ScienceDirect. Behavior of Integral Curves of the Quasilinear Second Order Differential Equations. Alma Omerspahic *

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1 Avalable onlne a wwwscencedeccom ScenceDec oceda Engneeng h DAAAM Inenaonal Smposum on Inellgen Manufacung and Auomaon Behavo of Inegal Cuves of he uaslnea Second Ode Dffeenal Equaons Alma Omespahc * Unves of Saajevo acul of Mechancal Engneeng n Saajevo Vlsonovo sealse 9 7 Saajevo Bosna and ezegovna Absac Ths pape deals wh cean classes of Cauch s soluons of quaslnea second ode dffeenal equaons n geneal fom Van de ol's dffeenal equaon whch s used n he heo of elecc ccus and agesom's dffeenal equaons whch s used n asmpoc eamen of vscous flow pas a sold a low Renolds numbe Behavou of negal cuves n he neghbouhoods of an aba o negal cuve s consdeed Obaned esuls esablsh suffcen condons fo he esence and asmpoc behavou of he obseved equaons The obaned esuls conan he answe o he queson on appomaon of soluons whose esence s esablshed The eos of he appomaon ae defned b funcons ha can be suffcenl small The qualave analss heo and opologcal eacon mehods wee used 4 The Auhos ublshed b Elseve d 4 The Auhos ublshed b Elseve d Open access unde CC BY-NC-ND lcense Selecon and pee-evew unde esponsbl of DAAAM Inenaonal Venna Selecon and pee-evew unde esponsbl of DAAAM Inenaonal Venna Kewods: quaslnea dffeenal equaon; behavo of soluons; appomaon of soluons Inoducon Man pocesses n scence and echnque ae descbed wh he quaslnea dffeenal equaons whose soluon s no alwas possble o fnd [5] Mehods of qualave analss of dffeenal equaons allow o deemne he esence and asmpoc behavou of soluons of hese equaons he sabl of he soluon and f s possble o appomael deemne he equesed soluon [][4] o eample snce noduced n he 95s b A agesom he model of agesom s equaon whee suded b man auhos wh he help of vaaon echnques [][9] The auhos opovc and Szmolan used n [9] geomecal appoach ee we shall use he qualave analss heo of dffeenal equaons and opologcal eacon mehod [][6] [7] [8] [] [] [] * Coespondng auho Tel: ; fa: E-mal addess: almaomespahc@mefunsaba The Auhos ublshed b Elseve d Open access unde CC BY-NC-ND lcense Selecon and pee-evew unde esponsbl of DAAAM Inenaonal Venna do: 6/jpoeng46

2 Alma Omespahc / oceda Engneeng Noaon fϵci f connuous funcon on neval I=a fϵc I f and f' connuous funcons on neval I fϵc I f f' and f" connuous funcons on neval I ψ =ѱ ϵ I = ϵ I ' =' ϵ I S p I p ϵ {} class of soluons defned on I whch depends on p paamees S I ess a leas one soluons defned on I = pschz s consans The behavou of he soluons of quaslnea second ode dffeenal equaon n geneal fom whee connuous funcon on R I I=a a ϵ R n he neghbouhood of an aba cuve ae consdeed As specal cases we consde agesom s equaon and Van de ol s equaon [5] e R I I whee φϵc I s an aba cuve n R I e ϵ C I > > on I and le he soluons of equaons sasf on I ehe he condons o Usng subsuon = whee = s a new unknown funcon equaon s ansfomed no a quaslnea ssem of equaons 4 e whee be an aba negal cuve of ssem 4 and le Ω=R I We shall consde he behavou of he negal cuve of ssem 4 wh espec o he ses: :

3 854 Alma Omespahc / oceda Engneeng The bounda sufaces of σ and ω ae especvel e us denoe he angen veco feld o an negal cuve φ ψ of ssem 4 b T T The vecos and ae he oue nomals on sufaces and especvel: B means of scala poducs: T T T on sufaces and especvel we esablsh he esence and behavou of negal cuves of 4 wh espec and ω The esuls of hs pape ae based on he followng emmas see [6]-[8] emma If fo he ssem 4 he scala poduc hen he ssem 4 has a class of soluons belongng o he se ω fo all ϵi e emma If fo he ssem 4 he scala poduc hen he ssem 4 has a leas one soluonon I whose gaph belongs o he se ω fo all ϵi e emma If fo he ssem 4 he scala poduc and o evesel hen he ssem 4 has a class of soluons belongng o he se fo all e Accodng o emma he se s a se of pons of sc enance of negal cuves of he

4 Alma Omespahc / oceda Engneeng ssem 4 wh espec o he ses ω σ and Ω ence all soluons of ssem 4 whch sasf condon also sasf condons - - fo eve e Accodng o emma he se s a se of pons of sc e of negal cuves of he ssem 4 wh espec o he ses ω σ and Ω ence accodng o TWazewsk s eacon mehod [] ssem 4 has a leas one soluon belongng o se ω σ fo e Accodng o emma he se s a se of pons of sc enance and s a se of sc e o evesel of negal cuves of 4 wh espec o he ses and Ω ence accodng o eacon mehod ssem 4 has a one-paamee class of soluons belongng o se σ fo e The man esuls Theoem e ϵr I sasf he condons: 5 whee ϵr I and ϵc I > > Then: a If he condons 6 7 ae sasfed on hen all soluons of he poblem sasf he condons 8 > b If he condons 9 ae sasfed on hen a leas one soluons of he poblem sasfes he condons 8 c If he condons 6 and o 7 and 9 ae sasfed on hen he poblem has one-paamee class of soluons ha sasf he condons 8 Theoem e ϵr I and le he condons 5 be sasfed e ϵc I > > and

5 856 Alma Omespahc / oceda Engneeng Then: If > hen all soluons of he poblem sasf he condon > If 4 hen a leas one soluon of he poblem sasfes he condon oof of Theoem We shall consde he equaons hough he equvalen ssem 4 e us consde he negal cuves of he ssem 4 wh espec o he se σ o he scala poduc on and on we have: a Accodng o he condons 5 and 9 he followng esmaes fo on and on ae vald especvel: Accodngl se s a se of pons of sc enance of negal cuves of he ssem 4 wh espec o he ses σ and Ω ence all soluons of he ssem 4 whch sasf he condons also sasf condons > Snce n vew of = -φ = -ѱ all soluons of he poblem sasf he condons 8

6 857 Alma Omespahc / oceda Engneeng b Accodng o he condons 5 9 and he followng esmaes fo on and on ae vald especvel: > > Accodngl se s a se of pons of sc e of negal cuves of he ssem 4 wh espec o he ses σ and ence accodng o T Wazewsk s eacon mehod [] he ssem 4 has a leas one soluons belongng o he se σ fo all Consequenl he poblem has a leas one soluon whch sasfes he condons 8 c In hs case s a se of pon of sc e and s a se of pons of sc enance o evesel of negal cuves of he ssem 4 wh espec o he ses σ and Accodng o he eacon mehod he ssem 4 has one-paamee class of soluons belongng o he se σ fo all ence he poblem also has one-paamee class of soluons whch sasf he condons 8 oof of Theoem e us consde he negal cuves of he ssem 4 wh espec o he se ω o he scala poduc on he suface we have: If we noduce he noaon Y we have: Y Y In vew of 5 he followng esmaes fo ae vald: Y Y Y Y The gh-hand sdes of he above nequales ae he quadac smmec foms

7 858 Alma Omespahc / oceda Engneeng a a Y ay whee coespondng coeffcens a a a ae noduced Condons and 4 mpl a a a - a > whch accodng o Slvese s ceon means ha Consequenl se s a se of pons of sc enance of negal cuves of ssem 4 wh espec o he ses ω and ence all soluons of he ssem 4 whch sasf he condons 5 sasf he nequal > 6 Snce hen all soluons of he poblem sasf condon Condons 4 mpl a > a a - a > whch accodng o Slvese s ceon means ha Consequenl s a se of pons of sc e of negal cuves of he ssem 4 wh espec o he ses ω and Ω ence accodng o he eacon mehod he poblem 4 5 has a leas one soluon whch sasfes condon 6 Consequenl he poblem has a leas one soluon whch sasfes condon The applcaons Van de ol equaon o he Van de ol s equaon [] : and condon > 7 8 we can pove he followng: ln

8 Alma Omespahc / oceda Engneeng If funcon Φ> hen all soluons of he poblem 7 8 sasf he condon ln fo ϵ > Ths esul follows fom Theoem wh = =ln agesom s equaon In geneal fom agesom s equaon s gven b he non-auonomous second ode dffeenal equaon: n 9 n N n The cases n= and n= epesen he phscall elevan sengs of flow n wo and hee dmensons especvel o he agesom s equaon we can pove he followng: e Г be an aba cuve and ϵc IR + a If n n on hen all soluons of he poblem 9 sasf he condons b If > n n > 4 on hen a leas one soluon of he poblem 7 sasf he condons c If condons 4 o ae sasfed hen he poblem 7 has a one-paamee class of soluons ha sasf he condons 4 Concluson Ths pape deals wh esence and behavou of negal cuves of second ode quaslnea dffeenal equaons n geneal fom As specal cases Van de ol's dffeenal equaon and agesom's dffeenal equaons ae

9 86 Alma Omespahc / oceda Engneeng consdeed The obaned esuls esablsh suffcen condons fo he esence and asmpoc behavou of he obseved equaons n neghbouhoods of an aba o negal cuve n defnon doman Also n hs pape s pesened a opologcal eacon mehod as a ve useful mehod of qualave analss of dffeenal equaons The esuls also conan an answe o he queson on appomaon of soluons whose esence s esablshed o eample he eos of appomaon fo soluons and devave n Theoem ae defned b he funcon and whch end o zeo as as and < = ϵ I o eample we can use = αe -s and = βe -p s > p > and wh paamees α and β ha can be aba small In ha case cuve Г epesens a good appomaon of soluons n σ The obaned esuls also gve he possbl o dscuss he sabl nsabl of soluons of he ssem 4 o eample unde he condons of Theoem a eve soluons of 4 wh nal value n ω s -sable sable wh he funcons of sabl = f ends o zeo as and < = ϵ I oweve f we consde he case b hen s esablshed soluons n ω s -unsable n case whee > ϵ I The ne sep would be a numecal smulaon of soluons n obseved domans and compason wh obaned esuls Refeences [] Benede I Mguell Zecca Esence esuls fo genealzed vaaonal nequales va opologcal mehod Topologcal mehods n nonlnea analss Vol 9 No pp 7-56 [] Cohen DS okasa agesom A oof of some asmpoc esuls fo a model equaon fo low Renolds flow SIAM J ApplMah pp 87-7 [] alo R Bahuguna D ande D Esence and unqueness of soluons fo quas-lnea dffeenal equaons wh devang agumens Elecon Jounal of Dffeenal Equaons Vol No pp - [4] aman Odna Dffeenal equaons Wle New Jok 98 [5] Mnosk N Nonlnea Oscllacons publshed b D Van Nosand Compan Canada 96 [6] Omespahc A Reacon mehod n he qualave analss of he soluons of he quaslnea second ode dffeenal equaons In: Appled Mahemacs and Compuaon Depamen of Mahemacs Unves of Zageb Zageb Coaa pp 65-7 [7] Omespahc A Vdoljak B On paamee classes of soluons of a ssem of quaslnea dffeenal equaons oceedngs of he Confeence on Appled Mahemacs and Scenfc Compung Bjun eded b ZDmac MMausc and ZTuek Spnge Dodeh 5 pp 6-7 [8] Omespahc A Esence and behavo soluons of a ssem of quaslnea dffeenal equaons Ceave Mahemacs & Infomacs Vol 7 No 8 pp [9] opovc N Szmolan A geomc analss of he agesom model poblem J Dffeenal equaons 99 4 pp 9-5 [] Vdoljak B Omespahc A ualave analss of some soluons of quaslnea ssem of dffeenal equaons In: Appled Mahemacs and Scenfc Compung eded b MDmac Va Sopa ZTuek KVeselc Kluwe Academc/lenum ublshes New Jok Boson DodehondonMoscow pp - [] Vdoljak B Omespahc A Esence and appomaon of soluons of a ssem of dffeenal equaons of Volea pe Mahemacal Communcaons Vol 9 No 4 pp 5-9 [] Wazewsk T On a opologcal pncple of eamnng he asmpoc shape of he negals of odna dffeenal equaons Ann Soc olon Mah 947 pp 79-