iffracbtyia o tnh iscekm i-insfitnriitpe 187
5.5G enertahle oorfya nothsepre ceiqaualt ion 196
5.6Di rffaticnob ys triapnsds litosffin itwei dth 203
Miscellaenxeamopulsea sn dr esults 207
V
THE GENERALS OLUTIONO F THE BASIC
VI. WIENER-HOPFP ROBLEM
6.1I ntroduction 220
6.2T hee xascotl uotfic oenr tdauiainln e gtrale quations2 22
Miscelleaxnaemopualsne drs e suVlIt s 228
Bibliography 237
Inde 243
x
PREFA CE
Them ethoddess criinbt ehdib so osko lcveer tbaoiunn dary-value
probleomfsp ractiicmaplo rtainncveo lvpianrgt idailff erential
equatiAo ntsy.p ipcraolb lreemq uisroelsu toifto hnes teady-state
wavee quatiionfn r esep acweh ens emi-inbfionuintdea ries are
preseEnxta.m plaergsei vferno eml ectromatghneeoatrciyoc,u stics,
hydrodynamics,a ndepl oatsetntithcieiaotlry y .
Thet win aimosft hibso oakr et:ot akteh set udefnrtoo mr dinary
dogrseteu diinetsto h er eseafireclhcd o verbeytd h Wei ener-Hopf
techniaqnudet ,op rovitdheer eseawrocrhk weirt ah r easonably
comprehesnusmimvaero yfw hacta na ndw hacta nnboetd onaet
them omenbty ttheec hniqTuheer. e adeart'tse ntiisdo rna wn
particutloa rtlvhyae r iomuest hofdosra prpxoiamtseo lutoifo n
probleOmnseo. ft hree markfaebalteui rste hsre a ngofea pparently
unrelattoepdic cosv erbeydr amificaotfit ohnets e chniqIutie s.
hopetdh asotm eo ft hceo mmenitnts h e taenxdit ne xampmleasy
suggseusitt albilnfeeo sfr u rtrheesre arch.
TheW iener-Htoepcfh niwqausie n venatbeodu1 t9 3t1os olavne
integerqaula toifao s np ectiyapleD .u ritnhgwe a ri tw asn otebdy
J.S chwingeri nd(eapnedn dbeynE t.Tl .yC opsotnh)ap tr oblems
involving dbiysff ermaic-tinfiniitoen pcloaulndbe esf ormuliant ed
termosfi ntegerqaula tiwohniscc ho ulbdes olvbeydt heWi ener­
HoptfechniTqhuese o.l utdieopne nodnts h ues oef F ouriinetr egrals
too btaai cno mplveaxr iaebqluea twihoinc ihss olvbeyda nalytic
continuaMtoisootfnt .hi bso oiksb aseodna d ifferbeunettq uivalent
approdauceth o D . JoneFso.u rtirearn sfaorrem sa pdpilrieecd.t ly
S.
tot hpea rtdiiaffle reenqtuiaatlai nodtn h ceo mplveaxr iaebqluea tion
iso btaidnierde cwtiltyh otuhten ecessfiotrfy o rmulaotfia onn
integerqaula tiForno.m t hipso inotf v iewth eW iener-Hopf
technipqruoev iads eisg nifiacnadnn att uerxatl ensoifto hner ange
ofp robltehmasct a nb es olvbeydt hues eo fF ourieLra,p laacned
Melltirna nsfoIrs mtsa.rt theibdso owki tthh ien tenotfir ounn ning
thien tegerqaula tainodJn o nemse'tsh oadl ongesaicdohet hebru,t
as twrhiet inpgr ogreistss eede mepdo intlaensdcs o nfustion g
elabotrwaoet qeu ivamleetnhto dJso.n emse'tsh osde emssi mptloe r
mea ndh avien cluodnleysd u fficideentta oiftl hsie n tegerqaula tion
I
methotdoe nabtlhere e adteofr o lltohwle i terature.
Them ateriinta hli bso oskh oubleda ccesstioab nlyeo nweh oi s
familwiiatrh Ltahpel atcrea nsfiotrcsmo ,m plinevxe rsfioornm ula,
andi ntegriantt ihoceno mplpelxa nTeh.efi rscth aptiesinr t ended
tos upplemtehnuets uauln dergracdouurastieenc omplveaxr iable
vii
PREFACE
Vlll
theoarnytd, of amiltiharere iazdweei rtt hh ues oeft hFeo urier
transifnto hrcemo mpplleaxnA est. h bioso hka bse ewnr itftoern
workwehrosis net earrepesr tism airnai plpyl icoaftt hiteoh neso ry
rathtehrai nnt hteh eoirtyst ehlseft ,a ndard of rigour may not
sattishpfeuy rm ea thematthiocuiigstahh n o suludffif coor practical
purops.e s
It iimpso rttoea mnpth atshiafztre ot mhp eo ionfvt i eawd opted
int hbioso tkhe es seonfc Weti heen- eHrotpefncihqiutseh aitct an
beu setdoo btaniunm erviaclaufleo psrh ysqiucaanilte .is Ftor
variroeuassI oh nasvde e citdooe mdin tu mertiacbaallle tsh aousg h
faars p ossriebsluaelr gteis v ienna f orsmu itfaobnrlu em erical
computaantdrie orfnee ncaeres gtiov eftneh we exsiesttsi nogf
tabulvaatleuPder sa.c tniocd ailslcyui sss igointv hepenh yosfi cal
implicoaft iroenFssou erll tesc.t rotmhaegotnrhegyita sisp ch ould
befil lebdyt hveo luimne stehribsiy De s. JSo.n es.
Int heex amptlreesai tnte htdee xhta vcea rrtihaeend a lfyasri s
I
enouigneh a ccha tsoeo btaatil ne aosntree soufpl hty ssiicganli fi­
canicnse i mfprolmeT. h iisps a rttole yn coutrhabege eg iwnhnoe r
migthhti tnhkac to mplifcroamutlecadabe ne i nteropnlryew tietdh
thaei d oefl eacntc roomnpiuct er.
Thset imtuowrl iutste h bioso cka moer igifonrmaal c loyu orfs e
post-glreacdtusuautrgeeg seb syPt reodDf .C. .P acAkm.o nogt her
thingasmg ratteofP urloP fa.c fko r orgiadnewiaozlri knign g
I
condiitnhi iodsne sp artamtte hnReto yCaoll loefSg cei eanncde
TechnoGlloagsyg,oa wma. l gsroa tteoPf ruolf .S nedfdoorn
I I.N .
askmient gow ritev otlhufimosher i sse rainefdso ,hr e lllIp sfugges­
tioinncs o nnewxiitsohhn o rtaem nainnugs ctrhiawptat sm uch
longero rtihgaiennn avlgilesydaI .ti sp erhwaoprsmt ehn tioning
thath avIhe a dt oo mitc haa pdteearl wiintaghp plications of
Besfsnueclt diuoanil n teegqruaaltt iodo inpssrk o ebmlsA.r eferee
fotrh Per oCacm.bP .h l.iS o(cw.h onsaemc ea nnotb etn roawc ed)
suggetsomt ete hda t thesaerb epe trsaotcb klbleytem hdsWe i ener­
Hopmfe thHoadv.i wrnigt ttehnbi oso akm n oetn ticroenlvyi nced
I
thatths eu ggeissct oirornbe ucittna , s entshere e f'escr oemement
provotkhbeeod o Ik !a imn detbotP erdo f.J oDn.fe osvSr a. r ious
rerfeencaensdc orrespaonndtd oeD nrcW.e. E .W illiwahmos
carefcuhlelcCykh eadp Vt.eM ryt hanakrdseu teo ptrhien ftoerr s
acrcautwoer okna d ifficmualntu script.
B.
N.
12.57.
31.

SOME
BASICN OTATIONA ND RESULTS
FROM CHAPTER
1
Thef ollomwaiypn rgo uvsee furle ffeeorr.Ae ntcimfeca tor
ex(p- wi)ti uss etdh routghhbioousot k .
oc + iT k kl+ ik'2 (1k 0,k 2 0).
= (f : = > >
(2oc- k2)1/2= -iP( -OC)21/2. [(.11]4 )
Y=
(k OC)1-/2 i(-ock) 1/2 (-k oc-1)/2 -i(oc+ 1k2./) [(1.]1 2)
= : =
-iki foc locii foc i rse aanlld a r.g e
Y= = ° : y!":::i
-00
where
Wewr itfeo,ir n stance,- 00
whearneyo ft hefsroem sma yb eu seadc rcdoitnogc onvenience
provtidheatdth eirse r nioos fkc ounsfion.
IfI (c/>)\x ex(pT _aXsX) � +<Xl,<I> +(ocir)se guilnTa> r T _
< A }
If\c/> (x )\ ex(pT X)a sX� -<Xl,<I> _(ocir)se guilnTa rT
< B + < +
[§ 1.3]
Ifc/> x() ""a sXx �'1 + 0t,h e<I>n+( oc"") - oc'11-a soc �<Xl �nT > T_}
Ifc/> x() x""'1a s X � -0,t he<I>n_( oc"" )-oc '1-1a soc � <Xl T T+
III <
[C(f.1.47) ]
(j(..x..g).,( xa)sx meantsh ajt( x)g (x+)h (xw)h erhej g 0 as
x The n--+ ua mbmearyb ei nfi=n iOtceac.s ionaasli lnt--+y h ela st
p--+a ra.a grwaeup shje a.. ..t.,om ega jn = Cg+ hf osro mceo nstCa,in ft
thvea loufCe isn oitm ponrt.t)a
x
CHAPTER
I
COMPLEX VARIABLE AND FOURIER
TRANSFORMS
1.1 Introduction
Onoeft hree marfkaeatbulorefte hsme a themadetsiccraoilfp tion
natural pbhyme enaoonmfspe anrat diiaffle reeqnutiaatilist o hnes
compareaatswieiv teh whichc absneo o lbuttaiifonocnrees dr tain
geometsrhiacpsaeulsca ,hsc iracnliden sfi nsittsrebi, yp t hmee thod
ofs eparoaftv iaorni aibnlc eosn:t rcaosnts,i ddeirffiacbullet y is
usuaelnlcyo unitnfie nrdeisdno gl utfiosroh naspn eocsto vebrye d
thmee thoofsd e paroafvt airoina blWeise.nH eoTrp-htfee c hnique
provais diegsn iefixctaennotsf i ortnah neog fe prtohbaclteab mne
solbvyeF do urLiaeprl,aa ncMdee lilnitnle gsr.a
Toi llustthreraseteme a arnktdso r emitnhdre e aodfet rhr ee al­
variFaobulriein etrec gornaslti hdreperre o blceomnsn ewcittehd the
steady-setqautaet iwoanv e
(1.1)
Suppwoesw ei sthofi nda s oluotfit ohnie sq uatiinto hne semi­
infinrietgei- o00n < 00x, y <0 ,s ucthh arfot r epreasne nts
�
outgwoaivnaegti nfiniinet ayc ht horsfee ep acraastees
()i rfo f= ()x ony =O,- oo< x <oo;
()i i orfol=o yg ()x ony =O,- oo<x <oo;
rfo f= ()x ony =O, < xoo ,<
(ii)i ° } (1.2)
orfol=o ygx () ony =O,- 00< x <0.
Separation-soofl-uvteaixroiifnsaostb(r l1 .ei1sn)t hfeo rrfom=
X()Yx (y)w ith·
X(x= )e± i= Y(y)= e±Y1I, y= (ex2_ k2)1/2,
wheries p aar ameTtoegre.tw hietthrhf eca tt hatthr ea nogfxe
iinsfi exn itthesi usg guessoetft s hF eo uriinetrei gn-r 00a l < 00x, <
anidnf ca wte s hotwh tathfi er tswtpo r obcleabmness o levxeadc tly
byF ouriinetre gtrhatelh si:lr eda tdoes q uatwihoinccsah bn e
solbvyet dhW ei en-Heorptfe chnique.
1