JournalofAppliedMathematicsandPhysics,2019,7,3022-3030
https://www.scirp.org/journal/jamp
ISSNOnline:2327-4379
ISSNPrint:2327-4352
ExtrapolationofExplicitDIMSIMsofHigh
OrdertoSolvetheOrdinaryDifferential
Equations
AliJ.Kadhim,AnnieGorgey
MathematicsDepartment,FacultyofScienceandMathematics,UniversitiPendidikanSultanIdris,
TanjongMalim,Perak,Malaysia
Howtocitethispaper:Kadhim,A.J.and
Gorgey,A.(2019)ExtrapolationofExplicit
DIMSIMsofHighOrdertoSolvetheOrdi-
naryDifferentialEquations.JournalofAp-
pliedMathematicsandPhysics,7,3022-3030.
https://doi.org/10.4236/jamp.2019.712212
Received:November6,2019
Accepted:December9,2019
Published:December12,2019
Copyright
c
2019byauthor(s)andScientific
Research PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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sectetuernec,malesuadanec,elit.Aliquam
ornaretelluseuurna.Sednecmetus.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integer metus.Morbi pulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehicula purus.Fusce acnislaliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cumsociisnatoquepenatibus etmagnis
disparturientmontes, nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbi magna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubia nostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretiumnonummyfringilla,tortor lacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumacmetus dictumtempus.Nam ut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aenean egestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunt a,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabitur ullamcorper
ligula nec orci.Aliquam purus turpis,aliquam
id, ornarevitae,porttitornon,wisi.Maecenas
luctusportalorem.Donecvitae ligulaeuante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclaciniaauctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuadaegestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis,felis.Morbinulla nibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinar blandit.Nuncsit amet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel,metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac,mauris.Maecenaset elit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat, wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiamfermentum convallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarcu libero,malesuadavitae, ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurus sedsem
tempor rhoncus.Ut quamdui, fringilla at,dic-
tum eget,ultricies quis,quam.Etiam sem est,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sed scelerisque sus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareantenuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisiultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsumanisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisicommodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisque eratelit, viverrasit amet, sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputateelit.Pellentesque
sem.Praesent eusapien.Duiselitmagna,ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimetcommodo malesuada,tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultriciesaccumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eusapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendisse potenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisse consequat.Mauris a dui
et tortormattispretium.Sed nulla metus,vo-
lutpatid, aliquameget, ullamcorperut, ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Ut blanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretiumultricies.Fusce
temporpellentesqueodio.Vivamus auguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inportacongue,velit leo facilisis odio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,vel posuere nibhlacus nec nibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulum libero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sitameteuismod lectusfelis et
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pede cursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,variuseu,commodosed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis,sed pellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpat duimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquam consectetuervarius nul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velit at elit vehicula porttitor.Phasel-
lusnislmi,sagittisac,pulvinarid,gravida sit
amet,erat.Vestibulumest.Loremipsumdo-
lorsitamet,consectetueradipiscingelit.Cur-
abitur idsem elementum leo rutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreeteget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Mauris enimsem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisque lacus.Donec etipsumsit
amet elitnonummy aliquet.Sed viverranisl at
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendisse potenti.Sedidlec-
tus sitamet purusfaucibus vehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac,nunc.Maurismalesuadamalesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin,ultriciesid,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplatea dictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristiquequis,felis.Vestibulumante ip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris porta dui,vitaealiquamarcu turpisac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellusaliquam, tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementum dui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu, euismodac, tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna,blanditnec, iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnonarcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbiipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuadavestibu-
lum.Quisquesollicitudinipsumvel sem.Nul-
laenim.Proinnonummyfelisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum, justoat dignissim
auctor,wisi odiolobortis arcu,sedsollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nulla facilisi.Quisque
posuerebibendumpurus.Nulla quammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitur sollicitudinhen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in,dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper, dui ut
dignissim auctor, orciliberofringilla massa, b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tincidunt fermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netuset malesuadafames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulum dapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectus wisi,elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipitsitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringillatempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitant morbi tristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnasedenim.Donec egeterosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulum ante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semperat, metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputatepurus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,nonporttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vellectusut me-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatis vitae,vulputate a,pharetrasit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid,mi.Nullamnecnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nunc nulla pulvinarjusto,aeuismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus,eusodalesmagnarisusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integer purus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursusarcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donec feliselit,dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et, commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectusneque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In eget risusluctus wisiconva-
llis pulvinar.Vivamus sapien risus,tempor in,
viverra in, aliquetpellentesque, eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisquedignissim,lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligulasuscipiturna,vitaeportaodioer-
at quissapien.Proin luctusleoid erat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursussed,
justo.Phasellus nec lorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fusce vehicula, tortoret gravida porttitor, me-
tusnibhconguelorem,uttempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit,imperdietvitae,faucibus nec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbi fringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisseornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit,gravida ac,pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehiculaat, conguesit amet,
sem.Sedsagittispedeanisl.Sedtincidunt o-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nullaettellusnon felisimperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallis hendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsitametnibhnecpurusdignissimmat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpis diam feugiatar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulum elit.Nam ligulaarcu, volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisusatmetuspulvinarmalesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumanteipsumprimisinfaucibusorci
luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nunc bibendum augueluctus
diam.Etiam nec sem.Sed eros turpis, facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajustovehicula diam,eucondimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitametest.Vivamus pulvinarleonon
sapien.Aliquameratvolutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet,mattis suscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibh ullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamus loremdui,commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisque vestibulum pulvinar di-
am.Phasellustincidunt,leovitaetristiquefa-
cilisis,ipsumwisi interdum sem,dapibussem-
pernulla velit vellectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
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condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
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eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim,sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitae dolorsollicitudin adipiscing.
Crasaneque.Utrisusdui,interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsum primisin faucibusorciluctus etultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere,augue erosporttitor lec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
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ametneque.Sedcommodomalesuadalectus.
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ttacitisociosquadlitoratorquentperconu-
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pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
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ipsumdolorsitamet,consectetueradipiscing
elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
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lis.Vivamuspulvinar.Aliquameratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sit ametmauris.Cras commodo, urna ut port-
titor venenatis, arcu metussodales risus, vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultrices felis erat at lectus.In est massa, male-
suadanon,suscipitat,ullamcorpereu,elit.
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tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibhpretiumtellus,atdignissimleo eros
egetpede.Nullavenenatiseleifenderos.Ae-
nean ut odio dignissim augue rutrum faucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamus vehicula leoajusto.Quisquenecau-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenimsitametrisus.Curabiturvitaevelit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeunulla ut nulla interdum consectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
Thepurposeofthisresearchistoinvestigatetheefficiencyofex-
plicitdiagonallyimplicitmulti-stageintegrationmethodswithex-
trapolation.Theauthorgavedetailedexplanationofexplicitdi-
agonallyimplicitmulti-stage integration methodandcomparedthe
basemethodwithatechniqueknownasextrapolationtoimprove
the efficiency.Extrapolation forsymmetric Runge-Kuttamethodis
proventoimprovetheaccuracysincewithextrapolationthesolu-
tions exhibitasymptoticerror expansion, however for Generallinear
methods, itisnotknown whether extrapolation canimprove the ef-
ficiencyornot.Thereforethisresearchfocusesonthenumerical
experimentalresultsoftheexplicitdiagonallyimplicitmultistage
integrationwithandwithoutextrapolationforsolvingsomeordi-
narydifferentialequations.Thenumericalresultsshowedthatthe
basemethodwithextrapolationismoreefficientthanthemethod
withoutextrapolation.
Keywords
ExtrapolationTechnique,GeneralLinearMethods,Diagonally
ImplicitMuti-StageIntegration,ExplicitMethods
1.Introduction
Although general linear methods (GLMs) havebeen introduced about
40yearsago,manyresearchershavenotconsideredGLMsasprac-
ticalnumericalmethods.Themostcomplicatedforthesemethods
isidentifyingthepracticalmethods.Forthisreason,itisnecessary
tointroduceasubclassofGLMsknownasdiagonallyimplicitmuti-
stage integration, abbreviated as (DIMSIMs) [1].DIMSIMs have been
proventobeadvantageousthanRKmethodsandlinearmulti-step
methods.DIMSIMsalsohavebeenconsideredverypotentialforeffi-
cientimplementations.
DOI:10.4236/jamp.2019.712212Dec.12,20193022JournalofAppliedMathematicsandPhysics
A.J.Kadhim,A.Gorgey
Considerthefollowinginitial-valueproblems
y
0
(x) = f(y(x)),x[x
0
,X]
y(x
0
) = y
0
(1)
Generallinearmethods(GLMs)isgivenby
Y
i
=
s
X
j=1
a
ij
hf(Y
j
)+
r
X
j=1
u
ij
y
[n1]
j
,i= 1,2,···,s,
y
[n]
i
=
s
X
j=1
b
ij
hf(Y
j
)+
r
X
j=1
v
ij
y
[n1]
j
,i= 1,2,···,r.
(2)
whereY
i
canbeapproximatedasfollows
Y
i
= y(x
n1
+c
i
h)+O(h
q+1
),(3)
TheincomingandoutgoingquantitiesofGLMscanalsoapproxi-
matesto
y
[n1]
i
=
p
X
k=0
α
ik
y
(k)
(x
n1
)h
k
+O(h
p+1
),
y
[n]
i
=
p
X
k=0
α
ik
y
(k)
(x
n
)h
k
+O(h
p+1
).
(4)
The subclassofGLMs which isDIMSIMs hasthe leadingcoeffcient
matrix,A,thatcanhavefourdifferentstructuresknownastypes,
dependingonwhethertheintendedapplicationisnon-stifforstiff
and onwhetherthe computerarchitecture issequential orparallel[2].
DIMSIMscanbedividedintofourtypessuchasfollows:
TypeI.
ThecoefficientmatrixAof thistype consideredaslower triangu-
lar matrixand itsdiagonals equalto zero.The typeone required
tosolvethenon-stiffproblemonasequentialcomputer.
TypeII.
ThecoefficientmatrixAofthistypeconsideredaslowertrian-
gularmatrixanditsdiagonalsequaltoconstant.Thetypetwo
requiredtosolvethestiffproblemonasequentialcomputer.
TypeIII.
The coefficient matrix Agiven for this type as zero matrix.This
typerequiredfornon-stiffprobleminaparallelenvironment.
TypeIV.
ThecoefficientmatrixAisassumedinthistypeasdiagonal
matrix.Thistyperequiredforstiffprobleminaparallelenvi-
ronment.
TheclassofGLMsincludemanyspecialmethods,asgivenin[2]
and[3].Inparticular,thesemethodshavesubclassofDIMSIMscon-
sideredin[1]andfurtherinvestigatedin[4],[5]and[6],two-stepRK
methodsconsideredin[7]andfurtherinvestigatedin[8]and[9].The
DIMSIMsparticularlyhadbeeninvestigatedbyWillWrightinhis
PhDthesis[10].DIMSIMshavealsobeendiscussedinamonograph
givenbyJackiewicz[11],HairerandWanner[12]andareviewpaper
byButcher[13].
DOI:10.4236/jamp.2019.7122123023JournalofAppliedMathematicsandPhysics
A.J.Kadhim,A.Gorgey
Intherecentyears,ithasbeenchallengingtoconstructaneffi-
cientmethodsthataremoreaccurateandmoreefficient thanoriginal
methods.In[14],hedescribedtheconstructionofalgebraicallysta-
bleDIMSIMsbyintroducinganewideasuchasalgebraicstability
andstudyitsconsequences.In[15],theyintroducedanewclassof
implicit-explicitDIMSIMs,wherethenon-stiffpartistreatedbyan
explicit formula and the stiff part is treated by an implicit formula.In
thesameyear,FamelisandJackiewicz[16]gavethenewconstruction
ofDIMSIMsbasedonDifferentialEvolution.Inthispaper,thead-
vantagesofusingtheextrapolationtechniquewithexplicitDIMSIMs
methodsaregivenforsomenon-stiffproblems.
2.OrderConditions
Theorder conditionsof DIMSIMscanbe consideredbyusing thesteps
between x
n1
andx
n
withstep-sizeh.Theoutputapproximations of
DIMSIMsisassumedtohavethesameorder.Theorderpandstage
orderqareconsideredtobethesame.
RecalltheformofGLMsforasinglestep
Y
[n]
= Ahf
Y
[n]
+Uy
[n1]
y
[n]
= Bhf
Y
[n]
+Vy
[n1]
.
(5)
InordertosatisfytheorderconditionofDIMSIMs,itisnecessary
toconsiderthefollowing
exp(cz) = zAexp(cz)+UZ+O
h
p+1
exp(z)Z= zBexp(cz)+VZ+O
h
p+1
,
(6)
where
exp(cz) =
exp(c
1
z)
exp(c
2
z)
.
.
.
exp(c
s
z)
,Z=
1
z
.
.
.
z
p
.
(7)
3.ConstructionofTypeIDIMSIMs
ExplicitsolvershaveupdatedthetypeIDIMSIMswhichhasbeen
consideredfirstlyby[1].Themethodsconsideredarewithconditions
p= q= r= s.The stability functionduringthesemethods considered
by
p(ω,z) = ω
s1
(ωR(z)),(8)
where
R(z) = 1+z+
z
2
2!
+···+
z
s
s!
,(9)
whichisapproximatetotheexponentialfunctionexp(z).Thismeth-
odshavesimilarstabilityastheRKmethodsoforders.Byfollow-
ing[4],thestabilityfunctionisgivenby
p(ω,z) = ω
s
p
1
(z)ω
s1
+···+(1)
s1
p
s1
(z)ω+(1)
s
p
s
(z)ω,
(10)
DOI:10.4236/jamp.2019.7122123024JournalofAppliedMathematicsandPhysics
A.J.Kadhim,A.Gorgey
where
p
1
(z) = 1+p
11
z+p
12
z
2
+···+p
1s
z
s
,
p
2
(z) = p
21
z+p
22
z
2
+···+p
2s
z
s
,
.
.
.
p
s1
(z) = p
s1,s2
z
s2
+p
s1,s1
z
s1
+p
s1,s
z
s
,
p
s
(z) = p
s,s1
z
s1
+p
ss
z
s
,
wherep
i,j
arecoefficientsofpolynomialp
i
(z)relyingona
ij
,i =
1,2,···,s,j=1,2,···,i1andv
i
,i=1,2,···,,s1.Bysolving
theformof(s1)(s+2)/2fornon-linearproblems
p
kl
= 0,k= 2,3,···,s,l= k1,k,···,s,(11)
withrespecttothevalue(s1)(s+ 2)/2.ThecoefficientmatrixB
canbeobtainedbyusingthefollowingcondition:
B= B
0
AB
1
VB
2
+VA,(12)
Thecoefficientsp
ij
ontheotherhandcanbecomputedbyusing
theFourierseriesapproach.
AFourierseriesapproachcanbesummarizedasfollows.Consider
thatw
µ
µ= 1,2,...,N
1
,arecomplexnumbersspanneduniformlyon
theunitcircle,whereN
1
denotesasufficientlylargeinteger.Multi-
plyingthefollowingrelation
p(w
µ
,z) = w
s
µ
p
1
(z)w
s1
µ
+···+(1)
s1
p
s1
(z)w
µ
+(1)
s
p
s
(z)
(13)
byω
ks
µ
,k=1,2,···,s,andusingthesummationwithrespecttoµ,
then
(1)
k
p
k
(z) =
1
N
1
N1
X
µ=1
ω
ks
µ
p(ω
µ
,z).(14)
Inthesameway,multiplying
p
k
(z
v
) = p
k,k1
z
k1
v
+p
kk
z
k
v
+···+p
ks
z
s
v
,k= 2,3,···,s,(15)
byz
1
v
,andusingthesummationwithrespecttov,then
p
kl
=
1
N2
N
2
X
v=1
z
1
v
p
k
(z
v
),
= (1)
k
1
N
1
N
2
N
1
X
µ=1
N
2
X
v=1
ω
ks
µ
z
1
v
p(ω
µ,z
v
).
Thissystemcanbenumericallysolvedas
N
1
X
µ=1
N
2
X
v=1
ω
ks
µ
z
1
v
p(ω
µ,z
v
) = 0,(16)
Therefore, thepolynomial oftypeIDIMSIMscanbeobtainedfrom
thefollowingequation
p(ω,z) = det(Q(ω,z)),(17)
where
Q(ω,z) = ω(IzA)VzB
0
+zAB
1
+zVB
2
.(18)
andthedeterminantdetisascalarvaluethatcanbecomputedfrom
theelementsofasquarematrixandencodescertainpropertiesofthe
lineartransformation.
DOI:10.4236/jamp.2019.7122123025JournalofAppliedMathematicsandPhysics
A.J.Kadhim,A.Gorgey
4.ExtrapolationTechnique
Extrapolationtechniqueisapowerfulcomputationaltechniquethat
canbesuccessfullyappliedintheeffortstoimprovetheaccuracyof
theapproximatesolutionsinsolvingordinarydifferentialequations.
ExtrapolationwasfirstbeenintroducedbyRichardsonin1911.It
isanapproximationmethodinthenumericalsolutionofdifferential
equations.Richardsonshowedthattheapproximationsarisingfrom
finitedifferenceschemecanbeexpressedintheformof
T(h) = τ
0
+τ
2
h
2
+τ
4
h
4
+...+O
h
8
,(19)
wherehisstepsize,T(0)=τ
0
isanadjustableparameterandτ
2
4
areindependentofh.
TheextrapolationformulabytheAitken-Nevilleformulaisgiven
by
T
ij
= T
i,j1
+
T
i,j1
T
i1,j1
m
1
m
ij+1
1
,m= 1,2,...(20)
wherei= j= 2,...,n.
Therearetwomodesofextrapolationthatcanbeappliedinnu-
mericalapproach.Theyarepassiveandactiveextrapolations.When
extrapolatedvaluesarenotusedinanysubsequentcomputationof
otherextrapolatedvalues,theprocessiscalledpassiveextrapolation
whereasiftheextrapolatedvaluesareusedattheendofastepas
thestartingvalueforthenextstep,itiscalledactiveextrapolation.
Sincetheextrapolationcanincreasetheaccuracyandefficiencyof
the approximate solutions, many researchers during these recent years
arestillfindinganddevelopingthebestwaystousethistechnique.
Gorgey [17] showed that passive extrapolation by the two stage Gauss
methodismoreefficientthantheactiveextrapolationinsolvingthe
linearandnonlinearstiffproblems.Besides,thepaperin[18]ex-
pressedthecomputingtimespentbytheRichardsonextrapolation
forbothtypesactiveandpassiveismorethantentimessmallerthan
thecorrespondingcomputingtimebytheBackwardEulerFormula.
Byfollowingtheresearch byMona,LagziandHavasiasgivenin[18],
itisconcludedthatextrapolationisapowerfultoolforincreasingthe
accuracyandefficiencywithregardtothecomputationalcostespe-
ciallywhentheaccuracyrequirementisnotextremelylow.In[19],it
is shown that a new procedure to buildstabilized explicit RKmethods
basedonRichardsonextrapolationwithhigherorderisstudied.The
strategywaseasyandallowsderivingstabilizedexplicitRKmethods
withorderashighasdesired.However,forloworder,thestability
propertywerealittleshorterthanothermethodswithoutthistech-
nique.
In[20],theadvantagesofusingextrapolationaregivenasfollows:
Itispossibletousetheextrapolationtechniqueinordertoim-
provetheaccuracyofthenumericalexperiments.
Theactivetypeofextrapolationtechniquecombiningwith
Trapezoidalrolecanmakethecomputationsbecomesunstable.
Thecomputationalcostofextrapolationtechniqueislessthan
theunderlyingnumericalmethodsforprescribingtheaccuracy.
The application withapplying the Backward Euler formula with
activeactivetypeofextrapolationisleadingtoastableproce-
dure.
DOI:10.4236/jamp.2019.7122123026JournalofAppliedMathematicsandPhysics
A.J.Kadhim,A.Gorgey
Inthispaper,passive extrapolation technique isappliedtoimprove
theaccuracyoftheapproximatesolutionsbytheexplicitDIMSIMs
methods of higher order in solving non-stiffproblems.Inthe next sec-
tion,wewillconsiderthenumericalexperiments ofexplicitDIMSIMs
withextrapolationtechnique.
5.NumericalResults
Inthissection,wediscusstheresultsofnumericalexperimentsob-
tainedbytheMATLABcodedim18-extrap.m.Thiscodeisbased
onexplicitDIMSIMswithextrapolationtechniqueofhigherorder
p
max
=8..Tocompare,wealsopresenttheresultsobtainedbythe
MATLABcodedim18.m.ThiscodeisbasedonexplicitDIMSIM-
swithoutextrapolationofhighorderp
max
=8,asbeingdescribed
in[21].
Thedecisionaboutdeterminingtheneworderwithin1p8
relayingonthefollowingform
ratio=
kest(x
n
,p
n
)k
kest(x
n
,p
n
1)k
(21)
where
est(x
n
,p
n
1) = C(p
n
1)z
[n]
(p
n
m+1 : (p
n
+1)m)(22)
Nowifthepreviousstepwasnotrejectedandratio<r
min
,p
n
<
p
max
,theneworderwillbechooseasp
n+1
=p
n
+1.Besides,ifthe
ratio>r
max
andp
n
>1,thentheneworderinthiscaseleadsto
p
n+1
=p
n
1.Bycontractwiththiscase,theorderwillbenot
changed.Duringtheabovecodesdim18-extrap.manddim18.m,the
parameterschosenasr
min
=0.9,r
max
=1.1andthemaximaloreris
p
max
= 8.
Theresultswillshowwhichoneofthesemethodsisefficientin
solving non-stiff problems such as Van der Pol (VDP) and Brusselator
(BRUS) problems.The first test problem is Van derPol (VDP) which
isdefinedasfollows:
y
0
1
= y
2
,
y
0
2
= ((1y1
2
)y
2
y
1
)/,
(23)
wherethefirstequationdenotesthenon-stiffequationwhilethesecond
equationdenotesstiffequationwithincludingsmall.Inorderto
solvethenon-stiffproblem,thenconsideredhereisequaltoone.
TheinitialvaluesofVDParegivenasfollows
y
1
(0) = 2,
y
2
(0) =
2
3
+
10
81
292
2187
2
1814
19683
3
+O(
4
).
(24)
Theproblemisintegratedtox
n
= 3.
ThesecondtestproblemisBrusselator(BRUS)whichisanauto
catalyticoscillatingchemicalreactionproblem.Itisasystemoftwo
ordinarydifferentialequationsthataregivenby
y
0
1
= 1+y
2
1
y
2
4y
1
,
y
0
2
= 3y
1
y
2
1
y
2
.
(25)
where y
1
(0) = 1.5 andy
2
(0) = 3.The problemisintegrated to x
n
= 1
byusingh= 0.05.
DOI:10.4236/jamp.2019.7122123027JournalofAppliedMathematicsandPhysics
A.J.Kadhim,A.Gorgey
Forthetwoproblems,fourfiguresaregiven.Thesefiguresgive
thenumericalresultsofstepsizeandCPU-timeversusglobalerrors
fororder8DIMSIMswithandwithoutextrapolationtechnique.All
thenumericalcomputationalareperformedbyusingMATLAB.The
CPU-time ismeasured by applying ticand tocas given in MATLAB.
Thestartingabsoluteandrelativetolerancesaretol= 10
7
.
Figure1givesthenumericalresults forVDP testproblem.Fromthe
figure,wecanseetheextrapolationtechniquewithexplicitDIMSIMs
oforder-8givesbetterefficiencythanexplicitDIMSIMswithoutex-
trapolation.Furthermore, thenumericalresults given inFigure 2show
thattheextrapolationtechniquewithexplicitDIMSIMsoforder-8 also
givesgreaterefficiencythanexplicitDIMSIMswithoutextrapolation
forBRUSStestproblem.
Figure1.NumericalresultsforVDPproblembyorder-8DIMSIMwith
andwithoutextrapolation.
Figure2.NumericalresultsforBRUSproblembyorder-8DIMSIMswith
andwithoutextrapolation.
6.Conclusions
Inthispaper, wediscusssomeissues relatedthe developmentofexplic-
itDIMSIMsforthenumericalsolutionofnon-stiffdifferentialequa-
tions.Thispaperalsogivestheapplicationofpassiveextrapolation
bytheorder-8explicitDIMSIMs.Aswecanseefromnumericalre-
sults,theextrapolationtechniquewithexplicitDIMSIMsoforder-8
givesbetterefficiencythanexplicitDIMSIMswithoutextrapolation
insolvingVDPandBRUStestproblems.
Futureworkwill involve verifying theefficiency ofimplicit DIMSIMs
ofhigherorderwithextrapolation insolvingstiffdifferentialequations.
DOI:10.4236/jamp.2019.7122123028JournalofAppliedMathematicsandPhysics
A.J.Kadhim,A.Gorgey
Acknowledgements
TheauthorswouldliketoextendtheirgratitudetotheMinistryof
Higher Educationin Malaysia forproviding theresearch grant FRGS,
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ofthispaper.
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