Prof. Dr. Syod lkrrm Tirnizi

MESHLESS METHOD OF LINES FOR NUMERICAL
SOLUTIONS OF NONLINEAR TIME DEPENDENT
PARTIAL DIFFERENTIAL EQUATiONS
By
Nrginr Bibi
Supenbd by
Irr. Sinjul Hrq
Prof. Dr. Syod lkrrm
A
dism.ion submiEn
iD
p.nid ill,lllndt
of
lh.
Tirnizi
Equimdt
of
th. D.sre. of Dclot of
PhilMphy (PhD) in EnsiMring S.i.n@s
M'Y 201I
Chulam Ishaq Khan InstiNr€ ofEngin€cring Sciences and Technolos/,
Topi. Swabi, Pakist4
/&
(+
+
2
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+,
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t-.
w
(v/
Inih. iLm€ otAllaft,th. Mo6t
..r
uL
$€ Mort &n.fic.||t
Dedicated to my Parents
A,
My Sisterc
dt*<rrf rL.6j,ii:rre!.JrruDLllr+
GhuLm bh.q Khrn In ltut of
Engh..rlng ScLnccc and T.chnology
Certiflcate of A
roval
-Meshle€s lilathod of Lings for l{rlnedcal Solutions of ilonlingar
Time Dependent partial Difrerential Eouations',
phD ft6-b b, N.si!. 8ibi
q
is of a very sood qur,ry. she
l:TllT3 dete*'€d
ry]t.th
'l$(esrurry
'l'j :q,r
h.r PhD TlEs's on " M{y.201t, h rlr Feully ot Enqlr.rin!_
\,.4es.urK Inst,rute u,e srotrgly Ecomend d. awrd of phD Dese lo the ctudid;k.
VJq"^-.-t A{.*
Prof, D!.
hf,@d
**-
ttdd r@
\\'
,.'^ \-za)
,'n)A L r/,t1r4i6,,
PreLE..
sy.nt anrnntzt ul
)/l
Dsdco.suEtuisr
Appov.d by
tcRflor
tAcadctucs)
;---t,utlt-
Ac*ioeledgema6
thBis ca@t be the sinslchrnd.d etron of d individ@I. Th. sinc@ h.lp
m of i'i@n*
dnd c@!en1io! of all thoe in clos @nt!41 wifi the
in ils
valw. I wish to exrend Dy h€art-fclr Srdirude to .ll tho* who helped
1o
wne
a
l!ffhq
n
ud foenost, in all hmilily I thrl ny Alhn for massinS sill-pov€r dd
@Mg. in m. lo tulfiU this daulirg t8t in rhe fae of ndy ch3lldg6. I ow. ad
illinitum ro by Grrcios lad fd mting m. @! this fd in pww@ of my
Fi!$
My sup€Fisoi, Dr. Si6j-ulHaq. who Srotly hclFd me b th..nli@ poccss oftbdis
complelion ed rude me b€liwe in my pot nlial wilh his positive ed en@daging
My gntil|lle is rc l.s lo ny @supdvi$r, Pofes. Di Sycd lt@ Tidizi. hs
b..n . @.sbrn bsror, guide and suppon to me rhrcughour ttis disnario.. I
indcd, ind.bl€d to him fo. loding ! pdicd .d ro my dicli€s ed problem ad thc
fNiltul disusioc on $c ropic always cad my last.
u,
danls ae due lo Prof€ssor Johr Bulch€r, Unire6ily of Arcklmd (New
Z.{lad) for exlcndiDs inlaluble euidse in the slFF ofFqsidins dieu$ions. H.
is a peMn of grel tnowledge and inr.ller. I m also gdlrful ro Pmf€ssr Srd
Son, Mdshal UnivcBity for previdilg @ with .xtr€mely uscftn nar.rid Elded o
ny Effih wort. I sould d$ like to thanl all my t lchc.s to! lh€ intidl lhey e!v.
Specidl
lo ne dulng rh€
cow eql,
I m .lso $rni(ful to Mr. Jehegn Ba8h!' (R@ror GIKI) ed Prol Dr. Fdl Ahnad
Khalid (SI PGR4lor, academics, CIKI) for th€i. ad$iristniiv. md noral suppon
.luing Dy sludis .1 GlKl.
No
hour
of
rhEtftlns
@uldjusdry
d. oL
played by my loving paanls (S. M.
Has ud R Jd) End my swer sis&u (R ai and B!bli). Th.i. p6ys ad coopcEtion r€mdin€d a coNlet snid€ ed a b.eon oflidt o. lhis ardmu jom€y Md
rcplenished ny spirils, Without their luppon ud modle-boosting, I would havc
n.vr b.o succc$ful. 'nr€ glid in rheh eyes spaks of th.b mlold joy on lhh
Mey thlrks io my dd oes cul+S.tu, Madiha Malili" Ohoai. saee4 shas!ff!
Tahir, Robi@ Sulta. dd Inm Hasd for.ielys supponing ed en@unginc nc in
I d thdltul
ro
Deparrrn
nr ot Highr Educalion, Govem.nl of Khyb€r
dd HEc Paldslan for limcial dsislancc
PathiunklNa for grer ol study leare
duing ny PhD prcgrmnq
Th. codenb oI this dkstution arc my oiginal *olk
lhis
acknosledsemem is given,
pln
rc Nny orh.r
dis
.raF
*heE sFcific
uion h6 not bed subniled in wholc or in
Uriwuirt. Cdrai. aters of rhis dissctulion
nave
ben pubhhcn
Plblhh.d /Subtr r&d Joumal P.p.6l
IrI
Si6jDl
Munmd U3nb, M6hl!* nahod ol
x'h ia oJ seroti&z Ktu
siwh,l*) .q@iq ,
Ha, N.3lq Btbi,
tiB lo. wiat
S. L
A, Timizi and
^gdif,l
MrrhdMis e Compldio. vol. 2t1 \201O).9p.24t*ulJ.
l2l N{inr Brbr, s. L A. Timi, sd sndrl
'l.tq,
Mdh'd aI tk6 ualbhed ||ik Ronizt
36l; Fu.ti4 (RBF,lt Nwti.ol Sahtim of KMhM,pe eq@iM
Marh.Miq Vol. 2 (201 ), pp. ffii13.
Applied
I
l3l
N.th
Blbl, Sinjul 8,4 and S,l,A Timi4
"Nwriat Soh*r
oJ
&!tutlar b! Mesh|s Melul ol IiNt Applied M{hcmdicll Modelling
lal Nreiu Bibi Sinjul
gq
(
Eutl
Su
wiah
bn ittcd).
Timit, ''tktlod al LiM Jd tt* sohtim
tudittd .tt,.t ,itxh |MEE) .q@r!, ai'la tdiot tail jtuiN , luiel
Ensinedns M.0rtrics (Submidd)
I51
ord S.l.A.
NqfM a|bf, S.LA. Timizi ud SirErl Hq M6^1es nahad of liret lor salrinE
nunk u s.ttodinea.q@kn , )oudl of conpnhod Mdne@tus (subhirEd).
Table of Contents
NetuxrnsF6l.oiMJ*Eiruuwo
iudEtua lsorw|on of
Equ[wm
iioDtrEo
Ee![ wm IMMI €qu nor urrM
N|. tuid
t a.2 htidiqoItro:o oaNa$
w'
5.. t hn.rtb olkt|
t.a.t
Shtt.
'rtibd
'01hry
nG noru,iEra s.NRoor,ic$ EauAr
Ij'st o! Figuns
tis.
2.2: stwri
Fitw
2..:
tbdr
2.s: rh.
w Ponas ot @.$tusrtutY 4di.r
Foltt
btnq w. Fopaqtunot e.Eoti,.d
*irg.M
@
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k'.rd r
sMtukv 48bttu tit6
'd!tbn)
chod. totahn ol th. $4!onor *ttttttt c.lrbr htt'd.otditu
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t@tu.q
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7:
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helt.3.1st spedtrn
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t46.at
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fttd2 tllil; stobt2.i@tot6l&5ng[ tedt w ac2q l4 Ma@ e k)tMa
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h4w'ehdtlld,rtoarrtud.
tuory w!6dth dlflRnt oqpth'h'
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touc t.2 spdpl d. 4.@aa.d. ald E attk t I t . ,t
rda. t.t: R.rtE l@ M,4i.d tu tsn.Edia
tou.
t. Wd @4w4ed!e Eqt r2t=s
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ol.q|.ryE. dt lu Ex@k r,r t - 5
rabb J t hn4qrJq turtud al@ tut'ttu ld Etqapk t.1
fabk t.3: tihtuA ld hbtut'd al thn. tu M lu Ea,qta r.t
fqut t.o tn\aiq, h, hE4!i@ ottb4,olnn tu, tunpt. t 6
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Abstract
Much ot uselul wo.l beln! don€ todsy aor num€d61 $lurion of garlial difieEhlirl equations
inwlws
nonlanear ec u.laons .risins in dilteEnr
usd num.
ol
ii.lds 6l si€n@ and .nlane.dn,
r'6t
widerv
t.diniquer at€ 6nit€ diffden..3, fintte element', sp.dral d€rho&
3nd
@llftation m.tho&. Howd€. these n€lhod! fa.e bme limitnions like @n*rudion ol re8ular
gad for irf.gu
,.
anorher cla$
ol
and con pler
methods
Beom*net
knNn
:,
s
ow conve Gen
e
Ere
sla
bilitY
.nd low
accu
rdv
m6h free methods k expe.ted to b€ supeior lhan
convenrion.l mesh ba*d methods in prdidlng mor. aGuEt. and ltable num.rkal solulEn
rnY.onn..r!. mesh. M.lhles d€lhods using hdi:l basis fundioni a.. mo'€ flerible
with hish 6nw.!.nc. 6re. Tn€e nerhod! pBid. v.ry.cd.are nune ol elurio^ with row
withour
Jhe
.4.aich pRlented
in
thh dlssen.lb.
ls
brs.d on
m6hl4
m.thod ol linee using
Edialbasn fundons for numencalblulloB of nonlinear tlme dependent pafti:ldifieremial
equations {PoErl namelx Generalten (lEfrolo Slvash nskv (6rG) equttlon KaMhad tvpe
equarbh,, €qual Wldth (Ew)€quation, Moditied Equalwidth {MEw)equation and No'linea'
giwn PDE in ro
s.hrcdined (N|J)€quatio Fidthesparialde Erlv6 are dkcreiize ro'wniru
tiur ord.r odinary difid€ntial .qutrions (ooEs), sharh is lh.i slwd bv 'lt$i"l
pro9'ni's ol rhe
Jourd, oder Runs.-lGtra {Rx-4)sheme Eishv.l!€ stabilitv and.onre4.nce
€'ior
method are ako di*u$ed. Ac.uracv ot the method i! mea5ued in t€.fis ol ir and fa svsrem ol
norms and conrerualve oropefti€s ot mast mofientum and enetgv The pEsent method is
smples ava'l.bl€
dh
ertsri4 num..al re.hh'ques
inplen.nt.tion a^d
il lh'
lteGru
e
rh. n€$od shs rup'no roEcY
€fiiciencY of m.5hle$
n.thod
€d:e
or
Chapter 7
IntToductlon
1.1 Lit€rstur€ SurveY
of numc.ical rerhorb for th. solution of nonlircar Paftial diffecntial
tlo tm fmn bolh
equadons hd ctjoy.d u intere period of &rivilv ovd the lsl
$@crical dd pncd.al poinr of vi.v lnPDvemnls itr nuDcrical t'chniqucs, rogeoer
wi$ rhe mpid advanss in compul.r lcchnologv hav. neant oul mnv of Panial
'"The nudy
difloE.tial equatio.s .tising Aon .igin4rinS and $ientific applications' which *ee
pEviously inkacublc, c& no* b€ rcldrcly solvcd Ge Milch.ll and Gifinns [1])
\onlned
panial d'ff.Rndal equat'oru dc{nbrng
panicul& mhaty vavcs
die
in
vuio*
plsm physics, guanrw Physics, dd
most ess, rhe cx6ct slution of th*
$lution ol
liclds such
nonlre
d
$avc Plrmmna dnd in
mnlid optrcs hydrodvnmics
(s
Dcbnalt [22]) ln
so one
ncds ro find thc
oplcat fibd @municanoi
cqu.ions h .ol availabl.,
such eouations nunencaUv
Numerical meuods
sEh N nnne diffeEnce method (FDM) fi.ite elercm
mrhod GEM). finitc volure rethod (FvM) and bound{v €le@f,t ft$od (BEM) havc
ber usd lo solvc a widc nnge of panid dilfcFrtial equ*ions Thce rthods di*Eriz
Oe
doMn
'
or
m.$, gnd or a {r ol
G€ne6lly, $e fornaion of
a
PornE tqurnng
t liicd
conn4rion dnong
fien
squm ot t.clan8lld gnd is equiEd in FDM Nhich is lery
dd FvM ac fleiible in d€aling
vith @npLx gcomuy bur fmtion of ! *cll suiled ftsh in 3-D sltucluE of $e dam
involvod ud coDpuar progming is difficull (D.ntowicz .l .1. l2l tid Rachowicz a
chalLnging vhen problem dondn is nor regulu FEM
o/. t3l).
ftee @
sitlatioN wh@
E cEucdo o
enlargcneDl ol m'sh ov'r
tine
is
ned€d, like pbblems of cnck Ptupat.tion ot fngrentadon Also ii boundav fEc
problems, arising in fluid mhanics dnd finMce mod€ling, lne donaiD b€i'8 Pan or
a.P!4
b. DEdicled in.dvo@. In continuous mhuics one h410 ddl wilh
fic snudon whcE ftsh may beat down bc.aue of d.fom ion. ln all th* dss it is
soludon c@or
v.ry rougn lo deal wirh a strtltuEd mrh. Apan fron difficulry of
.noti{
gensaion,
prcblen is achieve@nt of accutrcy, which is linked with oder of approxinaton
schemes used.
apprcxi@rions
In order to aroid polyiomial snating
prcblems only
u.
acb$ dE msh bul nol ils
used to appoxihatc
d.
funclion oDly
low ord{
e@c
Fd tuE
.pForimtion of derivativcs, hi€b order *hcms
neesery which involve addition l computalion l @st. Whil. using low ordcr
panial dqilatircs.
e
mh
she@s ecurscy c.n be inpovcd by lclining lhe @sh but again at lhe expense of
incr€ased conpuutional cosr I4l.
Merhod
of lines (MOL) is eothcr well es6btsn d .u@ticd r*bniquc
so[ine pdial diffcrcnt.l equatons IPDE5). The
us.d by mth.nadcians for solvins bourdary
merho<l
for
sas orieinally developed and
vdE pmbL6 in physics (z,f@lhn l5l,
caer dd Hinds 16l aid schiess [7]) yhich w6
furrher d.velop€d by PEsla.r al. lE,
9, l0l fot pmblems dising in elatrcmgnetism. Tne rethod oi lincs is a special form ol
finit
difreEnce m.thod which is
noE efficienl thm rgular FDM with Especl
to
aeurcy dd @nput timal c6t. Tbis is a emi{r*cie minod wbiclr is convenic.t aid
quit @liable. In 0!is ncfiod by dicrtjzinS th. spadal d€dvativ.s only dd ldving tim
vciable co.tinuous, th€ original PDE is convcrted inro a system of odin,ry diflerntial
eqlatioos (ODES), which is dEn
properties of
and
i
egrat
d io tim. The Brability and conlergcnce
tn @dod used in MoL .4 4sily b€ e$rblishcd
tire disEriation. Preglzlmire .ffon
ce
b€
conpe.d
Tbe dilTicultics
dd emci.ncy of diflftnt
aily by slving ODES I I I ].
b.i.8
spfc
is ato Edu€€d by nakjng use of st n&rd
ODE soltc6. Thc accuncy
derivatiyes
by sepdating rhe
rpprcxinatjons tor spadal
faced in Gsh-b.6ed nelhods noivated $e
res@he^ to
d altemliv. ro tmditional grid-based nu@ical D.lhods. So a Dew ficld of
n4hlN netbodr dergd ard thc fid n€shl€$ mcthod 'sn@rh Panicle
lmk fo.
Hydrodynmicj v.s pBncd ir
fo. lh.
simuladon
ll2j dd Luct ll3j
of atrophysics poblcms. The mcahlcas mcthorl! for n!rcrical
1977 by Gingold
ed
MonaShan
.slutioo ol PDEe hale becotu very oppealing and .chieved a enarkable prcgles ovc!
the lat No decadcs, TIE min s@l of thcs. mthods vas !o elimitut€ lne ditriculties rhlt
conpLr ge.neuies wheE lomtion of con@cled nesh.s or c1'@nt6 $ a malor
Drebl.n vhil. usiig radltimal dcsh bas.d m.lhods lit finile diff'Fnce mthod
ely
on
(FvM)' vdious
GDM), nnire.le@ rethod (FEM) .nd linile tolum tulhod
nBhlss tuthods n ve been develoFd *hich cd bc dssmed into dE greups;
(l) d6h f@ Paniclc mthods in qhich a finii. nunb.r oI discele panicld tr u$d ro
describe tlE
slat of a systen
and to
@o.d tn. novcNnl oi tne svdemr
qdk fom of PDES;
(3) nethods basd on collMlion technique dd *ork o! sbng iom of PDES Tnc ke,
tqtuB of mcah fE. ft$ods is ro poaidc a $abL nu@dcal elutio fot PDE9 or
irl.gral equadotu *ilh all titrd of p6sible boudary cond,rions usinS a sct of sndEd
nodcs or eanictes wiino dY mh.
A *idc cl6s of neshless @$ods in prcgrcss bdav is bs*d on colldation
(2) we.k oeshlcss @thods rhai work on
neshless method using adial bdsis functions,
methods using ladial basis functio.s
solurion
oi panial diff.Ential
al$ knoM 6
(RBF hav. bsn
me$od Mcsh tEe
equations involving multidiDensio.al complex doroins
alPonm.do
involrcs pan
poinrs so RaFs rcrhodr suffer no @nplicarion whil€
in $e
el4ion
b'l{en
lhe drta
dd neiibilitv
wi*
dislanc'
wqting {ith highfi dimDsional
@ RBFS appdiimtion e@s bacr to thc v6t of R L fttdv
inrclpolatron of $au.rcd d.t involved in geogEphical surftces. ud
pDbl.hs. Thc
Il4l for $e
s
usal suc@$fullv fot nunerical
RBFS metho<b haw advdtag€ of exponential conv.ry€rce
of nodal l@ation. As RBFs
Ksa
liLElft
RBns for
sraliralio.al and ms.eric donalies. I. earlv 90t E.l K@a [15. 16] used
solving elliptic, p@bolic dd hypetuolic PDES Golb€r8 ed Chen i30l cxPtinenDllv
prcvcd thar RBF l.chrqne in strong fom is vcry useful for elvinS Fnhl diff'ential d
equtids. Ealy sort ot Xris I|6l od Zemu*ar 'r'zl Po*"and Chet
llTl shopcd 6!r $e Fd()!m. of RBF bled rethodr is mrch b€llct tlts FDM
Ll6son and Fohberg tlSl md. t conporisn b.tw@n RBFS nc odsr FDM $d
well
s
intcStal
Pseudosoeclrll mthods md lhev lound thal high
odr 4mcv
is mhicvcd by RBFS
ne$od. A dircd quantiBrive conpeison b€t*een RBF derhods aid nnne dement
retnorh was oafied ou by Jichun.,4l ll9l, Poving suFrior acu'acv of RBF
ne|hods. ln F6.ke's Eview papq l20l duldquadric Bdial bdis tunction
'Ppox'mton
of Mo
for
inl.Aolation
of
rclhods
nub€r
b6t
dong
a
ldgc
re$od w.s dt d s th€
din€nsional scattered dlila baed on then stability,
&.mcv,
efticiencv, ease of
inplcmnkrion a.d mDory rcquite'n nt, MiccheUi {2ll Prcvid€d a oathedaicd
fond.rion to tua@te nonsinguldnt of thc RBF coll@rion oatix. Th. eri$eme.
uniq@ne$ and coove.g.re of RBFS wts Ptolcd bv Micch€lli l2ll, Po*ell t231.
Madych
dd Nllsn
t241, and schaback 1251, for
mullivdide sereF!
dlt
inrctPolalion.
l27l lnd Wentnand [28] .slablished rhce f6crs ror
$e $lurior of PDEI. Two importa asPccts of RBFS nerhod had ben obstved in Lhes
while
w!
l)
t261,
Fnnlc
and scnaback
ruty rcshlcss altodthn in qhich coleation Poims can b€
elcct d fEly withdt ey cdmctrvity, h.mc @Dplid.d ftsh fomatioD is
Th. m€thod is
2) ft
a
is indcpcnde of
O(lr'),
wheE
dincnsion.
spaial
I!
in the sme lnar th. conrcrgcrce is of
sFtill ditunion
n is densiry of coll@orion
oiher words ord.r of convergence
dircdon
*ith low nunbq
of the
pobl.h
and hcrce
or @ll@adon Poi
!.
tu
points and
I
is
spatial
incrces *ith incMs in
accutocv crn D. pr.eryed
In panicul{ Golb€rg and Chen
1291.
direNiond Pobsn €quario! with d v 60 tudomlt dislribded
nod6 ro gd $e sm 4cufuy !s sith FEM redDd usinS 01 000) lined
solved
$c
Resedchcs like Fombclg Dd Dn$ou
ard Tanalc 1341,
sd
t3 U, Hon and
wn [32l, Chen and Hon (33] chei
ch€n d, al. I35l aho conuibut€d a lot
KlNt n
an
usinB RBFS to solve PDE9
H.mile lTe collddtion mcthod Laler
on Hon !r 4i. funhct wid€n the RBFS n.rnod fd soldng difr.Enl tyP.s of ddinary od
panial difi.Endtl cqutions imludine Scncal inidal value Problcns I3?l ionlind
Bugdt .qBlion wiih shNk ww IIEL boud..v lie pDbl.hs likc thc Amencd
F6shaler [36] nodined lhe
thod to a
id!8uld boudsnes sucn as shallow warer
equation for tidc and cuftnt sinuhtion l4ll. A clss of Kort€w€dg.'dc_Vties equalion
option pncint 139,401 and prcblens with
ionliner s.hrodinse. cquiion [45, 461, couple Sinc cotdon equaton
Md
l4?l ed rcSuldiz€d long *!ve iqudion. t*o dinensional CouPled Burgs cqualion
MGrlincnsionrl R6rion-Diftusion Boss.lalo. Svslen (se L48l) hs be'n studed
using cotldltion apprMh bas<t o ndigl bsis auncrios Ovq atd Wnghl l49l usd
t42, 43 a.d 441,
tire
&BF @rhod for $€ fmr
and
to
$lve shlllw w{Lr €qdiiotu
Pict l50l sohrd mv@tivc
in
[5
sd esy i.
Fomberg
], 52].
of lifts witi ddi.l bah luNrio6
CouplinS rethod
ncrhod nore acc@r
sn
ed
uing RBFS appmximioB. Funher
PDE on splPE
d€hih csddins RBF rcthodolosy cd b.
on sPlEE.
(s
[?8, ?91)
ner.s
tnc
impl.n€nration as no nesh is equiGd in thc problcm
domajo, Hi81l quatuy ODE elveB ca. b. u&d to obtain solution fot the systcm of ODES
obaircd
d
a rcsulr of RBF appbximtion of spalial d€nvrdves. Thc liEratuE for
nshlds tuthod
l2
of lires l4hdquc i. lcry limit d for
Method of
LtH
depen.lent
nonlit.d
PDE5.
usiDg RBFS
Befor de$ribinS fic rEtlrod
l.2J
tim
s
16l prEs.nt thc folowing d.fioirions;
Definlllotr (RDrs)
A Eql lalucd
dishnc.
d:!8'rS,(d=1,2,3),
runction
ES'
b.tNai
ond sooc nrcd point
whose value depends only oo he
,j €t8', ./ =1,...,x (knorn
as ccDrre), so
rha,rG,,,) =r(1,-,,1)=r(,). *hc,!
. is Scnehly tI€ EElidm disoncc-
fi.
r vnal blc in then application, ehich is dcfincd
positivc dcfinitcms of RBFS pl.ys
1.2.2 D.ndlion
A continuous function
/:ry J$,h srictly pGitive deflnile of order u if for ev.ry ser
ofdistnrl poi s r,,r:,....r"€
$i,i.rrlll,
ror
-,.1
a|l,i,,4,...,rx
9'.
l,o
satisfying
24Pt''t=o
for all polynooi.ls p€
also
c.lled
d
r:
.
A @idnionauy pGitive definite function ot order
r=0
posilive detuiL furlioD.
ln ord.r to apply
msl cs
mefiod of liEs
iahniqe E mrde!
PDE of rh.
fom
h
$*r1,t=0. *o, 'a
o.l)
*1r". ,=,(r,r) sd L is spttirl d.riv.tive opedor' kr us slPPos' lh't
ir.r,,.,r,€ oc9r'.(d=1.2,3)be t giv.. st
ideas proposed in
$e
solutjon
r(r,r)
cai
.xP6sd
"(")=:,..i,(,v{ll,,,lj.
wh.E q is $ne adial bst
untDown ceflicients lo
domjn FolloNins
nethod I8l foi 6lime dependent PDE
Kee's
be
of ce.ten in prcbleh
te
apptoximale
as
(
tumrion,
t/'r
t.2)
@ anr6 dd t,(r),(i=r, ../v)
b. del.mired. similuly lle appoimte soluion fot
ft
sPatial
dcrivltivc opcnlor cm b€ Mitren ar
tui,)=$ r,r,)lL,lll' .
Thc abov€ apploxi@t ons (1.2)
dd ( l3)
,{r=".
r =[,, t).,,(,).....,,
al. = .(u
)
can
b. wtid€! in mtdx fom
a
(,)]'
I1.4)
(1.5)
whcrc A hs cntries of $e
fm
/,
=,/(l', -,,11). i,i ='...,,{
and
,
is
(t.l)
ll)1.
lnrisynnerric matrix wnh cnnics of
,,, =r.y(ll'-',1l).,, i.i=
the
fom
t,....N.
usint (1.4) dd (l-5) wc set
L(u)=(t')o.
Equ.tion (!.6)
m
be
(
mido
6
l,(u)=Du,
After die€lization in
smi dncrefted
1.6)
(1.?)
spd vith sditl
sysrcm eiven by
baih functions, PDE
(l l) is rosiomed
in to
a
0.8)
The sysrem of ODES (l.S)
ce
lhen b€ solved
wi$ mv
have
ued classical folnh o!d.r Rung€-Kutta mcthod
IlE
sumcient condniotu for
t/(r)
to
ODE solv€t
gladte m!{ingolditv
l'
lhis thesis we
A wed
or rhc marix
Schc'btte
ri6c by Sch@b.tg t53l btcr on Miehelli tzl I
'rttntlcd
idea so lhat a ldSe class of functions could b. @nsidcFd'
w€ fisl siatc som elevmt dennitiois b.foe Soine to schoenb'B 8d Micch'ui
given for Oe
fid
ftfiridm r3
A fumlon
(i)
(ComPleEly Monobnc Furcdon)
r. k sid
lo bc @mplerely mnolorc
'/€c10,6),
(,ii)
(
l)'sL"(')>0,ror.>0ed
D.fiDiriu r.4 (Positiv.
symeuic dd r'A r r
cigenvahcs of
Tbeorem
con$ant
,4 wid'
oi
0 for every
Posilivc
rschcnbe'g l5jlr
[0,@) , lncn ror
ouis
a,t
(.)
dy st
=l(lr,-rrll)
lf @r''-d.t' a comPkklv
,
or
distirct poinb
foloving
r....,
'}.
RBFs
e
$c nxn
{r/ i
= 1. .
mono@ne bu' nol
r}
lne
ix'
matix
n pcirivc definn (od theeroe no4ingula)
tt (t)= ((.1;). c"lo *),t'(r)> 0 for
'/
is conplctely morctore bur not cons|Ant
eoDs {r,. i =
The
@uix ,4 is Fshiv. dqfinite il n is a
,iimtuiond 6lumn varor rto. or if aI tlE
Definite Mattix) A
tn tutrix ,{ N
l.l
I =0,1,2,..
Theor€m 1.2 (Miccherri I2U)
v/
if:
mtir
^
oi
(0
@)
wi'i cir.i6 zr
&eqEndy uscd,n
li|.ra{w:
thcn for
=d
dv
{li, - 4l)
r>o.and
scr or n
is
distincl
noFsinsure
Tru.lJ
ryFs /(r), f>0
Idbli.ly doo& RBf.
M"ltiquldric (MQ)
tnv.*
mlltiquldric (lMQ)
PL..rlrc .Do$ tAFt
Trin pl.E
splinc.s
(PS)
ft d 6ily bc pm!.d that inv.Gc Ellli{urddc oMQ). inE$c quldic (lQ) .d
GaNiln (cA) ddid t6i! tfttiG snitfy tlE suffci.nt conditi@ of thc Thc.Em
(l.l), whilc multiquadric (MQ) sltitfy thc aufiici4! coiditi@ oltlF nE$!n (1.2). dd
lE@ for ilFe typct of RBFS cqultid (1.3) is uiqucly slvrble for 4y 3d of dktiltt
TIF
slullcy of
RBFS
n
thod! d.pcn& or thc choii. of
oplMl
val@ of thc
rlEF p€r.iDct.l inlolvcd in idiritcly 3oooih RBF3 lik!, Multiquldtic (MQ) Oau$is
(GA) and li6c oolti$ldric (IMQ ndid boi! fuictioN By oPlind vduc. q. 6@
rhc rloc th.t poducca th. Mt @usic rllult! ln Flctice, th! vduc of ih. shlpe
pmeter deiircd
.
nust be adjusted with the nunb.t of enrc^ so rhat he
inr.rpoladon n.rrix A is wcll condniomd .rcugh to b. inv€ned in linic pr.cision
silhnetic. Ii
has
do.sny of poinrs
as
ben shown
d
th
th. accu@y of nuldquadric int .polalion is rclated to
well as th. valuc of rhe !hap. Pu|neter, The@foc, aEUrcy oi the
nunerical apprcxi@tion c.n be inct ascd by cithe. incrcNing nunber of couocation
points (i.e. by d4reasiry mcrh siz.) or by incrcasin8 tbe value of the shaF
Td*arr
[54] fouid thal lhc rcol m.an squae of enor dccF.es up lo corlain
Oe' inceass npidly when
opplied
$e
p@oct
r.
techniquc
FontE€
stably conputing
€r
fic
.+*.
of o6s
linil a.d
[56] atd Hickerell and Hon 156l
validadon lo obllin an oPtidl !.1u. of th€ sbap€
Golb.q er
ai. I5?l 4lablish.d
a
41.
Contour-P.d( aPpoach which is caPable of
RBF lppoximation for
develop€d RBF QR algori$m ro
lll
c>O
olcrcom ill-conditioninB of
ln order ro inpove condirioniry ol RBFs colledi@
>
pmter.
Fomberg ahd
PiE
1501.
RBF8 interpolation on
n.triG, th.
de
folloving altemalvcs
Conpedy suppottcd ridial basis functions (csRBF have b.cn inmduc.d by
Buhmm [60j) in q{tq to obbin a sPaE
ini.rFbtion Datrix dd th. F6sibilitt ol6Pid cvaluarion of fuetions. Unlike
audlo6
0l.dl&d
goballt supponcd
fixed
dimal d
I5E, 591,
.d
RBFS, CsRaFs
1581,
ducared polynomial,
nE
G
rc
srticdy p.sitiv. dcfinne oD
9'
onlt fd
a
bed on a
aftitrlry dcSne of snelhMs
CSRBF co.strucEd by Wendlad 1591,
pcnirc dcnnnc having
Cr, 16ll. wu t62l offcrcd dorhcr tcchniqu. to consrtuct the sde csRBFs. buI
pnliding a high.r d.t@ polyiomial lor a giv.n l.ycl of snoolhfts and
>
PEconditionins techdqu€ (B€|6oi er al. 1631, Ling and Kansa
&l
b6ed on appoximte cadinal bash functions (ACBF have
effectively
>
i.
which is
bai
used
inErpolation and in PDE stting.
overlapping and non.ovcrlappiit
don.in deonp6idon
methods (DDM)
coupl€d wnh meshlc$ RBFS nethod on Datcbing and non-nalching grid have
becn dcleloped. TIF ovqhppins DDM
less spltial
Bohion
qith RBFS [65] p.rrormed bener wirh
as compaftd wnh FDM .nd FEM.
Liig
and Kaisa t66l
vqified dut combi.inS ACBF pEcondhidi.s .nd domiin &@mpGilion ma*es
lst6 s .mpdd ft) di@t i.vdsio
Lealiz€d RBF netho& in vhich ndy stull ovctl|pping RBF sysldns m usd to
oblain a fle$ble solltio for lal8. sllc prcblcms wnh m is@s ol conditioning. Liu
sd his collsgus 16?, 68, 6q 70, 7l iid 721 lllst inl@duced l@l RBFS appmrch in
invdsion ollbe
>
Ndll
systcn
io elv. fEe vibradon of
radial poinl intcPol*ion mclhod
2_D solids and de
incompEssibb nuid flow 3inllation, lnd difieMtial qudBluE oollocation nethod
for
Defidiilon
elving $o dincsional inMpesibl. Nrvierstoles
1.5
eq@lioDs rcspect'vely
Vetor Norn
r is . @l Posnivc numbo Stving n.6uG of lhe tiz'
velor and is de.oied by lJl . x musl $rhfy the followins 4bn'
(i) lLd>o if r*red l{l=0 ir r-0.
Th. nom ol a veclor
ci) lql= kll4 rd ! Ed
(iii) lt+rl<l{+lrl.
o, @nphx
civ€n the
terorr=(r,,!,,--r,), $h.
D.fitrition
1.6
The
l-mm ofttF rdlor
of the
s.dr ..
r
of 0E
rcll kNM vdbr mm ryps re
h thc aum ofthe moduli oflhe @mpo*nb
rhe
of
l't, =t',1.t', *.....*,. =Il',1
sm of $e squd€s of dle moduli of the @mporchts of
n =lr'+l4l'+
''
The 2-nom Sives
+
r-'l'/
r
Mtof
the
is muinun of lhe noduli of
rhe
Dennition t.7The2-oom (orEudidsn nom)ofthe vcclor
r
is lhe
squE
i.e,
) h'
L;
l
=L
lsgrh' of lhe volor.
DefDitior 1.8 InUniiy (a) mm of th. wctoi
l4=s,,
x
kt /(r)€c[a,r]
D.Itbition r.9
k,rl . o / G). t, k,rl
ti t,
=lJ[/c)]
t'
d,
).
In
oder
work
done
on
usd the following approxiMtion for
I,
ercr nom of tlte
ir>
and
{-
cxetind numncal dunots rcsPectvely,
lsis
sbbilily of RBF trrthod
ro undeEland
h6 her
nrdiom
I
I
Sbbility An
continuous
l
r
l3
dl
space of
rrrc rcar varucd runcdo' dcnned by
have
c=lJ"l,,*,
(rllc lin@
s
for m
[73, 14, and ?51. A $ablc
a very
hlle
dm i.qtuion $he@ Equrs
thar
d.Fndenr
PDES
o.ly
ha rll th. cie.nv.lues wnn m4ositivc Eal pan. A
sFctrun having Eal pdn of all eiS.nvalws in $c l.ft half ple is known as. skbL
thc spetrum of tte
sFdrum
1751.
diecri&d
A ole of
PDE
6unb for sl.biliry of Nthod of lim
is tlEt "the eigenvahes of
tim slep At, lie io fi€ st biliiy egion of
ir sotu ces rh! d.tals of fie stability de norc
in $h disenation ve @ dealine wirh nonlincd
the spatial disEtized openlor, ealed by thc
time{iscerizarion opc6ror, alftouSh
rcchnicol
ed
rcsrriclive I?61". Sinc.
PDES, so after spatial
dhcFliation we havc a systeE of mnlined
solved by cldsicsl Runge-Kuttr
rquies
that all eigenvalucs
scnem i.€ for
s
d
n
thod of
odd fou
(RX4).
fte
ODES, which is then
snhllry of $e rcthod
ol thc Jacobian mat ix sadsry stability condirion of RK4
sufficiently stull li@ st
!
& , {c have (-2.78
<
a&
< 0,
fd oll a).
sm lim
RBF nethods cannot bc
@Mt sd
un*tuinry pdrciple
Th. condirion numb.r ot llE @ll@arion @rix n dcfin d by
r(,4)=l.l[,a'1.
[251.
ecll cmdidomd.I dE
due to tlE
1.4 Solitons ard Solitsry wevts
Solitary waves @ wav. p6ct $ or
pller
lhat popaSlt. in nonlin@ disFBive nEdia-
ir
fluid @chsics, pldna physica. solid statc
A wide y&iety of phemnena eising
lircrics and g.Ghcmist y de descnb.d by elilary
pbysics, oflical fibe6, ch.mical
wayes. Sohon is a sp€cial typc of solit!ry wav. that
dain
ns shape whilc it travels al a
consht sFed. Solirons ar produced duc ro cancell.tion of nonline& snd dhpersive
effecE in thc nediun "Dezin !.d lohnsoi l7?l de$ribed the followins $r€e ProP€nies
. Wavc of pemoent ioml
. t calied wilhin a &Cion;
. Inrenct wit[ olhcr slilons and.ftree
exepr for
a
phe
aftcr collhion witnor chmge in shaF,
shift".
A elilon wilh ldSEr anplitude novcs m@ apidly 6 conpaFd to a soliton wnh
ploc.s of prbP.8ldon lwo such wavcs witn diffeEnl
anplitude getl clscr. @llidc ed $cn ep@rc into two $litlry wav6 dint ining then
odginal shrp.s and rclcitics.
smalle! @plitude. Duing lre
1.5 Overvi€w of th€ diss€dxrion
o. th. d.rclopncnt and .volution of I6Nss refiod ol
lies usine Edial b6is furctions for t lving nonlind qave equ.aons like dE
Tbis dissnarion is pivorcd
genelalized K@oro-Sivasni.sty .quation, KawalEa .quation, equd wadlh (Ew)
equation md nodified
precne
nftri.ll
.qu.l qidth (MEW) cq!.tion lor obtaining high ecmc]
and
solltions of these prcblens. ntc dkscnador is dividcd into 5 chapren.
In Chaprer 2, the nuhencal sldion of thc generalizd Kuonotesiv6hi.sky
equation is pesented by using neshlcss method of lires (MOL). we havc usod globolly
supported ininitely RBFS for rhis
plaos.
Mde in l.ms of global elativc.rcr
tc'
RBFS nedod of lines. Stobilily of rh.
in spae dd
titu h also calculaFd
The compuison with lhe exhring techniqle is
povc lh. limple applicdbility and superio.iry or
poposd shene h also discused. Conv€ryerce
showing th. .fficiency of the schene.
In ChaF.r 3, RBFS-MOL tahnique i5 fomulalcd
kwanm eqution. Nutuncll
Esults in l.ms of
tft
rhc
nu4ric.l elulion of
mr noffi I2,L re conpded virh
liEnn!!. TllE c!|gvcd quetitics ot n ss, drty $d nom.nmm
,l! .ls Edli.d. Thc inlrdioo of wo rid tlulc .olibB is ltco discu$.d.
h Chapcr 4. s pEet ru@i.d io|lfiio of .qurt eidth (Ew) cqu.tior
Srbihy of dE rEtlad i. inEti8.t d by cdcrbrnt !E .itctv.locs of dE Jaobi$
rhc
lslts
mE
in the
of $c ilsultin8 sy3l!n.
lilc telDiqt
ndid bcsis fuFtions for dE
mMicd iolurioi of rlE Dodificd Gqrd widur (MEW) .{u.!io!. Soulit} daly3is is
rlso csLlblhh.d dd protld by gnphicol llPGsttrio of cig.nedu6. TlE mr .oths
Thc Chapl.l 5 cont!i63 rit$h
I,,
Le
using
cdculrtld rld coN.fltrivc poFnie, ol tn!
s,
mmnlum nd cncr5,
e lle
cquttd by msh lcts mthod or liF.
Moaiod of singl. solitotr, ircr.irioo of t*o sliioG &d Mrl(elllid initid condinon fll
bould 9r|! of elirom e di!.u3!.d Nuincricd Gult! e cooPlr.d with cri![ry
l!
ChaDcr 6,
s
elvc
d iEe S.hrcdi.gcr
Chapter 2
Meshless method of lines for the numerical
solution of generatized Kuramoto-Sivashinsl<y
equotion
2.1 Introducaion
The sen Blired Kumot,J'Sivshinsky (GKS) €quariot is a mnlined cvolution equaton
=+,=+/r -+v -+, -=0,
vh€E
d
(2.l)
I r<r., >0
/. v ed, @ calcosla.B.
This is tlE nonlined .quario thar Eainr lhq .sential clcren|3 of
involved in vave evolutio.. Th€ tems
instobiliry .nd enrgy production,
*
sy no.lirc&
,* k lnc doninmr noilined
for sr.bility
od
tem,
ener8, disipadoi,
*
ud
#
is $e
dispesion t€m. This equation was dciived by Kuranolo lo study dissiParjve structurc ol
re@tion-diffusion
1801.
fth
cquaion was d.rived ind.Fndenrly by sivahinsky whilc
sodying fi@e front pmpagalion in Djld codbusion process
For !=0. Eq. (2.1) is
!
18
U.
norlircE.vohdon equation called lhe
Sivahisky (KS) €quatior *hich desfiibca a vuiery of physi€al
Kuramolo_
Phenomena
disi.g
in
savs oi rhc incrf&e betwn two visus n!id. l82l
and unsbbL &iff wav6 in plaM .rd @don ditruiot s,stetu t83)Chaoric h.hrvior h.vitrg F.vcling wavc s rcl ions of p.maftnt shnp. is
obscry€d when Eq. i2.l) is int Stlt d orct a finil. donan with p.riodic boundary
lons wavcs on $ln
filN.
long
condiriohs 1841. Thrce clsses
oi fdleling qavc solutions temed
ds Ilgular shocks,
osillatory shocks and solitary wales for KS cquarion wet studied nufrericallv hv
H@per and Crinshaw t851. The travcling wav. slutions wcr. aho studicd bv Yms
v=0. nis u*d |o dcsnbe tbe Patt n fomtion on unsbble
flmc ftortl od Lhitr htdrodymic nltr 189. mt
Nutuncal l4hriques like tsal Ds@ninuous Galerldtr nethod l9lI Chebvshc!
t861. For
r=4=l
and
sp4ral coll@aion rcthod
1921, Inplicn_ExPlicir
Boltznann method (LBM) t94l md msh
fc.
BDF n€rhod [93], rhe lattce
collocatioo mthod I95l have beei used to
sody Eq. (2.I ). TadmoE 196l poved the wcll'p*d.ess or Ks equat'on
ln this cbapLr, w€ d€velop nesht ss ncrhod of In4 with ditfcrc tvFs of
infinitely smeth dditl bsG funclid3 fot dE nun ncal solulion of 0E Sendlized
Kudooro-sivashinsty equation. R6t of thc chapt r is organiz.d 6 follo*s: l. slion
2.2, we
briefly.iplain akontnn. In Secion
2 3,
sLbilirv in tems ofeigenvdlues of nne
iitegation schene h discused. In s*ion 24, wc p@sent sone nunencal exanpres
Elaled lo diffc& q?4 oi CKS eguation. Scclion 2 5 @tBins discussion @gddrn8
sel6tion ot shaF pdaftler
od
tale of conv.lSencc ln Salion 2 6. wc
sulmane $e
2.2 D€icrtplion of Method
coosid€r
YY+
tlt cKS Eq. (2.1)
u"!:+ u
i\+,
"=i+ n
wifi rh. following initid
il
=0,
o <
rs r.,:0
and boundary condnions
(22)
<r<4
!(r,0)=/r(r).
a
zk.,)=4t),
!(t,,)=4t)
Q3)
To appl) m.shless mthod of lines,
*c
first use ladial basis functions to approximte
poblen domin
ta,
bl n divided into nodes,,.i = l.
space derivativ.s. The
of$ese poinc
{,
i=
2.3,...,lV-l dc incriot
loinll- Thc appreirut rurction of
!(r)
is
Poinls
2
.
N
Out
vhile r' and rN aE ln. bolod2tv
dqot d bv
/'G)
la
u I t)
=
>)i !'lr' )14 = Y lr)t
(2.41
wha.t'r @ untnoMtn d.Fnd.nt $8titi6.nd P $m ndirl
Y
{4=Ls, (r), yr(r), -,
w. hr$ u$d
{,"(r)l
hosis
turctd t d
.
tlE followhg RBFS
y,(r)=!(,-r,) +.:, {r0)
r, {r)=
cr
r(-. (r - r,)').
(0,{ )
IINQ)
whc|! j
=L
!d .
Z ..., iv
r,l !'(t,)={,
b.ing 6h.tc
dEn ir mtrir-v.ctor
!.ntEar.
notltio
\2 5)
whccu ={4(,).
v'(',
t)t' ed r=tr, (,), rh(,)......1,(r)l'
- v,(,,)
'!,1\l ,r,\',) - ,/,(\)
v,(,,) v,G,)
)
Y'G.)
Y'k,
!,(,),..., !,
)
'
r,,(,") r/,('") ... e"G,)
F om Bq. (2.4)
rnd E9. (2.5)
i
folows dEt
r'(.r). vr(r),r-'o
=
w (r)0,
(2.6)
*h@ w(r)=Y'(r)r:'.
Applyint Eq. ( 2.6) to Eq.
{2. | ,.
ad
@llocadnS .t
4h
node
/(l..(r.'u I r, (W,,,tr ,)!) +
r=1.2.1...,1v,
l+.. ir{$.,,.tr,,!,=0,
l:jr+,.rW.tr.)u)
+
12.1)
l7
w,(J,) - Iryr,(',)w:,(r,)...Pr,
lr,,tr,)=;*r(r,),
(r,
)l
i = r.2....,
^,
v.,(r,) = Iy,,, (r, ) r' 1,, ( r' ).,,x' r., (rr )l
t(' t), i=t,2,...,N.
w
,,\r,)=
In
onl.|toMite lne.bow sy$eni!
i;v
u=k,,,...,,f
.
l'rn
of tic colum
@tos,
w,
=[w,(r)]".,
w. = lwb (.,)]*.w* =[w,* (r)]*,,
w- =[r^(r)]*.
dM,
q.
(17)
@
-+U.(w.U)
|
he
Miro a folb*6:
/W"U)+(W_U)+'(W_U)=0.
(2.8)
R.yiri.r &. e.8) a
9
-
"
,u
,.
t2.st
fl (u )= -(u { (}Y,u) r r(w,,U ) + y(w_U ) + ?(w,-u))
ad th. sFDol ( * ) delo(6 cory.nr bt cdEpoent hdtiplird@ of two vcc!06. Ti.
U(,o)-[,'(a] !"(a)..-
!'(+r]
'd(aU
,
rtd nF bou!d.!y c@ditiE d.sibcd i! Eq. (2.3)
r, (r)= r,
('),!,l')= rr (r)
{2.r0,
rtl
(2.rr)
ofod.r
four ro solvc Eqns. (2.9I(2.1l) 8i!cn asl
i,)
r, (u').
r, = l' (u" +
+,(,
r(u^++r(,
=
\2 t2)
.]
a{u'+Arr,)
,J.r,
ct!. ol dssic.l Rungc-Kulla m.lhod (b.inl one srcP
$hcn.) no irui{biury n6d to b. f@d ptovidcd th.t lh. tiFe s|.p & is sumci.ndv
shall ccc Coltltz t tol j, pas. I 12) shich h s in N casc Thk is .PPlicabtc lo srren
of cqu.tions !s wcU s singlc cqu.don. Fd mG d.uik E.&s ce dso visil LrVcqE
It h
Ehe*ld 0ul
in tl|c
([02], pats l5G!60).
23 Srab ity
Following thc
rcfiod of tin. rppmrch airy ndi.l b6h tuncrio.t Eq {2
conven d !o. sysrn of ODES (2.9). W. wric
4=ir{u).0.,<r.
t
.borc sysGE
1).
ha
be.
a
(2 l3)
urot = u,
wh.E U. is inillal veror.
For srabihy @lysis wG
[i4
g:'l
stabiliry
bv
rim scD a,
-J, +1W,,
0). Thc
of lh. J&obie
r.lrir
{',2-a&} b.lonB! to $c Eeon or abelur.
our cs ld cbsical rounh ddd Runs._
Jrcobid
is
gis.
oftt milinwllrn
[>,{r)w, (t,r,*{r(r)4 {i t))..,.
1,.
lr(44(i,t).
Ir s4rioi
by
+vw,,,1rrw,,..
whce J, i! th. rr.obid
J. =
d
"."lcd
or tirc inictnrio s.h.nlc, whi.h i.
Kuru rct||od is c2-?8,
J=
will pov. lhrt .ll .iSdvdGs
(2.4) bchlvio. ot cig.nvtlues for
(2.14t
U*(rry,U) d.dEd
ror d'Asona]
a
cr.mnb
fo no'tdj€dal.lelmts
difidlnt nu@dcrl .smPl6
2,4 Numericrl Applietion
In thh seclio. we apply lhe abolc mention.d ehere for nun€iical solution of
OcncaliEd Xumoto-sivdhiBky cqudion Thc tesults @ comp€Ed *nh the exacl
soludor s*t $N nention d in I92, 951 Tt'c Slobal @lativc c@ (cR.E) h calcultt'd bv
rh.
fmula
d.fin€d in
[9t
zl!4 tr'.'J-!* t,r.,,1
OiE=r
, j=1,2....,/v
,
ii
t'l
=i
LV'-\''
whee
!a ed rq, m aPprcxim.i4
t
Fot p = a =
^.d
(2 l5)
solution and ex&l
v = 0, Eg. (2.1) becomcs
slutio. Bpetivelv.
th. KS equarion [96]
au" Au a'u . a'u
Jr )r Jrr
,
{2.16)
dr'
(,.,r', . *\Ft', *''
T|E
hdldlry @ttditios
e
(- (,
siEn
-,.))-' trii lr (r - r.))l
(2 l?)
bY
f \Et',""''(- (4-,,-,.))-qtrii (t(, Dr-,.))l
(2 rE)
,(,.,)-,.i+rF1,,","(-(,-,,',.))-,tr,i(*(,- D1'.))l
(2.r9)
-frffl|l*r'r-(r- D,-,.))-,nri(,(,,'-,.))l
(2.20)
, (,.,)
=,.
,t".,r=,
For
nurctiol
cotr pulation
numricd eutt\
N
v.
h.ve scd MQ. GA and IMQ Edial bdis functions Thc
obtdftd wirh tE pe.amE6
D=5r-+J"!.\--l2
over
'h'
dotuin ta bl = t-3o' 301 ulins 1v=t2l Thc tlobrl tlat!rc @r (cRE) is calculal.d ar
, = l,2,1.4 wirh dre srp &=o.ml dd tlE Flults e lhdn in Table 2 r' Thc oflimal
valuc of $c sh'pe p6rame. c for MQ b found to b. iaid. ,E inteflal (0.7, 3) B shown
ubh fic rcsulB givcn in I92l using r$icc Bohmam wlhod with
lV=600 .nd Ar=0.0001 e al$ shopn for conpdison. Wc ctn se tal our mdhl$s
N$od of lin6 (usi.g MQ RBD F.fom ve[- Ir'e eguld .hck wav€ prcpagalion is
.lcarly visible at ditrcEfi dm levels in Fi& 2.2.
In ord.r 10 wdly sl.biliry of rb. m.thod for the abov. mcntioned p.obl.m, nsr
in Fig. 2.1. In this
we fix rhe nunb€r of .odcs
{
obsefred lhat all eigcnvalues
. within the inleryal (0,?, 3), ll is
lie in th€ lcft hrlf plane a long as c lies in a rcighbodood
md changc the value ol
of optinal value of |hc shrF psl"@r.r. So ln. rethdl Etuitu rtable in dis
shown in Fig. 2.8.
81,
h
l2l andc=2.5,
lh.
sond
ce E
kccp
.
fixed &d v..y /v. In this
all eiCenvalues hav. n.gative
(e
ce,
as
for /v = 41.
Ed pan rs shown i. Fig.2.9, So.gain
ncthod is st2ble. We know thal the accuncy of RBFS sohdon depends on valuc of the
shlF pa@tcr
c
dd
c
ud
number of nodes lV
/V, but at the
.d
of
al$, Accu@y cm bc improved by iicrclsins
conditioniB, This f&r is shou in Trble 2.2.
Er.mpl€ 2.4,
Consid.r $e cKs cquation [%]
a! au a'{ 1-+-.0.
J'a J'a
=-r-r:-+r
'
3r
<
r. r:0
!('.0) =D+e r5[ldh(](r-r,))+'dh, (r(I-r.))-r.ih'GG-r"))]
o 2l)
\2 22)
,(,.4=D+,-ri[hhtrk-Dl r.))+bn,(r(, D' r.)) dh,lrk D, r"))] c.2r)
,(r.r)=D+e r5[6h(r(D-Dr-r,))+hh,(^(b-Dr-r.D-hhi(r(b-&,,r] \2.24.)
Thc cracl solurion for this
eimple h
given by
r.))] (2.25)
Thc GRE is compuGd ar l= l,2, I and4 wirh tin sFpA r=o.ml d
p&amt€n valEs e sl*led a: D = 6, I = +, r. = -10, o€r $c conpur.liotul dotujn
!(I.r)=D+e rs[b$(r(r-r-,")+bnh,tG-Dl-r"))-@h'(i(r,Dl
h, bl = ['30, 30] whh N:121. TIE @ulB ac shoM in Table 2.3. Fron rhis rable it is
cld tltat dE Esulr ouaincd by ou @shl.$ Nthod of lin s ($ine GA, MQ dnd IMQ
RBF3)
e
bed.r
4
compoEd ro d€ Esultt given in [92] with /V = &N ttd Z | = 0.01)01
Th. optinEl value of th. shaF pam@cr
lu
MQ
li6 insi&
th. in|ryd (0.5, 2) at
sbown in Fig, 2,3. ln this c6e th€ solit'ty wave prcpagadon of GKS €quation at difteent
rine levels
is shown in Fig.2.4.
For this poblem wh.n
,{ is fix.d .nd c
cieervalus stisly stability conditi@ of RX4
difreE
val@s of tlE shatc
pdmt
bclongs
etlFd. Ir
r is shown. Fq
to (0.5, 2), dEn all
Fig. 2.10. shble sp€.ttuD ror
! 6red
c, and
diffcdt valEs
stable nurcrical rcludon.
eiplcn RunSc.Klua ne$od Frfom wcll giving
lhe
fte
of /v.
scrLd
eigehvalues for this ca8. de shown in Fig. 2. I L
Esmpl€ 2.4.3
In this exmplc w. consider the XS.4uation tE9l,
#.,*q.#-#=n
a<x<b. t>o
Q-N)
This examplc rep@enls thc siEpLst nonliaer partiar difredntill equarion showin8
cnaodc behrvior when sDatial
dotrEn is fiDit., with the Caussian initirl condnion,
\2.21)
and lne
boundlry conditoE,
!(d.,)=0.!(r.')=0.
t2.28)
The conpuntional do@in is [a b] = {.30,301wilh
and Fig. 2.6, wc
cm ob€fr€
th€
AJ=
=0.m1.In Fig.2.5
converSent nlnedcal Bults by our m.shless ne$ols
0.5 and A |
of lines with compl.tc chaotic behavio. at | = 5 and r = 20 Esp€ctivdly.
Stability sprctium for ditr@nt nunbd of no<les lv and shapc
Fie.2.U ad Fi8,2,13. Frcm ltcF figurcs $e
w
parh.cr
c is showo in
tnat fcw cigetrvolues have s@ll
pGitivc Eal pan showing a snall .xpoEntial grewrh in po6itivc h.lf plde bnt
erhod wotu w.ll in $is
tE
c& alF.
In this €ianple, wo consider the case with 4 = 0 and regative siSn before
I
. ln lhis case
CKS equltion is converred in to (dv-Burg.r equadon [97, 98],
t-,*-,*-,*=0.
asrs6,,>o
Q.29J
!t).n\=)o-----j!-
(2.r0)
lr+.\e(r) , )l'
and the boundary
condino6
Ir
| .re 1,, {" -,!))]'
(2.31)
(r, (, -
(2.32)
+
Ii
Tn..4t
solution
i5
..e
eiven
' ))]'
!
6
(2.13)
l0v
25v
For nu@icd c.lculnios
dr=0.1 .nd &
= 0,01
fte cRE calculltcd
2.4. Fom thc tlble
-
w
Th.
ta&. 0E comput{iorl domin
\ElB
of th.
bl =
la. al
wilh
p!F!eie6 re t=0.0009 dd v=0.000@.
ar , = 50. 150. 250 bv mcshl€s
*e cd se tlDt
Ir
turhod of lincs is shoM in Table
rhe resulB obtained by
ou nethod (usine MQ RBF
to 0!. Fsul$ siv.n in [9sl *ith !=E07. Gnphic.l
Epr*ntation of Esults is shM in Fig. 2.7.
Fq sbbiliry ealysis se F.f@ two diflmr qpgitMls. F6t w. chdge the Bl@ of
conpoEd
the shrp€ pdam.t
r.
insi& fie
fixed. Wc oh.ryc that
$
long
i!l.d.l
s
(0.01. 0.6) by lccping the numb.r of nod€s
the vllue
of.
cnaitu iosidc this *lecled interv.l
conraining optimal value, au eigenvlhcs have negarive ical pan, when we iocEasc c
oltside lhh intcNd some eigenvalucs novcs to ngnr half planc showing very s@ll
e4onenrial grcwlll
val@ of
.
s
shom
i.
Fig. 2,14, In s@nd
is kcpl fir.d. A spenub sirh
sheme is showr in Fi!.2.15.
.xperiDc w. ch6ee x dd tlE
dl .igcnvalus imidc stdility Mge
of RK4
CUpE4
2.5 Dbcussion
Ihe
..
uu&y
ln lhis
oI RBFS elutions hcrvily depends on thc choie of th. shape
wo* $e
idea b€nind lhc
peamcr
Pm@cr is
b6is functions is in*niblc
$l4tion of lhc optimal v.lue of
th€ shaF
inBal for c in which |h. narix A of tadial
aftl tlEn to sler a value fom thal int.Pal giving fi. mst @mtc Esult3 ln our
nuBic.l exledrents wc $eh lhh vllE bv Plolling 0E Eladmship bcrwcn |n'
shaF panmte. wiih sEp o.0l atd lhe oaioun+ru' Fren Fig 2.l 4nd Fig 2 3 w'
ee $ar $cs vdB e *le.t d iion the int dals O 3). (0, l) ud (04' 22)
Esp.dively in 0!c cN of MQ for ExMPlcs 2 l_2.3 |f w. furth€r inc€& lhc valuc of
'
$c
dovn
because
bEats
RBFS
soludon
$e
Mtioned
in&ryals
th.n
fie
above
fon
cocflicie mafiix A becomes highly ill{onditioncd, fte rade^ d€ Monnend'tl lo
to fiBt find the
rcad Madych 1241. wendland 1991. chens Il0Ol ard
cgdditrg pobleos of conditioning
Tbc point
vie
3tcp
!
Lherein
lor morc dek'h
and accutacv
rac of convergen@ in spoe and timc de cdcltar.d using lhc following
'*.q1iJ&r,^l)-"
whcrc
RteRms
*ffifr#
EprdenE.mct slution ed U{ cPGscrts !h.
(2.32)
Nmicd
elurions *ith sParial
sia,r,. we calculate spatial Btc of convsgene by ktrPing nme srP A'=0001
fixed and varyi4 the nunb€r of collocadon poinrs
Table 2,5 we cm
sik, In
se
thal the conlcrgcnc.
rat
('^r
= 30 60 120 240) Frem lhc
incrcases
witt the snallet spalial steP
tire rat of convcrg.ncc wc tcep the nunbet of coU@atio'
dne steP si4 & = 0.001, 0 OOO5, 0 00025, 0 0001. It can be nolcd
order to calcubte
points fixed
fion Tablc
ud
vary
2,6 thal tbe
tinc BE of co.rcrteic. &cca*s with @dla dme $ep'
T.U.t r: ConF.is old rErt &*in[y2]fd{nPh2l
otE
tnl
T.U.2l: Cdniuo nudtd
of A
vi.hdiftrs v.l6
CIE
Co.didddiha
I l.23ao(lor 2.ms?tld
.
2 1.1655!l('1
3 r.6arxtor
7.lp35xtd
3.il935tlo'l
T.bL rJ: Cor|Fris
of
N
,v
,rt
6r
l2l
nErh.dt wnh [92] fG
of
c!iNford.ntlc2l
GtA
65t26rtoj
Cddrion
i!n$.t
9.l.lgtrld
f.303&l0J 2.t24rlt
4m26xlor 1lo4xl0n
.mplc l
2
te2l
25
rrbk
2.4:
conpds
ot
N
n*rhods with
1951.
ld 4Nnpl.
2v)
hbL2J:
spdial
l.Dl.
Tire
|
-5,
2.6:
dt
*
iv= 24o,.It
of
of
mvsge${E5
dvd$re
lr3l5
2
4
!
a..F*'r
ItE
FtscXA
SheL
r.rr E&|{L 2.l rin
N
F!fiL *rwp&fitdd otlt{tt|M'Sl
rd&!.d IiF
!'|tdiu.rd !i fllslol
r I2l
rhiaty
oqdio.{Solid
blditr n pdiElt).
lird
I
3|F!||fu'l.l*.
rta
ttE
43.
B[taL
2,2
{ith
N=
l2l.
rE Fltgio olldrlr.d /@nbstiycliEtt lqdio
(soldriE't!.En.dDhrid.n.idFlirl'r,Fielsl{ir)2-A saliEy
Flsc 2l: I'. Cladc
6dM
F4a l&
TIE
odih
Sollnjd ofri. r\S
Ec. G.27,
.
r
clsic ltol'rid
Eq. o,2?)
I
lqdid
w
h th.
. 5 b, iEdtL. isiod
ofdF
r = 20 by
rls.qM
crsir
or
IiE.
rnh di
lEr'ld dhoa
of
iniri.l
c.a
lic.
iiirirl
rtn
a7:
S.ldo c@
ad
sludon
for
ydv
Bfls .q{dd
rd i' .[.vlu numdci
lsold
liE nrr.sr
$ludon).
e
€
B
I
B
s
€
1
Ite
i.b
2l! Fi8dds dLd
ol
by
d@.u,
frF Fr@ Icd{
N
.
A
rr
0,()01,
ailt
l2l (t!.d), ft.
ditu
BrqL
z,
vdEl
.,
3!
oEooo
o
oooooo@(Eaa@DrlrMn o
-r.2 n
46 4.6 4.4 {.2 0
flg@2J: Egavrlua*d.d bytlm&pAr=0O0l, forBffipL 2.
din Eorvrlu6ofN'r.ting'. 6red.
BiDg
d2
o6
g-"o o o o o ooooB
02
@
!"
E
E"'
-oo ^-^no.Qj
46
R..l ulc
ilN
2.r0:
E!ffns
sLd
orttlr.
bt
rir
F|.dg
ep a,
ftr
= 0-q)1,
Esqk 2e
tu
ditud
r.ls
F
{,6
{,3
n
-1.2 n
llctt}
{.6 46 lra {.2
d6t
&!oEl6 FLd
bt
dE
&
a,
=
0or,
o
hdlti.d
:ta
3
l0
E
I
n
{,3 {.6 {,4 4.2
0
0,2
F€rl{2)
rtd
1ta EisordEr add b, tim .L! A, = 0i01,l8 dilft'at
of
r[+. Fl@c ? 6r
E
qL
dc
2l-
35
I'E 2l
.6 -1,4 n,2 n +! {),4 .0.4 {,2 0
F.dF)
ftlEt
13
EaNlr.rLd
otN
!, de
q
|f Strlb
Ar=
23.
Oml
tudift.d
dq
0,2
E
-l
{:6
{.C
a.Ota .o0l 4d
0
0G
F.d{z1
ItnA+BtFv -
sr.d
olai.tc
ht
td a{ A t - o.ot, tudfrd v!6
c 6.Er.4L 2.a.
a7
N=A
-l
{i-7a.54€{-rO1
hd{z)
r
r{
rtn ll!: Etovds sL.l by iiD r.p A, r 0All. ft.dlbd vdE
2.6 Corcluslon
In this cMptcr, wc hsvc rpplicd mclhLss
n
thod of
liM
(using MQ, GA .nd IMQ
Fdial bsi! furcnoN) lor thc nun ricd lolulioo of &n dizcd Kuuoto'Shivashinskv
.qurion dd thc FNllts a$ v.ry cl* to th! .x&t $lution. Fom rlE nudEncal
crFritunl!, *. cln c 0|!t ou rcthod b€lcd on MQ 6dirl bair lsltctions p.rrm
w.ll s coqor.d to o0s trDthods giv.n i. tlE lit'nnlf.. ft ha to bc .npnlsi4d th.t
shlF plnmt r fd .ll th. calculnids Ftfom.d in this Pr!6 *.r. roultd
dp.rilMnlly. ln all nuo.ricd crmpLN . tt bl! lFcEun cln bc gn TlE PrePo*d
@thod h ustul iD $lving tin|. &pc!&ni @nlin.r PDE3 AIS tha lrdial b€re
tusd@ t v. l|. Flp.tty io tivc lMh higEr .clEy with low nmb.r of sPsid
no<b. Tw n jor !dvr!trt!. o{ d rthod at! ttl. Bltlets proFny rtd us of ODE
elwB of h4I gldity ad tlFir .oib to rpli@h {E solutioo of PDE!.
Chapter 3
Method of tines combined with Radial Bo,sis
Functions (RBFi) lor Numerical Solution
of
Kawahara Ape equotions
3.1 Inaraduction
l.
lhis Cha0er. we .heuss followin8
du ttu du du
9-,9.i.*-*=n
fom
(l.t)
a<r<h.
(3.2)
'zo
(3.3)
d<I<,,.>o
in pl6m {103i,
wirh suirae Ension 0041 &d capillaty sraYity *alel waves tl05l
physical ph.no@D
shallow ware. wayes
th€
^
3.*9.*-*=0.
A vdiety of
Kaw.iua type egutioos of
lik, na$clo loustic tares
se desrib€d by l(awanda €quarioi (l.l)
and
nodificd Kawanda equatioi (3.2). Kdv_
Kawahda equation (3.3) is used to dcenb€ the one di@nsionll evolution of small bul
tinite amphudc 1o.g wav.s in vdious Pobl.n!
specilic fom or Benney-ut equadon [ 106, l0?l
i.
nuid dynuics Thn €quaion is a
DifieEnt dalytic and nuhcricat rctho<h inchding
$.
Enn_function nethod
Il08l, Adc'mid decompcition fttnod ll09l, siic@sie relhod [ll0], varialonal
iEradon D.ihod. honoiopy Frturl i6 n.thod [lll], Cr.nl-Ni@len DifieFntjal
quadrarE
ncrhod
lbonlntu lll2t. Pedioor coreld @thodr
tll4l
ard RBF @ll€atiot mlhod
rswah@ rtT€ eq@tros.
[ll5]
[l131. Dual-Pctrov calc.kin
h.ve ben ptoposed for solving lhc
In lnis chaptcr, wc prc&n1 m.sh
t
c num.dcal nerhod fot solution of above
equdons. Conenadve Prcp.fii.s of lhc Kawahda lquation Elaled to
tus,
monentun
aid erergy @ ale iivcs!8aled numaically,
ln se.tio. 2 thc nu@rical schede h
rhc nuNrical examPlcs fot the jlslinc.ioi of te
This cbaptcr is oqanied in th@ se.tions
oxplained md Fclion 3 contaitu
nelnod and ve @nclude in &clion 4,
3.2 Numcrical Method
For inpl€nentrtion of nurcdcal @thod, wc consder
wi$
the
fi.
modificd Krsahm Eq. (1.2)
following initid dd bounddy condirions
(3
u(a,t)= 8,(r. u(b.t\=
tn pE.d@
!,(I)=
Y
r,,, (r)= v
!,,,,,
(3.5)
&\n
Following
atlogcd in Satio. 2.2, wc rtPorimare 3podal d.avatirs
1(r),t-'u = 0, (r)i.
1.,
(r),{
(3.6)
-'! = 0,,, (')o.
(3-?)
{r). v j,,.,1r)r''i . a,,,,, {,)!,
using Eqns. (3.Gl.E) in Eq. (3.2) lnd collcrtnrg
*+{!,
)'(4,(r,)r)
+(a*ia)!)-(Q*(r,)!)
(3.8)
ar
=
ach
o
Q,({)=Lq,(',) q,(t,)'-' 0*,(r,)l ,i=1,2,.-,d,
Q,\',)=iQ)o,),
i,
4)
j=t, 2.,.,. N.
r
sPadal
=1,2
rcde
.N
(3.9)
0-(r)=[g-(r)'o&(r,} ., q-(j,t.
=t'2, 'N'
Q-(.r,) = [q-(.\), 0l-(..,), - , gb
a@@=#ap,)'. j=t'2 .. N,
oBG)=rta'G).
u=1,'.,' ,'. .,,.
=
x
t. i
Now w. us. th. followiDg
o.
i=1, 2....
(r' )1, t = r,
?
.
x,
mlum wctoB not tim
" ""J'.
[o" ("' )],",
a- =[a*(1)]*.
o- =[o* ({)]*
Eq. (3.9)
k th.n conren
d to
ih! fom
#+u:'(a,u+(a-u)-(a*u=0.
Aborc calatio can be
win n i!
the
fm
4=yo
wh.E v (u)
(3 ro)
(l.tl)
=
-u'
+
(Q,u)
-(Q-u) + (Qsu)
aid (he synbol (*) d€ioEs coDpo@nt by cornporc nultiPucation of lwo ve'tm
ftntiord cdlid i!
sation 2 2
u('")=[!'(',) ,"(4)
...
,'k,-,) ,"(',]]'.
F oE rhc bounduy coidhions
q(r)=c,{,)
Nov
&snh.d
in (3 5) wc
(312)
g.t
-d,,(,)=c'(,).
o.rr)
w sc clasicai founh odT Rurgc'Kulla slm
(3.13).
N!rcdcal sheN for
foloeing dt .bos
a
orher two c$adons
siE
(212) to solvc EqM (3ll)_
in se.tiot (31) ctn be obancd
Pd*,
12
33 Sbbility An lysis
ln rhis scrion, shbility of tn. first oder systEn of oDEs (3 l l). obtai.ed
RBFS
appmnma on
r"-bhn Dr.tJ
=-J,
Q*
=
fd
spari.t d*ivativs is
([o'(u)]=[iP]) c.4
dalv4d
as a
by calcul.tinS eiSenvalucs of the
bv,
-(O--a*),
r,{ ', md Q*
=
rcaul or
(3.r4)
c.4',
qh.E A antl C m 4!4
m[ics whoe cnri6 @ of $c
fom
a',2(,- rl) - a'*{1"-,,1)
-4=-r''"=_-F
ti
Eq. (1. | 4). J, is th. .l&obian of nonliicd Em Ur '
lt
3l!tr rr J
lk (r)' r,
(r,
t
l'. .(?!l,1rl*ls.li.*i)
).
(f
U),
which is siren bv
. lor dt'8onal clcmnb.
for no.diasonar crcmnts
In the n xt srction wc will fhd $ability sFctrum for differcnr nurcrical exmPles
ad
di$us $abiliy of thc rethod,
3.4 Num$icslrpplication
In this sdon rhc numdcal Esulls fot K.wah@, lEdified Kavthe and KdV
Kawdhan €quadons e Pesented. Fo! Kavalea eqution (3 1) the lovesl tuee
@nsp.d quritiA Elaled to nN. e8!y dd notuntum delincd ii [115] @
r,=
ld'.
t,=l+u'dt.t.-l l++ *r'++#rJaI
consider rhe Kawshda equaion (3 1)
a! ar a'! a'!
^
wnh th€ following inidtl md boundary cotditjoB
!(!o)=15sd/,'{*(t-4D
(3.15)
!k,r)=0;!(r,?)=0
(3.16)
The exrct solurior giv.n in
,(,.4 =-@s/"
ehetr 1=
For
(i--!,,i)).
(*
(l
nurencd computatio. we t l@ [a,r]
using MQ ard GA
tirc l=
ndid
thc neigbborhood of
$c
l?)
l r.
is cMi.d otrt up lo
in
Il12l is.
4
25. r? atd
L
basis functions
as shown
=
[-20,30],
nom
e
a
= 2 and ,v
calculalcd
vith th. valoe of
shaPe
,t
:
51 rhe sinuladon
r:0 5 15 dd 25.
par.ncl.r found
m be
in
in Fi8.3, I lnd sinildly fot GA $e value is fond to be
neigbborhood of 0,3. we have seafh.d $e
optirul vah. of the
sbape Paflmercr
bt plorting naximun edor 864 shaF pslnel.r wi$ st P 00l Th. lhr.. consrycd
qudtiris e ats shoq in the TrbL l,l,Thc upliru&s and Fir po6nioi of $.
sotiary vava e.lso cnlculdt€d. Thc ttulls wih preent m.lhod 6ing MQ @ betcr
thu the polynomial basl dinemtial qudr.tuE (PDQ mthod Il | 2) and re v€rv closc
to cosine expansion
fovdd
b.sd
differcntial quad6ru@ (CDQ) nerhod [l 12]
notion of lie slitary wav€ in conP!tuon qnh
qst
InFig32rh.
$lution (3.17)
ar
ditfetnt
$abifty spcculn lc this preblen is shown in Fig 3 l0 ud Fig,3 l I for diffcEnl
values or sbape pdmcter and nunbet of nodcs Especdvdv The physical benavior of
cigenvalu€s has nild exponential grc*th, which is cxFcted in nun.rical apPrcimtions
b
rhc @thod rcnains appli@ble in
tl s ce. Th.
Poinl
*is
tat. of convdgpn@ in
so@ is calculat d by Lcping ti@ saP Ar=0.m1 nxed and vatyiig de nunber ot
collarion Doinls (N = 20, 40. 80), from $c Table 32 {e can se€ $al the order o'
co.lergence d.cE!$s wilh
have
usd MQ
and
lh. smller
sp6tial slcP sirc' ln dll nunericol €xanPles we
CA in order to calculac ord.t of conv€rgence
coNider d€ modifi.d Klwaha.a
.qutioi
(3.2)
-+r -+---=q
wirh rh€ followin8 initid md boundary cordnions
!G.o)=Dsr,1(rG)
(3.r8)
!k.4= D*ci:0k-rr)r!{r,4= Decrl(r(r-84)
(3.19)
'Ihe boundry conditions
e
extiacGd ftotn thc cxact
rlution (giv.n
in {l16l),
r('.')=Dsr:(tir-84)
',The calculatids
@
(l.m)
1l
tri.d
out by ErinS 1., bl = l-30. 301 wirfi
t
=
6l.wc Ds Mq cA
ard lMQ ndial b.sis widr sh.p. paElrscl found !o bc in fic migl|to.h@d of 3 ror MQ
s
slb{n in Fig.3-3.
For CA
Espedvcly. The a, dd
L
dd
IMQ tnc vatuc of
nms c.lcubt
d {r t
-
.
lies in vicinity of 0.5 and 0.0m1
0. 5. 15. 25
@ shom in Table 3.3.
TIE ordcr of convdgcne in spd d.c@Fs *ith imMsing lva show. in Table 3.4.
'DE $litary wlvc profilc at diff@nl dN lcv€lr ii.ompdisi with the cxet solutaoo is
In this cas
shape
ssLd
eiS.nvalues with
pcarcter md nunbd ot
nodcs
d
sndl pGitirc Ml pan for diffeGnt
shoei in
considd lhe Kdv- Kawalaa cqurtion (3.3)
wilh the followinS inidal Md bound.ry cddnions
valnes
Fig.3. | 2 rnd Fis.3.13, @sp€ctively.
of
,(-.0)=-qs4/,'
(+(. { ))
(3,21)
(22)
("i;(.-#,-t))
,(,.r)=-ese,,'(t!(r--4,-)it
,(,,.)=-s*r
The initid conditiod
dd bdrndary cddnions
(3 23)
N ext&t
d fiom dE ex&t $lution (8iv.n
inlll?l),
))
' 160 t2Jrr( :!:,-,
"',."=tls*r(-!L
r6a ".JJ
rl2al
The simulaion n perfoned by uking Ia, bl =
dd
10. 2001
A, = 1. The dis.r€t
ot
l-
@ calculaled using MQ. CA
and IMQ 6dial basis furcrions up ro tihe, = 5. F on tbe rsulti shopn in Tablc 3,5, wc
clD se dut the both MQ and CA !E showing very good .grenenl vith Lhe ex&t
m.an sq!re €ror nom Lr
maximun
sludon, The spao.l Etc of @Dv.rgcnc.
d.cEes
by increGing
olldltion
crcr nom
wilh
is shown in Table 3.6. Th€ otdet of conlcrgcDc.
poiou for a tiied time srp A t = 0.001. Thc
fo
sd
!t dificcnt mc lcv.h in @npanson Mth th. extl
elution (3.24) is shom in Fis.3.6 (s.@ s in [18]).
Eigenvalu* ofthc rfobie mat ix with snall pGitive E l pdt hav€ ben shown
movcmefl oi lhe soha.y wavc
in Fi&3.14 and Fig.3. | 5 EsFctivcly but it d@s not
ha&
lne
R(4 mrhod utupplicablc.
For intenction of two pcitive solitary qavcs of equ.tion (3.1). we corsitu th.
following initial cohdltron
,1-.0)=ia,*/,, (jF(,-,,r)
we solle fie problem by usint MQ
la,
rl
ro
tirc
=
[-50.100] wirh
r=
cqladon
x = 2ol
and
GA Bdial bsis tunctiom. The sFdal intctvd
is scl4r.d. rhc nutuncd erpe'irent is
50 wirh dne sGp a r = o@1. Thc vrlucs of Oc
e
choen
s:
r, = o,
4 =N. o = 4l
Jr(,f.,
csicd
our up
pom1e6 usd in dr abov.
4 =ro/o an 4=E 6
$li!.ry wav* prepapt roe.lde righr s the tie prcga6, The p@ss of
im.rutior is shoM in Fig. 1,7. DunnS thb prec$ dE lueer wave cakh.s uP lhc
The teo
snallet orc a.d tlEn th. bolh warcs
scpec frcn .!ch
othe. m.intaining their @gimr
sb4e. Frcm Table 3.?, w. can se tlut th. inv&iants of moridn €min.lmon consened
.s ram incEases. The vriation in th. fiEe cons.rycd qutnlides is found to be in $e
lL
140.509259<
MQ
<40.4838916.1
140.509259<
t,3 40.412842,
l
s 45.84J648
r< /. <45.8509184.I. GA
i4r.8jor4
145.8361413Ir
L-J.2 J7:ro< r, < -32.15a9r
fl2l?2189st.<-J2
I
Fo. intc@fon of
,',.0,=
f@
I.
14082201
solihty wN.s we cotuida .qution (3 1) vith the following
i,c '*r'l::ru-r
\=
rl.
we solve tlE p'oblcD by usins MQ
!d
ar=o ?5 TlE.alcularion h cdi.d our
val@s oi rh. pal"@t E ued
aA fsnctions orins [4, t]=[-30, r20] wid
tmc r =50 with drc slep & = 0.001. The
rh. abov. equation @ choen 6:
up lo
in
=.xi, 1=0, 4 =zq o = tf ,!tu, e, =rq ". e, =u1" .a e, =01".
Thc rhe solibry wa6 FopaSaa owards ddr, The pta.s ofi etution is shown in
Fig3.8. Dring this proc.s tlE ralld qavc mv.t fast r dd .archcs up the sn ld
n
vavd
and
rh.n rh. th@ wav* scpatate
vavs aler coubion
motion
Edin
fon @h
o$er. Tte shapc of tlE
is mlnlaimd, F oD Tabl. 3.8
alnost consesld
w.
can
s |h
6 tim inc@e' Th. vdialion
$@
solitary
de inv&imts of
the
in |he
consrycd
qumdri€s is found ro bc in lh. rdgc:
<514?]E98.
I
[51.4726313/,
<51.r04t2.
MQ j5r.roo4ars /. s5r r5838.
CA
l.
151.100491<t,
I.
/. <-l]069?3J
<-Jl.3036l
l5r.a726jl3r
s5r.568591,
I
[-jr.6]8063
In this exanple we
[-3]63806sI.
qill
I
show thc phenomena ot wale Senention for equalion
considd$e iollowing iniial condidon
,(,,0)=/r.r'(#(,-a)).
(l.l). we
Conptatlonal domin [-40, l30j with ,r = 0.625 is coftideEd. The sh@ is Nn upto
tim, = 18. W. tal@ /=10. IlE 3i!sL 0olit ry w.vc aplits b 10 thE€ rclit$y waves.
Thc
cqed.d q@!ti.!, h.idt &d pdition
of
thc els e
cd@lat d
.l
veious
tirc lcrcls d slbwn ir T.blc 3.9, With th. D$srge of tinE dE l@ditrg wrE owinS ro n5
f!.rd v.l@ity gcts f& t@ ttc othd two er!6 a 3bown in Fig3.9. veiaid in ftE
coNcrvld{llD iries n fdDd to b. in 0r follosiry tu&:
<r,<96.14165[
I
196.1a365r
I
3 r: 3 329.65039,
GA
<
l2c.6503ec,
196.14165rs/,<%.lij65r.
MQ j
329.65O39e
-<
1329.65039
l.
'r
r,
s-70r.426ElEl
r,
<-?06.E
E96j
1,875.4160413
[.875.416123<
l.
chopet 3
Trbt
3.
l: R6ulr ld Krudle €qunion
in
@mDtisn whn
I
t2f250
0.15362
0.163t3
1272502
25
PDQTII?]
2,5(t
CDQII
D]
12]
t
IrbkJ2:Sp.ri.lr .of @rrc4EeartuEspl.3.1 t=25
fi'd.
MA
Tdlc !3t RBult3 for Modified Kawah@.qution
L-
_l
1j;Ta
l5
5
0,311
o,m3
TlbL J'+ Spdi.l rE
oI
mqgc|* r
for ExmPL 3 2 |
:
5
LL
6.7te3
T.bL 15: R6!lt! fo. r\dv
los.hn
cqurdd .l E5
3266
59m0
59ti.8
t.n8
T.tL IG SFd.l re
of
@u8de {
T.tL
lq
in|*rid
3?r trvMsts
Ib.
of r*o
GtnPL Sl
I=5
tolihr fd EEfpk
I
t
ahopEt 1
T.nL 3r!
Irwieb
nr
inldid
or
|nE
rljhr
f6 Fwnph ]5
-31.6]306
51,472631
-ll.l90t5
r.8rt962
51,63t932
5t.tcx53
51.r5333
r.bh !J: lnveieB fq rnE@io
of
6E oliloB
tor
EN|Pk r.6
MA
1.5
t2.5
li:11.
m
tl* 1l &d ffid.F F.rec rrSurbl.l.
FllrE 31. Tr.slung wrs toldid
.howlni .rEl tolutid
of
&r*rlm lqurioi (!olk llmr
{d tlhowiig Nrt&i
toludon)
5,1
d|qPatt'
nr! lt!
rtn:ltar
So|t.y
&E
tE
16.ondd
dd rtid(.tl lh.
d.F
ot
!ao'
F,.dr c b
B!!p& 3r,
!.dn d v$br..$ao
qd.rri.
&.t
lr
or!.id
en
t tldl'l mElcd .l'tin),
Itc!J! &ff |!..bFFn|F.
I{a- !4
SorL,
do{
rE dd.a
r.dv
bl44lt3
3,
xrwls.qdbr (dn uc
4d ot|bn !d t'iFvia .uEial
$ldid0.
liaEr &t! bEed.n orrs
.liM
.qu!b,r, BUDDb 34.
ror
K.*rhra
ctpoE
ItF3t rrffdr oad6.lx.!
fa BLrd. 3J
S
ItE:t': Gdid olffi
ft.
Er4&
J,6
59
-9-
I
ls
oo."
tlr.llc $68
lo
t.,
tu
Erqb
11,
s8:
ftNarrr
SaLd
iaEnls b Br4L
t.
t,
a4g1
i
.;x
ItN:irtr s!6n
I
F
!
h. BrMd.
3.2.
*
1":fru"'--"
IlFaDr SrLd.ia.vd6
6.
E
,aL 32
61
s
t
e ..
fd.-
"
r|g!F
iL.
d
il':.'
EI3.lvdar ft. EiEd.
.
3-3
I
€
ryEi
ar3: SDdE ft.
&.!rL
13
62
35
Cotrclusion
In Oir chrpi.r, w. harc u$d @rh l€ss m.tbod of lims
fd nu@ic.l
Noluton or
d.sibile oomn of singlc sliiity
wav$ dd pl'.no@ra of wave Scn.rtion
Xaqahda typc .quali@. Th. nu@dcal r.lults
wav., inLnctio! of two .nd $re. soliury
be! disused. Th. &cu..y of thc aolurion d.Flds upo. Ue cnoic of th. rh.p.
pmnlcr, shich h.s b@ s.lccr.d o(primntlly. TIE .unericd ada $iry MQ &d
GA td Krwrl& eqlnid N b.!.r tho th. Crut Nicols dif@ttlitl qu&|tuE
dsqirhD 0!21. rl|e inv.rirnb of mtid F@i!.d cd*nld duiot dF PMlss or
cdrPur.*io fd a[ c'$.
hlvc
Chapter 4
Numerical Solution of Equal width Equation by
Meshless Method of Lines
4.1
Inhduction
Equal width (Ew) cquanon was suggested by Motriso. et al.
tll
as a
nodd to deetibe
v.v6- This .quadon b D alt madve fom of w ircar disPe6'v'
waves ro qell kno{n egririz.d long wrvc (RIW) cquatro. and Konoseg_ de vries
nonliM
dispe6ive
(KdV) equalion. Thee equations hav.
slit
or packe$. Th6e wltcs popagatc in
mt
{
shble wave
tv wave slutions in th. fom of wave Pul*s
itr
disp.sivc nedit Thee wavcs na|nlan
even atur inletution baause of balaDcing of nonlined ud
geneEl
EW equarion rls tav. eharv s.v. elutaons blt ot l's
fom
disF^ivc.ftets.
ryp€. Thc EW equaion dErived fot prcpagation of
a! d!
J
lar,
-+!--r=tsheE
!
is lhe
dDlnud. of th. wav.
t
Oily a fcw dalyiicd Echniqua
numnc.t fludy otEw equation got
)t=u,^
is a posidve
ot
Lhe
'
lotg saves in positive i-di@ion
(4.1)
padmrr
'nd
!ro
avanabb for EW equdrion
aucrdotr of
has
6' r..mlE^
merhods nlve been opplied to solve thh equalon CardDer LRT
as
rt1_
Th'relbe $c
DiRcEnl nutrncal
ettl disu$ed elitary
wavs of Ew eq@ion Il2ol. Archila $.d spcclral mtbod to elve this e{uadon [l2ll
Zrli uFd $e nethod of leasl squ@s linne .kncnl *ith sPace{ne lin'd finit
quanic BiPliie tl2ll Ranos
elemenB ll22l. Rasl.n ued coucation mlhod using
a
tr€d exDlicn finite diffcEne @thod for EW cqudjon [124] Saka ll25! @nstucd
fini& elenent solution for th. EW equaion. Sat! cl al {1261 lsed qlanic B splinc
Grl.*rn nediod (QBGM). diff.Endal quaddt@ Ethod (DQM) 6d mshlcss Etho<l
S.}r I l27l
Dag and
also solved EW equaion by linn€
ln this chapter we dev€loP a mesh
widrh cquatio.. The chapter is
fomllrtion of nesh
fe
ftc
ehnent melhod
sheme for $e numencll slution ot equal
orghidd
s
follows:
I!
setion 4.2, w. Sive
@tnod. S4noi 4.3 conuins $abilnv molvsis and S.ction
contains iuherical exmPles of lhe Pmposd
trrthod
l!
ection
4
5
we
the
4
4
sumnez. dE
4.2 D€s.rtptton of rh€ method
Consid.t lllc Ew eqlation
(4.2)
wi$
the following innial md houndary condnions
(4.3)
u@,n =
\!),
u(b,t)
=,,o
To apply resnbs nelnod of lircs, vc n6r use 6dial bash functlons lo
lppoximatc spa@ denvadv4. wc will use muldquadric (MQ), eausian (cA) ed
iivdsc multiquadnc 0MQ) 6dial
{,i=1,2,,,.,,
xr
=r.
basis functions. In
oder ro
implen.
be the collocation points in thc inleNal [a,
In meshl.ss rethod of li.es
v.
b]
tne tuthod, let
such lhat
lr=d
a.d
use the followi.S apPoxinadons for spdtial
!,(r)= Yl(r1.1 'r = M,(r)u.
(4.5)
u,,(r)=vl.(r).{ '! = M.,(r)u.
(4.6)
whe@
M,(r)= YIG)i L md M-(r)=
Yl(I)/
r,
$
we get the follorins disclized
@1
f;.n1u.111"1-,(x-tnr*)=n
I!dd.rbirit
i!
Eq (4.7)
'-,t ".n
6n
ctu v.cr| mlr[ldt L!
M,=[nr(r]L,
u-+.(4[-'
x.(r,)-[r.,,(r,) rr,(r,)-x,,(',)1,
r,G,).*r,(,,),
J=!.2...
f
x -(r,).lx,,.(r,) L :-(,,).-. |,,.,(r,)1,
u,,r,,l=
I&E
$t,t,,r.
8q, (4J)
i=t,,,...,".
b.c!@.,
gL*'t"."1-,("-#)-0.
(4s)
1r-,r-1ff--p.rr.u1,
(4.e
#=-*
c(tD
F
(1.10)
|
=(r-6ir.f (u.(M,u))
@ |!c ilitid
rd or hddly caddo.
d.$dD.d b
Eqr
u(6)=[ro6],r'6),...,!"(',.r!'(,.f
,,,()-,,('),,,(')-",(')
(43)
$it (44)
(4.u)
(1-t2l
To solve Eqns.
(4.10 4,12), we use dalshallounl orde. Rung€-Kulta nethod:
U"'=U'+
r,=o(u').K,=c[u'.+r
,J
r,
=c[u'.fx,
).r.
=
c(u'-a, r,)
43 St bilitJ
In this
stio
we
syskm (4 l0r. RewntinB E9.
pEsnl sBbility
4=-rliU,tM u)ll
wh.e.=(/-,M_)
(4.l])
i,a@t ix with Nnstant
c@fficienB. The Jacobi4
natii
ior
this nonlired synem k given by
a(-.(u'(M .u)))
(4.t4)
AU
allu'(M un)
--:.:-----j---j-- $d its.t.ftrrs eolthe
fm
l(rr(t)M.(r.t)) +>(!(r)M.(i,r)),
-11,141".1'
"
o;
In ow ou@rical eMbples,
4.4 Nume.ical T€sts
rd
re
wiu di!c!$ $ability of $e mfiod hv showing th.
Results
Nurcricd 6'nts of lfi! aborc shcm fq EW .quatior
savc, int tudon of rwo
slibry vav6
.nd undul$
bd.,
6
discusr€d
fo
singlc soliury
The Ew equation poss6ses thc folowing con6.nalvc laws for nass, mom.nrun and
(4.1t
r..l[4' .4,^)')d. =J'
For singlc
slilon slurio thc
(4.16)
,(r.0) =3dscft ' {p['-i,]).
(hc boundary condilions
an
(4 t7)
a(0.r=!(30,')-0
The
end elurion s Srtn in Il25l,
is
(4 l8)
!(x.r)=&scir'(ptj ],-d'l)
This equation
epdF
ts a singlc
soliary
The values of lhe Pdanele.s involled
Tho spadal intedal t0.301
wilhn=
cxperituturion- The @Ufuy of dE
*!rc
!c
or
dPlnud?
ld
velcitv d and
$lcded asr d=01003
o15 @d
P@
I.
-l- '
'=land{=t0
aI=0o5 h slectcd
$herc in lcm of
P=
for numencal
aid
L
noms N
d al tl26l Th. lolilary wavc PDPagates rhe
spadal inledal t0, 301 duting th€ dme incdtl is [0 801. In ou nume'ical expenmnrs
4 4 dd | '
wc sld de oPtmrl val@ ol tlE shtpe Pusrclds in the neidborhood ol 0
compaEd
{ith
lhe
Es!l$ givei by
for MQ, GA antl IMQ.
quan i.s for
.tur roms
6
shosn in Fig
d = O.l and d =
by MQ
saka
4l
The
emt nom
003 de listcd in Table 4
m b.ttq lhd
n is cleu $al while consideriry
l
and
and Tablc 4
th€ tcsuxs Siven bv Saka
't
$Ee conFRed
2 Wc
see that thc
al (126l Fom
!, dd I. , $e Esulb of MQ @ betr.r fin
the tables
lhai of
cA
Thc solitary
and lMQ for d = 0.1 and CA is 3howing bclt€r performdncc for d = 0 03
motenent of
wlve soludons fd drE RBFS m shown in Fig 4 2 Fon this figre
$'
norion lor a
solir,ry watcs sith no choSc in shaF is obsefled Th' lhru invdidls of
soliBry wave of anplitude 3d and width dlpeoding on p,
(as
my
be
.valulcd Mlldcallv
siver ir Il26l)
64
t2d1 4au',E
pp
Far d =0.1, /, = 1,2, 1r =0.2E8,1r = 0,05?6,
Fot d =A.$, I | = 0.36, I1 =0.92592. rr = 0.00156.
Fon
ln
Tablca 4.1 md 4.2 we can scc
d. ir
thce inldimt!
v!ry
Sood
N
nurudcrl valks .nd fi@Etical valws oi
agr.crcnt. In Fig. 4.3, qc havc Plolt d ln€ saled
tnt
Jeobiu mtrix fd diff.|lil mbd of nodes t@ping the lallc of
shaF ptrs@ter fi-d. Fd MQ ! slobtc apcctruD b obbircd whil! ir cs oa GA and
IMQ rcw ciscnttlB lic in pcidrc hrr pl.nc wry cloF !o am as shoM in Fi8. 4.4. h
cigenvalu€s of rh.
ou wond crp.rimdt
E
kep
insid€ srabihy cej@ of RK'4
/V
lixcd ad chs8c .. ln
sh.m 6
furthd incras c fic aF.uun shows
tlE vduc of
.
ce
is
ol MQ all eig.nvalucs
li.
lf
n
clos. !o opliMl valuc.
. 6ild .xponcntial trcwtt
in posni* half
Pld.
Sinile b.havid is ob$Feil for IMQ whilc fr GA strul!
of shaF paFna.r tullts in stlblc nod.s &d inclts in vrluc of c cNes utr|oble
which do.s mr affect $rbilily.
valuc
nod.s. TIE sLbility rpeEum is shoM in Fi8- 4.5.
To
obFrc
rhe
ini.rdtid
of
t*! slillry
!(x,o)=!,+a,
u, = 3d,.@
h' I e,
\'
waws wc coNidd lh. innial condidon
(4.re)
-
x,
- d,)),
(.mJ
ed bolrd4ry condnions N
!(0,,)-!(80,')=0.
14.2t)
Tbe valles of tlrc pEmelers
.=1.
Th€
p,
=05.
Esrl$ @
pr
@ tllen
s:
=0.5,I =10, J: : 25. 4 =1.5 dd 4 =0.75.
compured
*ith
Ar=0.l,Ar=0.1 at timc r=0,15,30, the pronb
is
|*o waves e moving rowards
ngh wnh whitiA thrt &pcnd d drh dSrtrtudet dd fl ccnln slagc tlE ld86 waw
caEhes op 0E snaller on. ar|d bofi lhe qavcs unite $d tlFr sepmt frnm dch othr
shown in th. Fig. 4
6.
!r is cleu from thc figurc thar ihe
teir odginll shape. The prccess of energenc. .nd sepdaion lakes Place
berween O. trnes r=10 ard r=20. Th. valua of th€ consmcd quaniri€s in
nainaining
comprism eidt qlanic B.sPline GaL*rn rcthod (QBCM)
The rcsul1s 6E obrtned by using MQ
TrE
dalttcal valEs of
rhc |h@
ndill
invuidt!
basis tunction
e srH
[
126]
t*ing
@ given in Table
4 3.
the valuc of tlE shaF
bclos bv usitrg fomula givs in
It26S
t,=rz(d,+a)=zt, r, =2E.E(d,:+ll)=8L
Frcm rh. T.blc 43, n is clc& rhat tlE
MMLaF
rhe
very
r, =5?
6(li +d:J=2t8.1.
nu@nol valEs of tlE @NP.d qu.ntnies bv
clos to dalyticd vahes
in conpdison with QBGM 1126l. showing that
prcsnt shcm. is fairlY conervcd.
10 study
undutation the initial condition n @Nid.Ed as
wr!.
,(r,o)=o.5t4l
/
\
I
/.-. \\
|!nhl:---:c ll
\.,./
&d bolndary conditions
(4.22)
e
(4.2r)
vhere ,o is the anpliturL of w.ter
the slop€
betwn
simulation is
abov€ dE
equilibrim l€vel at initial tine and d h
th€ stitiondy wacr and .lccp.r
@ied out lalint
vatei !n Eq, (4.21). !0=0,1, Tnc
6=0.1666667,rn =0,
At=l
md Ar=0.2, oter $c
in dal-20<J<50. IIE .onpot tjoo is cuutcd up to dme t=Em and rhe
&rclop@ of utduld polil. is shown in fi$. 4.4-4.5. wc E odtiquadric RBF
lMQ) with shlpe pamcler .=0.45. For wo diffctnt valEs ot d=5 dd d=2, rhc
rcsults for
thm iDvtridrs,
pcak position
.!d uplitude of dE
l€ading undulalioD
givcn in Table 4.4 and Tdble 4.5. Fom thc Tablc 4,4 md Tablc 4.5, ir is
mximum value of amplitude is 0.18J030
for d =2, which
slier
e
alnosr $e sme
a
sah tl25l,
at
r
cler
= 44 8 for d =5 and 0.186?4 ar
in Ef.Ence
[26].
l.
@
thar the
r
= 45.8
Tables 4.6 and 4.? lhe
and Dag and sala
[27] are
lisred md fmn nutuol conpuien w. sE that our mthod is capable of giving the ene
ectmy *iih low spadal r.shtion and a ld8.r tim step, Our me$od Nunerical
vdianon in the q@ttia r,, r,, r, e calculacd f|om T$16 4.4-4.5 using fic fomdd
EsulE of
studies by
Saka et .1.
[26]
given in [126],
I
,ld tiM
=8..]lJ)-
I
,ld tirc =0)
Th.* nuftrical vdiatiors
$eoEncal valu6 siven iD
e
shoen in Tabl. 4.E, and
[26],
ii*=i':=-'"'.
!i0 * *:) !.:
a, =
=
o
f;i.t*=!':-.o*".
aaoouo".
e
in very good
agrene vi0r
T.bb
4l: lnvlmB rd .m m
for rlE
,ir8L olir.ry wlE
QBGMll26lh=0.15
d=
0.1,
Trhle
h= 0.15,I t=
0.05
{2: Invdios old mr nms
for iitrgl. solil,ry
;
o0t43l
d
- o.oi h=0.t5,tt:0.05
wpc
T.bL
r|.3:
hveo6 Id
'nhdior
ot tvo
slitt, v.€
QBGMIr26l
2'8.7028!
;
26,9998!
2ts.702tx
213.70130
213,69966
269991t
Tbbh
44: D.FlopMt
of
u
undula
boi
xflmt3
3,0(x)$4
iorlr16l
o,t76r27
05630t5
!2.N
T.hL
'L5:
D.wlopnEr ord u.dllr
b.c
MQ.MOL
0.r8955
o.t9t27
MM t126l
0J3t62
T.bL
3OIr27l
a.6:
Drv.lopmr
0{,05)
3etr25l (lF4.0n
of
d ddulr
hoa (d=5)
ll.6
T.U.,Lt: D.vdo0|Mr
of
u udd& bd! (&2)
0.723!40
0:tbrD
6,00243
0,tztall
T.bh 4E:
Nmridl v.rl.!on
ln
conr.d o@tiri.s
Itn'lLhr4@..tuA!b{l,
t5
chdot
o2
0.05
Ilm a* soliey d€
lofils rith .rylibd. ol.
Pr.bLo 4,1,
rl
I
i
s
t
d
I
I
IEG,Lt Std.of
Lcobilt
tdir
nt drE
s or..da Pr.bld4l
7A
I
t
"l
I
IlGa.+ Spd. ofr..otie dn tr &* y.!E of d!r. Fr@.
Plobld a,l.
n|Dar:rddCoottw.lic, sl$lFbd
bM
a0
c
FirrE a6: Uiduldioi pbnhs,r dlllEm
op@ 4
It-a?. urdb ,.ld|. d dtftrd riD LEL tae2.
45 Cotrdusbn
Nwdcal t chriqE h.$d
bEn
od msh L.{5 ntthod of
ibpi.|Ml.d fd th. !l|@i6l $lulion
liB ui.8 Edid tois fidcric
ha
of Err .qonid. Tbc o€ibod pEenls dE
posiiu ald v.l@ity of $ryb 3olitlty ewc. TrE tslis fo tlE itrl.ledo4 of
r*o $lir.ty waEs &d slmdr d.rcloDdHt of urd rr b@ otrtrm sdid sludi!
Stability !"lysis is p€Ifom.d by cdcuLtinS dE .itlNrlB of dt Jeobi& urir.
.Dplisrde,
Tlrce
iNeidt'
pFsenr work.
of nodon
Eflicidt
.r.
found to bc
numerical
cNblr fonll
thc
cM
|. ult! at! obtlircd lad @np@d
wolk U24 l2?, l28l avdau€ in thc lit ratu..
d.$dbed in the
witn dE publish.d
Chspter 5
Method of lines for the solution of modifred equal
width (MEW equation using radial
bssis f?u.nctions
5,1 Intmduction
Thc nodified equal
*id$
dhp€sion process has the
Au
equriion (MEW) Il l9l, dising
fbn
Lhe
nonllnear nedia
fom
#.*(#)='
whcrc v is
a
(5.t)
pcitivc pan@io ald
a few $alytic elurions wilh
available
fd
modificd
egnldizd
thc sub6c.ipb
r ed v denot dificEntiarion.
. cuicl.d et of houndary dd
Only
initial mndilions
ft
nodifred equal width (MEW) cqoation. Thc MEw equrior is elaled !o
long w{vc (MRLW).quation
(Abdullcv.t
nodilied Kon weg de,Vries (MKDV) cqurior (Cordicr
of MEW eqmrion wilh
intercsr
wi$
0r vdious
foi the @semhe^,
E6e. and Kutluay
er al.
al.
[30]. Finite.l.rcnt Nthods
based
the
Il29l). Nunenc.l srudy
bouodary and innial conditions has
These hdrhods include finite
ll28l) md
ban
the ropic
of
difieene ne$od given by
on collcalio. a.d Galerkin
dd l34l bave ben
Y, DcEli ll35l usd iesl e$ n€lhod basd
nethods usi.8 qladndc, cubic and quintic B4pliies
nuftncd study of MEW equatoD.
on cou@adon t@h qne to solve Dodificd .qual
used for
ull,
112, 133,
width equalion.
In this chrprer, qc solve nodifcd cqurl widrh (MEw) equation by @shl.ss mcthod of
liDcs.
nshlss @tnod of lin6 (MMOL) for MEW
Setio. 5.3 is d.vocd to st biuiy ebsis. Scction 54 contaiN dE .u@rical
ln S4tion 5.2 se
cquaaon.
describe thc
exmples pmviding the validation of dE @thod md we €omlude in S6tion 5.4.
cr@to
52 l'trcrlpdou of th.
5
dtod
Cci.b XBw.q!dd
9*r,
dt i-"91{4l=q
dx d\u-)
"s,s!,
wirh
tu
folowinS ioitirl
!(ro)=s(r),
ud hdd&t cooditi@
a3r3r,
!(., r=s,(r), dr,
ForiopL@.t
(52)
'>o
(5.3)
r=rr(t.
dotr of tlE
(5.4)
r.ht$ ntu
of li'E!
s!
divid.
tlr 4xdd ifl.w.l
ta,
,l
[4]: t|}iIg { = a .tdt' = r. wc t!. F4a (14) a'd (25) rnd
sprdd d.dv.dvcs by ! .in€lc nltir by vcct|r muliPliqliotr
into co[ocldor poim!
,Cbc.
r,
(r)r,
(5.5)
l.(')r -'. = 8,, (r).,
(5.6)
(r)- Y l(r),{''!
!,. (I)=
Y
=
s,
$/ht q(r)=Ylr,rr'!d s-G)=Yl(,)^".
Nd
couocrdoo of 8q.
(t2) tt aclt
rd.
++3',,'(s,(r,)o)-'[s.(',)41=0.
u)
4
\
U!i68 tlE @lum rldt roLlid,
rj , Siv.8 tlE folowiry
fdm
r.ra...,^,.
1s.7,
s.=tr,(1)I-.
s-=[s"(r)]*,
s.(r,). Is,,(r,)s,,(,,)...s,,(,,)l
J,{r,). ;;J,(,,), i = I,2,...,f,
ali
)s1,.
(, ).. ri,,(r,)l
)=;;'s,(r I
i=1.2,..,x,
s,, (r I =
s,,,
(,
lJ,,(J
IJL. :,1u . u ;.
dt
N
1s
,u 11- " f s.,
I
o
{l=
dt)
(r.3)
=-rtu,
wh€'c F(u)
L) 9)
=(.I-vs-)'' (3(u1u)*(s,u)).
The initial @ndirios
e
U'-l!"rr,). !"(r,).....,"1r,-,).
fon
r'rr")l
dedbcd
boundary conditions
r5.ur)
in (5.4)
6
,,(,)=a,(,),,. (,)=a,t)
NN clNical fdnt
$lv.
tlE stsEm (5-9)-
*!!1j1{{!1!1
u-'
= u.
r,
r(u').x,
=
odhr Rugc-Kut|r (RK4) mthod is ued to
)
=
rIu'+]r,
J
r, = FIu'++,(,
=
r(u"+^, (,)
,),,(.
dd
the approxinar. soludon at
fte pfen
rethod
can
dy
hlndl.
point in th.
the
incnal
nonlinor t
r]
[a,
can
m arr, pEent
lineariation. TlF f&t h thal at r - 4 a can be touid frcm
'Ir'en rcsr is m.E elemc wisc nultiplicrdoi or vato .
th€
b. roud
ar
qch ti@
in €qn. (5.2) withoul
tilen iniial
condi rron.
5.3 St{bility Analysis
Fof sBbility malysh wc coNidd .qultion (5.9)
ltd rc*dle
s
35
4=
-3p({u,r{s,u))). o<, sr.
vlr.E
P=(r-vs-)]
b
a
squc mEix vith
Thc subility of tlE mthod is
of nBl
(r.12)
u(o) = u
kmwD clcmnts.
d$u4ed by cdculating ciS.nslud of tF ,sobirn
n
trix
ddd sysGh (5.12) dcli|ld a,
a(-rP
(u"(s,!
a{(!'.(s,u)))
)))
(5.13)
au
AU
(5.r4)
TIF
r,
dcIEG ot./r .r. oloc tom
[(r1,1r11's,1r,r1)
-{
l(,(4)'s,
Wc will aDlyz.
+>(2!(t!(r)s.(i.r)). rddi.sorrdeEnB
(i,t),
.igcn!.h*
rd mrdi.sorr.hEnrs
of tlE
,rco0i6 mEix
J iD
n.xt s.ction.
5,4 Numedc.l Exrmdcs
Th. MEw
cmryt
cqutid p6s.!g
niw
tlE following cons.
hws
rd nasi nom un
and
{* ll3ll)
Bpeiv.ly
,=i*'=i[,'."(*)h'',=!''l^
(5.t5)
Th. co|rwatio! of tlcs. iovariant shoes dE stibility of tl|c rul|i.ricd $lEnc.
tar $qh lola.rt erY. Dodo
Thc inirid co.dirion
r(r.o)=
f6
dE mrion of sitrr|. rolit ry
q!rc i!
dq
As/r(t(t-r))
*ith dE boondrry dlditioB
Th. cncr solirary wrE
i! I l3ll
(5.16)
rJ0
solltid
for
a
l{+-.
cqurtio (5.2)
(si6
in I 13l
l)
,{r4-,1si(r(!-e,-r))
whcE
t=*, l
is thc mpliardc aid p
k
thc
wkrity of th. sintlc elit ry s.v.
h
or rhe
ou sultr wnn
order to €onpde
paliftres
invotv.d
ja
s: !
dlicr
= l, A = 0.25, P
qort. w.
Publislul
=+,&
=
30
dd
clrc
the
vatM
spadal donaiD La. bl
=
= 0.1. Th. sinulrtion is run for 0<,<20 *ith tine step
& =0.05. The rcsulls obt in.d by n6hlc$ mthod of li.es using MQ, GA ud IMQ
Ddial bals tunctons m @nP.cd wi$ th. tesllrs PEse.i in ll29l, ll30l. Il3ll md
IO, 801
I
win scp
r
l32l- In Table 5.l. tlE cdor noms,
l3l I md
1321.
t
$M
invlnan6 of mlion al t = 20 de shown Fom
fie results by ncshlcss nethod of lines de betle. than [l30l.
we navc aho conpdd our Flults qith finitc ehieir se$od [ 133] ror
Table 5.1 we can
t
d
se
thar
lnplillde 3 showi ii
thrce differenr vallcs of $e
rale with
MQ at time ,=20 is
Tablc 5 2. Single soliDry
diff@nt amphudes in compeison with the cxacl soldion using
shown in FiC, 5,2. Th. optitul voluc of shtF paraml.. h sel4t€d tom the inrenah
(0.1,
l),(2.5,8)ed(0.1, l)fotMQ,GAandIMQEsFctivelyasshownioFis 5.l
The saled eigenvalu€s of lhc J&obian
dd
IMQ. Fi^t wc
nodes
mtdx
lir . (sh.F p8!trct.r) dd
(x =200,400
(5. 14)
Epc.t
e
inv.stiSrted using MQ. CA
.xperim
and 800). Fom thq Fi8. 5,4{i), we
c{ w
for diffeEnt nnnber or
thlr all
eagcnvalucs
wirhin sbbility Egion of RK4 $hcDa In lhc scond cas we keP /v fired
value
of.
spetrun
The
is
i.sid. the int€Rsls co .iniry oPlimll val@s. ln this
obt ined
ddrric
, _'"
ii
as shown
valu.s or
lhe
0.785J98,/
chmee
again a stablc
Fig. 5.4(iD).
itrvdidts givcn in I l30l:
=:::- ::::=0
IE
vslues of dese invsiants
aod
N
lrso
(e
dd
lic
16666?./r =
=0.00520811.
@din ddost consta dung all
ii lery Sood aCr4ment
conPurer siDllallots
wilh tlre abovc analylical varues.
Tne rate ot onvergence jn space is calculotcd using thc
bc," (1,.-., - uq
:^,
l/ 1,.-. - ua,
|
iomtla
deflned etrlier as:
)
lo8, (At, /Ai,r )
whe@
aa
r€pEsents exacl solution
Ar, as sprtiat stet size.
Ar
w.
=0.05 fi xed and varyins
md Uq tPescnts lhc numerical sohrions Pitb
cdculaE spatial taL of convcryencc by lcepinS line step
sparial
st p (d r = 0.8.
0
4. 0 2, 0
l)
Frcm $e Table 5 3 w€
qn
s.c thar thc mnvdge@ nls dect!&ss
conctg.nce
I
5.4.2
tu shown in
witl
the snDller spdtial stcP
Fi8 53
e.rctloD of aro solrl.Y
e.ve
Tnc initiol conditio. for intetelion of two positive solit ry waves
,(
siz. Ratt of
d
denned in
..
domain lor tnis proucm is
up
E*i
and Kurtuy
I
t30l wc tate
0<r<80 wnh l=0-2 ald tlE
lo , =@ with tine st P
qnh amplitude I
stac of edergence at l = 35
thc vslues of th.
&=oz
At t =0 thc la4er {ave
dd
w!rc
is ot
!m
pdition r=
15
is
thcn sePar.t hom each oth€r and nove foMard with
m
io Fis.5.5. The
daltrcal valus ot rhtc invsiur.s @ giv.n il E.Fi
l, =
=4?r2388e,I? =
r,
Tl'c splnal
siduladon is run tor
tnc sanc ahaF as b€Ioe collision, Th. Phcno@oa of int tution
'(A,+A,)
is:
t( posirion ,=30 wilh anPlitude 0.5 As ne
noves fasre! rnd it catch!6 the smaller wave Bolh wav.s ae in
and (ne small€r
p6ses rh. lsger wave
l0l
'51?)
pu!ftl.R s v=1,&=0.5,r,=05,i,=15.1=3O /4!=| ar'd,4,=0.5
=0
I
=>4 *c,r(rr lr-r,ll
\.0)
ln ordci ro comFrc ou Bults with
,
I
j(4
Sraphic6llv shown
and
Kurluv
Il
sol
* 4 )= 0.3333331
=i(4*4)=r.a16666?.
FrcE the Tabtc 5.4 we
position a.d amplitude
s
olr Gsuln ae v€ry cl6e lo dalytic values Tte
or both wavs ar tirc r=60 (when rhe process inteEctio. is
ca
dtat
conplete) 6c also shown in T.ble 5,5.
5.4J lrtffacdotr of tb@ soltlst wrv6
Th. itritial condition rd intemi@ ot l{o p6irjv. solit ry wavs give. in I l34l h
/(r.o)=t,4,qr{*,1 r-r,
ln o'der to compuc
p=1,
t=i-,
ou EsuLs with D.cli ll35l, {e ta&
r,=15, r:=30,4:45, r{=1,
rongc fot this Pobl€n is
r=0
(5.r8)
l)
valu6 or rh.
!r=0.5 dd
4=0.25
r:0,35 ard rhe sinulation
Ar=0,1. Al r=0 rhe ldgesl wave
0<J<140 wilh
up to r=200 wirh time slep
tne
A
r
-
15 {/Xh
fr.
.nd
to[d]
rspliad. I .td 6. |0rlln mw i! rt Ftttb! ,
30
vi6 qlitudc
05
.dlbl rlrt ! d Fddo t'a5 rft EC&d.05 T' tn'rtid " ft!'
tr Er56
wtva .t dt6nd tio. Ln{.
i lox!
Ttu
dlbdcd rdu.{ ofth!.lrYri.lt
t\=
tt
r,
=
ivr[ in tr3al
+L+rr>- ss?Et - !(,f+,ti +.lj)=as
ltD
=t(,i'+,{+ri)-Lan&
r. co tc lh o rudb rl! trry clost b 'ldrlic v'\rGr ts
ompd to 11351. It lodn d ddibd! d tvo ffi a ilD t=2m (dd
From
lt
oqldlt
TrtL
tt.
5.5
F6
of
l6rcdd) G rbar i!tlbL
5_?'
!l(l
T.tL 5J: hvuiuls dd .ru mru fn
tinSL
$Ii.ry saw
st I = 20
t-
Irrol
trrll
0r?t
MA
IMA
Ir3ll
Il]rl
T.ble 5.2r lnvdimts od
Ir34l
Ir34l
MA
11341
.d
nom
ior sind.
elibry d difree.t vdue
ot
mplitda A
T.bL 5J: Sp.nd @
ofsv.B.N
for singl.
$lioty w!v.
l.
MA
5.t62t
r71315
IMA
0<r<E0,/4=0.25!rr= l0
Lhre
5.4:
lnwiors fq
two
$li.rry s.w
jncdiont
!.!n413
l.ttt2cx
Il3ll
T!Dl. 5J,
MA
IMA
Posihon
dd anplnld.
of
No w.vB .t
t = 60
LtL
s.6l
I'|wi:n6
for
||ic $ltury sv.t hldqld
tr36l
Dra
tr36l
Trble 5J: Posnion
{d
mplitudd of
$$. *iv.s .t t = 200
MA
Il36l
54,25
tr361
IMQ
I'51
93
tl.Gl'r:S||FF||6tfugu.
FICG5::
siq|.DliE
rE
*tl| dltod rdiun . { r20 (S.l'd liD
.a.lulibdq
R.ta!
tunql"rl dut r!,
it ed
Ctu Ers
5
I
ItN
l3: sr.rid @
oror!l@.
c
."-l
""
q
!
I
-{
i
g
IhE q0.
sdinry {cfun h. sio* oliEy
d r20i(.) MQ,O) oA (c) IMQ
uE
."{
I
I
I
t
.{
tl-r$S*Erbri$rd.
r&rwr|4(4XlOloAG)DaI
95
cloptet 5
1020siaoa)
It@
5.! LErcioo.a
re.ritty ma
'ith
dtu
i.4liuda
ntln It
hcrutlotr
rirh
ot
tlm olirty vrss
dftEd MplifitLo.
5.5
Co[.bdotr
t ta !.d fd
uraL.l .oldi. d uBw otdd. Tb D.tod Crs U|}ry ..@r tlrl! 'itar
IEirrdod of dE lollD.r !.n ilvolvld in & lGW .$rie llE D.d.o of siigb
$lirlry wrv!, i!@lio of |w .rd tbF .liEy r.v.. |.d iltlldt of maid e
itirdrd.EtitutlEbrurt3ffr.dilcqdnerhlt
FDlit d{tat
1Bo, l3r, l3e l33, l3a d latl.ydH! i! lb liElc
b rti. ctrF,
o.tnod
d
lilr
!ad.d dh nditl t dt fto.dG
Chspter 6
Meshless method of lines for solving
nonlineor Schriidiryer
equotion
6.1 Intmduction
The noolin
d
Sctu6din8.r (NLS) equaion
,! -a;,'v',
vh€E i=Jr-, a
a
=
".
-* < I<-,,
@l con$lnt md
als
known
d
cubic S.hro<ling.r eqnal'on
(6
>0.
a(rr)
h
a
t)
conPler valu.d runction which govens
weally nonlinw, strongly dispesiYe and almst nonochrcnaric wave 1136] Ixn
nonlinw oanial difleential .quatioN tPDEs) aPFe in a wide ldietv oI applicauons
(s
l3El). In tanicul&, ir Provid?t a Mthemtiql mod'l ror pltnorcna i' fte
tll7,
of water *dv.s, nonlire( optics, hvdrcnagnetrc and plasma wates'
prcpagation of hear pulss in solids md of Nilhtory wav6 in Piez@l4tric
evoludon
eniconilucros tl39l. Thc a$ltdc slution for initiat
!(r'0)
ranishs foi sufticie.tly lrtg. I I I ws silcn bv zath@!
and
of NtJ
thar
'4!anon
Shlbar I l54i in l9? I using
ioresc saltering D.lhod Fot oo@ Scn tal inidal conditions lheottical solutions or
ehh
rhis equtio. m nol knovn. TneefoF nanv studi6 in orde! to d'al
'umericar
TheF
conditions
solurion of Nt-S hare bc€n m.le fo diffcEnt bound!ry-initial
the
rechniques include rinire elenenr nethod
(FEM) nnnc diffcFne mrhod (FDM)'
sp€.ral rcthods ed collaation rerhod sin8 nditr basis functiotu
I 140_
1531
In order to rind rhe oum€rical solurion of Eq. (61) Ne considct a bonnded inlcryil
or rcal
liN with aninci'l bound{v
condnionlh !
condidons
!(a'4=r{t't)=0
to hodcl 0E phvsical
J0 6 r+l@. kl
(6.2)
u(at)=v(.'ct)+ir(at)
qkrc
(I,4 ed ,('')
dF following coupled
", ="-
arc rcalfunctions. By sub6tiruins Eq
(6.2)inFa (6.1)'wecet
s'6cn ot PDES
+a("'+"')",
(6.3)
\= "_-q(v'+*,)..
The bounddy condtnons
e
rk.r)=v(a.,)=0,
p(r,')= v(r.r=0.
(6.4)
This ch{p&r is org.nizcd is thtce ections. ln sation 62 rhe
explained and
[4,']
sdon
6.3 @ntaim thc numncal
€xmpl6 fo.
nftncd eheft
the
rs
justific{tion or th.
Delhod and wc mnclude in eclion 6 4
6.2 Nudertcal Scbeoe
o( nunocd rchc@ *e lint divi.lc sPatial domain la bl iiro
Out ofrhce Fints {,i=2,3 , N_l arc i'knor points while J
For implemmarior of
nodes
and
\
4.i=1,2,..,,N.
.E
thc bounddy Points The aPprcxim& turction of
v(r) dd w(r)is
deoor.d by
@l dd
indginarv pans
v'(t) ed,'(,) r,r
=>1 (,)r,, (4= v'(').r
"^G)
(6.5)
it (x)->tovlG)=v, (,)l
(6.6)
whd ,l'r.nd t/s G udoown tiift &Pcdlnt qoaniti* a l v/lotu ndid b4is
nEdd lrd Y'(r) is tlE s.e s &6i.d in PFvior cb4r6 w. harc u!.d MQ, cA
ed
IMQ
.!di.l b6is tuDdc il @ nuEicd cx.DPLs.
tar v"(a)=v, and
/
({ )= a,
rhcn in narix-v@lor notatio.
(6.7)
r=[r,
(r),,r" ('),...,
&t)]"
'=r,,(r),,:(,),...,v,(,)f
r =[r, (r), z,(r), -,
r"(r)]',
i=trt).'c(r),....*"t)t'.
,
,/,1,,1v,(\)
Y'(,')
Y'G,)
- e, \',)
v,l v.l4 - v,Q')
v,?N)
Frm
86.
(6.5), (6.6) ard (6.?) it foXoe! dDt
"(')=Y'
'
v',(a) - vtl'NJ
(,)l-'r
' ( r ) = Y I ( r ) ,{ - ' |
wh@
P
{!)- Y'{r),{ '.
(r)r,
(6.8)
=P(r)',
(6.e)
=P
Apptri.s Eq. (6.D on Eq. (6.9) lo @upLd sy.rd! of
(P,,(r,)v)+e(,i
+
PDE$
(6
3), dld
@lla!in8 d
v,:)',
(6.10)
t=,.r.r....,"1
-(P,,(r,)r)- {(ri +,1)r,,
s,tw
',O=',
P.(r)
=
ed 4()= 4
[4,c,)P,,(r,).-P&(,,)]
P"l',\=
*Pto,i, i =1,2....,N
r,,c,)
= [1i,,
P
t,,(',)
Using
=
(r,)
P
;i
v,...
v,1
-
t(, ,r,
th. follo*itrg
v =lv
P,
@llm
(
r, )... P,,, (r,
i -t,2,.
,N
lcctoB mt
lid
]l
for
lbow 3y5Ln' lct
.
w=[',,,-,,]'
[P" (',
)],., .
P- = [P. (,,)],., ,
r- = [r- (r )],", '
P- = [r* (,,)],,. .
P. =
tlE[
Eq. (6.10)
s
b.
wird a follos:
ro!
J*=,t,,t,-0,"'t*',t
lf,= -r"..* r- rtv "
I
(6.tD
w 'rw .
I
4IL= nrv.w,. 4!= c{v.w'.
dt
(6.r2)
dl
ls {v,lY )=(P,,r)+4(v'+w')v
lc (v.l{ )= -(P,,w )- 4(v'+ w')w,
&d injn.l
ooditic tq
dE cdplcd sysEo
w('")=[,"(a), w"(l),..., r,"(r".),
vp";=f"rar.
"'rar....,
I
J
@
"rG,)]',
,'r'"-,r.,"c,rf.
Flom the bound{y condiliou &s.rib.d
it
Eq. (6.4)
oi
Iv, (,)= 0,vr (r)= 0 I
I,,
(,)
= o.
"-
(, ) =
Noe EqN. (6.10).(6.14)
e
(6 13)
(6.
solrtd by RK4
*lm.
r4)
w.,
w.+a,((,,+2lxn+x,rl+4,.)
.,"
dr{x2L
+
2(x):
f::
r+
lj.)
(v,w).
H
.E
+
I
v'+1r,
,w
(6.tt
o[u
).,'=
'tr lv'*!r...w
2|
)",,=
'fi
=c(v"
2"
nIu
2
6.3 Numerical T€$ Problems
mndot.d schctu for nu@ncal eluiion of noolaNe
In fiis sation oe apply lhe above
schrodaneer
noncntun
(Nt-s) equation- Thc
m olcllatcd
by using
losrl
rwo coiefled
qwdtid of m6s dd
ttc f@ulls dcnncd in I l4?l
as:
t)*
The
eol,lic slurioi
of
nd in6
(6.
Schrodingq .qudion {6. | ) is givcn by
/{,,)=dli}'l.xp'[;pr-r(e'-a'Yl]sa1'a(t-p4
This equauon rcF6enls singlc
pl'J'ljmErs
is
d
slhon of mplilude a
chosn fbm lilcnr\rc
-2o < r < m. For a =
s.
q
I enleloF rcliton
=2.
p = 4,
and sPe.d
/
i6
'ftc v.lues of lhe
is Siven by,
I noving lowdds nght
with mmtrnr speed 4, For our nunc.ical cal.ula ons w. usc r = 0.25, 0.3125 and
ed
17)
d = 1.2 Nd sPatial domain
wbich EpEenls notion of a sinlle soliwy wave of anphudc
us
r6)
rin.
lMQ, md oflimal valuc of shap€ PdaDerd for
ttrce RBFS is seteied from lh. int ryals {0.t - 2.5), (l - l0) and i0.5 ' 3.5) t$pecuv.ly
srepAr = 0.0q25. we
MQ, GA
s showD i! Fi&6.1. In Tlblc 6.1 .nor mm rnd |9o coNmd qodtili.s.l r=l .nd
fd a = 1,2 e dF$r ln T.bL 6.2. w. haw slD*n dr Esulli comFrcd by ftsNN
Iie .p?@h &d *irh rhos. plsnr in rlE licnort 1142, 144, l5ll. wc can
$. rhfl dE p|!s.!t ncthod F o.N vcry *.ll with low sprirl F$lulion ..d bc|rer
rssutu e obtlirEd 3 @Dpad with dlid r|ctlods. TIE lDlfic vtlB ol i.vdiett
e: r, =2 ad li =?.33!333 fora=l. Frcm T.blc 6.1 ttd TrbL 6.2 it is clear tht
@rl|od or
slB
inEidt! @ Ery d@ to d.ltiic iolution .nd tl.rivc chd8.
with rlspet ro initi.l ritu is wry slDll by dr Bhles nEllDd. ln Fis.6.3 ql lnd
nutudcd
of lwo
inryin.ry c@poF i $d trrduLs of rrrclling loliioo
Thc point wie rllc of c@!e€c@ in sp.e
rd tirc
!
sp.tial lal. of
=
A
.m
I
is cdcqlct\?d by
Ep6cn$.x*t iolution.rd Ur fll U! aFrscnB th.
Fipccrivcly wilh sp.tal sap
(l'
3hom.
.Id nm
st.p
mwryeM E kep dm
0.5, O25) . Tltc
$ing O.
a"(1,- -u.l/1,-
4.(1,- -u,lil,- -u* l)
wh.E
e
siu
,r
slcp A,
romlla:
-u*,1)
nunErical
slltioN
t .id Ar,. ln orLr !o c.lcolat
=0.qt5 fircd
td
vory spec scp
odcr of coiv.rtEE is shoM in T!bl. 6.3. By dlcrasinS
,,
thlt RBFS b shown in Fig.6.2 by ploonS
tlciproc.l otx(tumbqorcouoca{o poinB) !.n6 nqi!'M.b$l!t .mr. No* td
tir nc of conrrgcrcc c.lcuLlion vG l.cp ,=0.25 {ixcd dd vlry tim stcp
(& =0.@5,0.0025, o0l2i 0'0@625). Fmn Tlblc 6.4 w s tEr tlE Frc or
is Educcd. Thc
@rcrylncc in tim
consltdrc. b.hlvid of
&o!!g
*ith smrlld tim scp.
PmbLn 632
ln
6b plo6l.n ir|.r.dioo
,(,.4
=
i",{r'
of t$o slhdre is sNdicd by
{-p,(+p,
{,-,,)[*,'", (,-,,)
lsiry dE initial @di[on
(6lE)
105
In odct !o coDpdc our rcsults with lbose p€s.nt iD the literalut we ch@se lhe valucs
oi thc pdancten
6r q=2.at=l,A=4.\=-10.d1=lpr=
_4
and&=l0
The
-20<r<20 vith n=0.2, ar = 0 005
tn Table 6.5 dd Table 6,6 the conercd quantitics ad their elarive chdges comput.d
by MQ. GA. tMQ ed ate by thc mthods pacnt in [14], 150, dd l5l) aE 3hown
nuncdcal rsulls
m
conputed over thc domain
Two solitons of equal mplitude tnvelting in oPPo$b dnedon couid. then $Paratc
fom crch olnd pNfring their qidnd sh.F. tlc whole p|@s of inte.4tion is
shoM in Fig.6.4, whlch is iI ag@Enl{ith lh! bchaviorshow. by rn aulhoEin[14],
150. 15l,
l.
ed
1521 bodr
rhd.rically
and
nuncncaly.
rhis example, we show the binh of soli|on fot 89. (6.1) which vas itvestiSated by
Delfour er al. t14?l using a sqwe wcll initial condinor C.rdner er al. [144]
als
studied
lh. eme plEmnenoo using lh. Mdw.Uian initial condnion
!(r.o)=,{qp(
;).
(6.re)
ln N nurcdol
cal€ulations
wc
sld tlrc valu.s of
the para@te6 6:
/'=0.25.dr=0.005 md s=2 with comPutllional domb '45<r<45. Accoding lo
J
lhcn a solibn n genent
,{ = I <
t,
o$eNie
soliton d4avs awav Fo!
binh or a sllnding slilon decaying aw.y wilh ti@ is sho*n jn Fis
G
while forA
dforavaluegt atd $ai
-
qhich is in
1.78 >
tt-
lgEEnt
1.7725. a shnding solrton of amphtode 2
wnn ficorr. Th.
t =A'J:, t,=+A'lr,!z cA')G
.
's
)hoM in
|
Mlytic valks of No Invemb for
'8
6 5,
66
initial
Nunericrl valucs of l*o
invair
s
e
in v€ry good a8t4runt with ard)li€ vdua
a
shown in Table 6.? and Tabl. 6.8-
lo econd sinutation wc afudy binh of mobil.
! (r.0) =
$lito!
usiry lh€ initial condition
A.rp(-i + 2t).
Thc spodal domin for thk
cxF ircnt k -305r<@ litiry
Thc situladon b run up to
tin. | -
A = 1.78
for
q.
s
.nd dr =0-005.
6. A hobilc soliton of @p|nu.L 2 is dcv.loP.d for
shown in Fi8.6.?. Nururical
(6.I ) $c folloeing
/' =0 25
vrlus ol invEirnts @
3hown in Tablc 6.9,
initi.l cdtditid
(6.U0)
!t!t
prcduce ab@nd
q=24'. 4=12
elhm if
of P
.
fi. droctic.l lolutio is troM (s.e Mil.s tl55l). The
slltion is.o{ Bble it 4 >3. w. F.fom M NMicd sibulatron usiig MQ ndial
basis tuictioi in thc tu8c -m<r<m bling /'=0.0625,&=0.001. TlEc ard t@r
Fo.
{
bound sote of solibns
at
dly
d
=3 and t =4 epccdwly
ri@s of sihuhdon. TIE plo$ ol nEdul6 of elution d shown in Fig.6.8 aid
bound solilons of snall
nmw
sElcruEs
dcv.loFd fo.
4
TlEs 6ulll e dc s.m 3 dcvdopd by Clder {1451 ald KottDaz
I l52l-TlE datlic v.h6 of inv&iurs for initid coidition {6.20) e 6 fouoss:
fi9.6.9.
Thc
sl!6
of boii invdimL
@ v.ry closc
!o
rh.@rical
Emin coened
6ult! a
showtr
i.
by our hdhod for 4 =
TabL 6.10.
18 and 4
er al.
= 32 , drd
LtL
Cl: Si!8h $lit . .t t = I
t
MQ
1,ttt33
t.33333
IMQ
|MA
TrhL
al
CoIp6ri$ oldld. slitor !r
t=
| ehh
@!, Fe*
in
lildr@
i.:riiixid
irQ
oll2J
IMQ
0.1125
cDQtl'l
0.3125
-t2136&to,
l -5,a9l2oxlo'
{,flm3
109
T.bk
6l
Rr(
or converg.nc.
'n
,p6e rtr sin8l.
eliM
dt=I
(Ma)
15.7139
0Mq)
T.bL 6a:
(MQ)
OMQ)
Ro& of
ony.r8.M
in
6ft fq singh eliton for N = 160.
T.bh 6& Consrv.d
qdi.i.s fq
t*o $lnos
:'.:')
3199999
T.bl. 66: Consncd qMiiri.s ftr
rwo
slihs .l t = 2 5
.5.41r3x10' -:65so'r0'
tMO
lMA
.r.ol3&4^ro"
cDQt1521
T.bL
6.7:
Bini of EEdinS
l.576rb/roi
$lib
i-I
l.25llll
o367rir6 r.2J3314
0.57157 r 253ll.
0,167037
0167174 l.25r]|,r
0.36?lao
0.167142 1.253314
0.36?109
0.167r{15
Trbl. 6J:
Binh of
ibdins sohor
,Ma
Dra
b.ir'3.e'rrdb
I
ro 39rmo
2.0 39?rdl0
.ae2t6it 3ti0o0 l-aert6r7
4.9256,17 r 9710m 4,925621
-4.925656 3.9?10@ -4.925625
I
r.0 t.971000 -{ 92J65r l.97rrm
-4.925562 3.97100)
'
.{.9256?3
-4
95549
t.o t.97tw1 49u1L5 l..tttu2 49uqn
6.0 1.97r0r3 4921521 t 19@93 j.gt otl
L
.,::::: --:'::::
::::
A= 1,73, Anall4 cv.lues:4= 3.97100, /:=-4 92562.
T.bL
6.9:
Biih of nEbrle sli@, A
MA
T.t'L
t
l;+--t
6.10: Bound sbE of
:
|
73.
--'i i
elution
I
iia
.1r.113333 1{ImD
r!2
tl|E6.r:i+t@d.u.
113
ito
0,5
o1
oo15 o,@ o.@5
Ilcal!
Ra
06
0G6 0.0,t o0a5 0.o5 (t65
ordvd3@ i!:pc.
1L
It|Er 6l! li!C,
O)
&116 ibl n-1, (r)
i= lr(.) lr2r
(rt)
r...
t-
0;
It6
6lr
llbdbr
of
ts.obts,
115
tgG
6J: Sddji8
6Uh
,
A=I
tL1
t|ra.4 rdt-bq
A-
l'r
fla
,l.r
a': ||.rab.ra
A=
l.7l.
119
FlsBt d& Bo&ul.[b
ol&lud4
q
=
l3
121)
choEtE
t rpd 6J: B.md
N
ol
bl{id.
q
- !2
121
6,4 CoDclusiotr
In this wo* we
have appli.d
G!hl.$
nonline$ Schr6dingd .quatiotr. Fou
mcthod
qT.! of
of
lir6
ror numdol slutiotr of
Prcblens includiry Doton of a sin8l.
lolitlly wav.s, Max*elie inilirl @ndition and bound
!t!td of solitoN have hen studicd. Th. nuitric.l 6ult! fc @r |ll]m .nd inwi{tg
of DotioD e .nom in @npdio eith drtid drnods .EilabL i! rh. lLiilr .Id it
h fould ltnt tlc Fcent rlhod tiB !u! .cqrr!r. Bdts i! tI truEicd siEultlions.
solitary wav.. inlcnclion of two
r.l! of cdvergEc in stc &d tn it ds d.eodnE4 In dl NtEiical
cxFdrcna N mdEd F.fot$ vcry s.ll.
TIE
122
Chqpter 7
Future Scenario
Th. resedch pr.senEd in
using radial
bsb
the
disenaion
is f@used on rhe use
furctions to d.velop nututical schem€s
lir.d dm dcFndclr partial difreEntiol equadons
have prcduced encouFeing Nme.ical
resh
td
l€ss mcthod of
the soluion oI
lin€s
no.lirc&
iiv.$i8rdotu cdicd out
al$ iniriarcd a tumb.t of oFn
(PDES). The
Bults but have
reseeh prcblcN. The ue of globdlly supponed RBFS Esults in dcnsc coefficienl
ill condilioned wheo nunber of coU@ation points n increa*d
and rn marix becones n.slvablc if nlmbet of poifts exceds 1000 lhis difficuhv
hiod.s thc ue of RBF n€thods for ld8c eale ptoblems Another probbm n rhat of
matrix which becomes
sbbiliry whilc usinB explicil tine intcgaion $henes These difficultjca can b€ avoided
by lollowing appr@chcs:
.:.
De.ohposing domdn in Io smallfi subdo@ins insrcad of glob.l lbtge donaan.
Only very few tesulrs de av.ilable fo. couPling donain deconposnion lechniquo
wi$ RBF nethods. A very useful work
an
I
can bc done by extendine lhis
ide! 1o ecl
insigh inro rhis technique s4 Dubal 11561and BeaBonll5?l
Ue of inplicil scheGs for timc iilegotion in
od{
to
orercom. lh. problem of
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