Regionalization Method for Nonlinear Differential Equation Systems In a Cartesian Plan

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1 Journl of Mthemtcs nd Sttstcs 4: ISSN Scence Pulctons Regonlzton Method for Nonlner Dfferentl Euton Systems In Crtesn Pln Bel Lekhmss Deprtment of Mthemtcs Fculty of Scences Unversty of Orn Es-Sen Alger Astrct: We propose regonlzton technue for nlyzng nonlner dfferentl euton systems where coeffcents re stndrd nd nonzero. The present work strts wth the study of nturl oject whch ws the mgc sures provdng us wth new wy to prtton the pln n regons. Key words: Internl Set Theory Regonlzton nonlner dfferentl eutons opertor INTRODUCTION In the present work we study the effects of delnerty of ox type n ortl geometry nd n ortl dynmcs. We strt ths work wth frst nturl oject round whch we orgnze our pln nmely the mgc sure MS gvng the prtton of the pln R nto 5 externl sets depcted n the followng heurstc dgrm Also t plys n essentl role n the nonclssc prtton regonlzton s followng R G U A _ U P U A U G of R. Here re some of the uestons tht cn e sked The pprton of the mgc sure The ehvour of the orts n the regons of the MS The trnston of the orts of regon hve the other The nture of the sngulr plce The strut wth the lner cse. The ojects: We re nterested y non-stndrd systems of nonlner dfferentl eutons n R provded wth crtesn coordntes. ff where the rels > > 6 the regon defned y the condtons re stndrd nd nonzero nfntely gret Corondng Author: Bel Lekhmss Déprtement de Mthémtues Fculté des Scences Unversté D Orn 464 [ f nd for [ f p we proceed to shrp delnerzton " of ox type delnerzton " of dfferentl system wth constnt coeffcents Wth conservng the lner eutons. Then the prolem s to evlut the effects. We cn lso see prolem of trnsent to the lmt In some fmles whch re lner when. The dfferentl system s dvded nto three fmles :F F F 3 F : m F : m d ; d F 3 : rng The technue: we use the technue of regonlzton to get some predctons reltve to the mcroscopc ehvour of orts nd the dynmcs long the orts. Some predctons. The mgc sure MS n the frst nlyss the pln R s prttoned n 5 externl regons where we denote y 5 3 the regon defned y the condtons pp - nd ff ff nd

2 J. Mth. & Stt. 4: nd ff nd ff 9 7 the regon defned y the condtons pp nd ff pp nd pp - the regon defned y the condtons pp - nd ff nd 4 8 the regon defned y the condtons - nd nd 5 the regon defned y the condtons pp nd pp nd the regon defned y the condtons pp - nd pp ff nd pp 5 the regon defned y the condtons - nd pp nd pp 5 the regon defned y the condtons pp - nd pp nd pp 3 the regon defned y the condtons pp nd pp 6 4 the regon defned y the condtons pp - nd ff nd 8 the regon defned y the condtons - nd nd In 3 the mcroscopc ehvour s determned y the rectlgne system In 9 7 the mcroscopc ehvour s determned y rectlgne system n 9 7 the mcroscopc ehvour s determned y the rectlgne system The refned mgc sure: to know the ort ehvour n the other regons We must refne the mgc sure MS y ntroducng the extensons of the regons ectvely y the followng condtons 3 : [ [ [ [ P nd P [ 9 : [ G nd [ [ [ 7 : G nd [ [ 7 : [ G nd [ [ 9 : [ G nd [ The fmly F : the functon F m s frst ntegrl. The non-sngulr orts re rectlgne nd wth the sme slope m the sngulr poston s gven y the euton [ [ wth the prme ntegrl F m dfferentl system of the fmly F nduce fmly wth one rel prmeter C C specfy the level strght of dfferentl euton of order nr. Nmely [ [ D C : m C the form of sngulr plce of F mplue some numer of furctons n the eutons D c when C follow R. The fmly F [ [ F [ [ m d d The sngulr plce s empty If d ws null we otn system of F wth sngulr emptness n the cse to dd d hunts the sngulr plce ut t sty wtness tht crete rver phenomen. The fmly F 3 : we dstngush three cses j. the vectors re prwse ndependent. jj. ; Proposton : In regon 3 of mgc sure MS the mcroscopc ehvour of orts s determned y the rectlgne system 465

3 J. Mth. & Stt. 4: Let D denote the dervton opertor s regrd nd stte the followng result : Lemm : In the regon 3 of the mgc sure MS the feld of vectors [ [ [ [ Y D D ssocted to the system s nfntely ner to the rectlgne feld vectors Y 3 D D Proof of lemm : consder the regon 3 defned y the condtons pp nd pp Or the feld vectors: [ [ [ [ Y D D Assocted to the system [ [ pp nd pp nd f f s re rel nfntude gret [ [ nd re nfntude smll. conseuently the feld of vector Y n the coordntes s nfntely ner to the rectlgne feld Y 3 D D Proof of proposton : By the lemm of short shdows the orts of hve n the regon 3 the sme hlo s the system orts Proposton : In the regons 9 nd 7 9 nd 7 of the mgc sure MS the mcroscopc of the orts ehvour hs nfntely smll fluctutons ner s determned y the rectlgne system Lemm : In the regons 9 nd 7 9 nd 7 of the mgc sure MS the feld of vectors [ [ Y [ D [ D [ [ Y [ D D [ nd Y Y s nfntely ner of the rectlgne feld of vectors Y D D Y D D Proof of lemm : In the regons 9 nd 7 9 nd 7 of the mgc sure MS [ [ s nfntely smll [ [ s nfntely gret snce f f re nfntely gret [ [ Tkng s fctor we cn wrte the feld of vectors : [ [ [ [ Y D D Under the form [ [ [ Y [ D [ D [ [ [ Y [ D [ D In the regons 9 nd 7 9 nd 7 of the mgc sure the feld Y hs the sme orts s the feld [ [ Z [ D [ D. [ [ Z [ D [ D Z The feld Z s nfntely ner of the rectlgne feld Z D D Z D. D s the unttes [ [ [ nd [ re nfntely smll. [ [ [ [ Proof of proposton : The proposton s n mmedte conseuence of the lemm of the short shdows s long s we hve lemm. Exmnton of the fmly F3: F 3 : rng [ [ [ [ Y D D Assocted to the system s nfntely ner of the rectlgne feld of vectors 466 We note. we cn wrte

4 J. Mth. & Stt. 4: β ; β f [ [ β [ [ β β f. the sngulr pont s reduced to the pont S β 3. the sngulr pont cn only e fnd n the hlo of corner the unt sure. 4. the slow regon s gven y β other hl nd hl β 5. f the slow regon s strctly contned n the hlo of sgn- sgn- β. f β the slow regon s gven y P nd sgn β β the slow regon s gven y. f P nd hl 6. the sngulr pont s n the externl sure ES [ [ defned y L L [ [ Then n the regon G curves C nd C dont ntersect. G the * the functon F β s frst ntegrl of system * The sngulr plce of the system s defned y the [ [ euton. f p ff ff then [ [ [ p [ s pprected nd we hve f p f Corollre : sngulr pont of the system tken n the regon defned y ff ff And n the regon A defned y the condtons [ nd [ A U A Proof of proposton 3. As p s not sngulr we hve: [ [ P [ [ P f p pp pp then the rel [ [ [ [ p nd p nd re nfntely smll. where the followng reltons nd p nd p As nd nd pprected we deduce the reltons [ [ p re. Proposton 3 sngulr pont of the system Gven. f p pp pp then: * p *. f p p pp pp then * β ut The pont 467 nd nd re stndrd. thus. the relton [ [ show p tht p sn ce s pprected

5 J. Mth. & Stt. 4: the otned relton n β β mplue the eulty conseuently hence the eulty [ [ [ [ P P As p we otn β β or s β β β Whch mply the relton So the functon F β s frst ntegrl of the system β we see tht s [ [ [ [ β conseuently the sngulr plce of the system s defned y the euton The pont If p ff ff the eulty [ [ p cn lso e wrtten [ [ p [ [ [ [ p p [ [ [ [ Show tht p cn not e nfntely smll or nfntely gret ecuse s stndrd nonnull nd [ [ nfntely smll. [ [ Thus [ p [ s pprected. [ [ s p nfntely gret then t must e [ [ the sme s p such tht the uotent s pprected. [ [ f p s nfntely gret then ff p ff. REFERENCES. Arnold 974. Eutons Dfférentelle. Edton MIR. Moscou.. Boo S Régonlstons et Vn-der pol épss. Chers mthémtues d Orn fsccule n. 3. Boo S Méthode des régonlstons. Quelues pplctons I.R.M.A Strsourg. 4. Cllot J.L. 98. Bfurcton du portrt de phse pour des éutons dfférentelles du second ordre Thèse Strsourg. 468