Impulsive Differential Equations. by using the Euler Method

Transcription

1 Applid Mahmaical Scincs Vol. 4 1 no Impulsiv Diffrnial Equaions by using h Eulr Mhod Nor Shamsidah B Amir Hamzah 1 Musafa bin Mama J. Kaviumar L Siaw Chong 4 and Noor ani B Ahmad Dparmn of Mahmaics Faculy of Scinc and Tchnology Univrsii Malaysia Trngganu 1 Kuala Trngganu Trngganu Malaysia mus@um.du.my Musafa bin Mama 4 Dparmn of Scinc and Mahmaics Faculy of Scinc Ars and Hriag Univrsii Tun Hussin Onn Malaysia 864 Pari Raja Johor Malaysia Absrac Th hory of impulsiv diffrnial quaions is mrging as an imporan ara of invsigaion sinc such quaions appar o rprsn a naural framwor for mahmaical modling of svral ral phnomna. Thr hav bn innsiv sudis on h qualiaiv bhavior of soluions of h impulsiv diffrnial quaions. Howvr many impulsiv diffrnial quaions canno b solvd analyically or hir solving is complicad. In his papr w rprsn h algorihm which follows h hory of impulsiv diffrnial quaions o solv h impulsiv diffrnial quaions by using h Eulr mhods. I is clarly shown h impulsiv opraors I ha acs a h momns influnc h rror. Finally h br convrgnc rsul of h numrical soluion is givn by solving h numrical ampls. Kywords: Diffrnial Equaions Impulsiv Diffrnial Equaions Fid impuls Impulsiv jump Eulr Mhod 1. Inroducion Many voluion procsss ar characrizd by h fac ha a crain momns of im hy princ a chang of sa abruply. This is du o shor rm prurbaions whos duraion is ngligibl in comparison wih h duraion of h

2 Nor Shamsidah B Amir Hamzah al procss. I is assum naurally ha hos prurbaions ac insananously in h form of impulss. Thus impulsiv diffrnial quaions by mans diffrnial quaions involving impuls ffcs ar sn as a naural dscripion of obsrvd voluion phnomnon of svral ral world problms. For ampl mchanical sysm wih impac biological phnomnon involving hrsholds bursing rhyhm modls in mdicin and biology opimal conrol modls in conomics pharmacoinics and indusrial roboics and many mor do hibi impulsiv ffcs. Thrfor i is bnficial o sudy h hory of impulsiv diffrnial quaions as a wll dsrvd disciplin du o h incras applicaions of impulsiv diffrnial quaions in various filds in h fuur. Th pionr paprs in his hory ar wrin by A. D. Myshis and V. D. Mil man in 196 s [9]. In spi of is imporanc many soluions rgardd o impulsiv diffrnial quaions ar don analyically. Som of h famous rsarchrs who prsnd significanc rsuls ar V. Lashmianham D. Bainov P. Simonov and many ohrs [ ]. Howvr many impulsiv diffrnial quaions canno b solvd analyically or if don hir solving is vry much complicad [11]. Thrfor numrical soluions of impulsiv diffrnial quaions has o b sudid and h rsuls has o b improvd. In his papr h numrical soluions of impulsiv diffrnial quaions ar sough by using h Eulr mhod. Th algorihm proposd is inrprd according o h hory of impulsiv diffrnial quaions wrin by V. Lashmianham. al [8]. Basd on h hory h br numrical soluion of h problm is illusrad in h ampls.. Impulsiv Diffrnial Equaions Basically impulsiv diffrnial quaions consis of hr componns. A coninuous-im diffrnial quaion which govrns h sa of h sysm bwn impulss an impuls quaion which modls an impulsiv jump dfind by a jump funcion a h insan an impuls occurs and a jump cririon which dfins a s of jump vns. Mahmaically h quaion as h form ' f Δ I Z 1... m.1 n n whr Z is any ral inrval f : Z R R is a givn funcion n n I : R R 1... m and Δ 1... m. Th numbrs ar calld insans or momns of impuls I rprsns h jump of sa a ach whras and rprsn h righ limi and h lf limi rspcivly of h sa a. Th momns of impuls mayb fid or dpndd

3 Impulsiv diffrnial quaions 1 on h sa of h sysm. In his papr w will b concrnd wih fid momns only. Morovr impulsiv diffrnial quaions can b classifid according o hs hr componns. 1. Sysms wih impuls a fid momns. Th quaions hav h following form ' f. Δ I whr < 1 <... < < 1 <... Z and for Δ whr lim h. W surly s ha any soluion of. saisfis h i ' f 1 and ii Δ I Sysms wih impuls a variabl ims. Th quaions hav h following form ' f τ. Δ I τ 1... whr τ Ω R Ω is h phas spac and τ < τ 1 Z Ω. Sysms wih variabl momns of impulsiv ffc involv mor difficul problms han sysms wih fid momns of impulsiv ffc. This is du o h fac ha h momns of impulsiv ffc of. dpnd on h soluion i.. τ for ach. Thrfor soluions a diffrn saring poins will hav diffrn poins of disconinuiy.. Auonomous sysms wih impuls. Th quaions a h form ' f Δ I M M.4 L h ss M M N N and h opraor A A b indpndn of and l A:M N b dfind by A I whr I: Ω Ω. Whnvr any soluion

4 Nor Shamsidah B Amir Hamzah al his h s M a som im h opraor A insanly ransfrs h poin M ino h poin y I N. Gnrally h soluions of h impulsiv diffrnial quaions ar picwis coninuous funcions wih poins of disconinuiy a h momns of h impuls ffc. In his papr w dno S { : Z} R whr < 1 for all Z whn and whn. If Ω R is any ral inrval w suppos ha [ 1 T... ] n is a vcor of unnown funcions and f : Ω R n R n f n f 1... n f f n 1... n n is coninuous funcion on vry s [ 1 R. ] Dfiniion.1 A sysm of diffrnial quaion of h form d d f.5 wih condiions Δ I n n whr I : R R ar coninuous opraors ± 1 ±... is calld impulsiv diffrnial quaion IDE a fid impuls.

5 Impulsiv diffrnial quaions. Propris of Soluions of IDEs Th problm of isnc and uniqunss of h soluions of impulsiv diffrnial quaions is similar o ha of corrsponding ordinary diffrnial quaions. Th coninuabiliy of soluions is affcd by h naur of h impulsiv acion. Dfiniion.1 A soluion of h IDE.5 mans a picwis coninuous picwis coninuous firs drivaiv such ha d 1. f τ d : J R wih. τ τ I ± 1 ±... τ Thorm.1 n L h funcion f : R Ω R b coninuous on h ss τ τ 1] Ω Z and for ach Z and Ω suppos hr iss h fini limi of f as y τ > τ. Thn for ach R Ω hr n iss T > and a soluion : T R of h problm.5 wih iniial condiion. Furhrmor if h funcion f is locally Lipschiz coninuous wih rspc o in R Ω hn his soluion is uniqu. L b h soluion of IDE.5 wih iniial condiion hn can b rprsnd as f s s ds f s s ds < < < < I I Ω Ω whr Ω and Ω ar h maimal inrvals on which h soluion can b coninud o h righ or o h lf of h poin rspcivly.

6 4 Nor Shamsidah B Amir Hamzah al Thorm. n E n Assum ha f C[ I E ] and saisfis d [ f u f v] Ld[ u v] L > for n u v I E. Thn h iniial valu problm.5 has a uniqu soluion u u u on I. W also nd h following nown [8] impulsiv diffrnial inqualiis rsul. For his purpos w l PC dno h class of picwis coninuous funcions from R o R wih disconinuous of h firs ind only a ; 1... W can now sa h ndd rsuls. Thorm. Assum ha A h squnc } saisfis < < < K < < Kwih as ; { 1 A 1 m PC' [ R R] and m is lf coninuous a 1... A 1 Kand D m g m m ψ m.1 m w whr g : R R is coninuous in 1 ] R and for ach w R lim g z g w iss and ψ R R : is non-dcrasing; z w A r r w is h maimal soluion of ising on [. Thn w g w w ψ w. w w m r. W rcall ha h maimal soluion r of. mans h following

7 Impulsiv diffrnial quaions 5 r w [ 1] r1 1 r 1 1 ] M r r r 1 1].4 M M whr ach ri i ri 1 i is h maimal soluion of. on h inrval i i 1] for ach i 1 K and r ψ r r. i 1 i i i1 i i1 i i1 4. Algorihm Suppos h IDE.5 wih sar condiion and h impulsiv opraors I Z is givn. Th impulsiv opraors ac a h momns of jump happn for all Z which ar dscribd by h quadra marics of dimnsions n n. Th numrical algorihm is diffrn only a h jump poin whr w hav o apply h opraors concrn wih h paricular poin. Ohr han ha w mploy h usual mannrs o solv h IDE using h numrical mhod chosn. 1. A h momn w apply h numrical mhod o h funcion wih h iniial valus. Th algorihm applis unil h firs jump poin by now w will g h valus for h lf limi.. A h jump poin w apply h opraors o find h valus of h righ limi.. Th firs sp is rpad unil h n jump poin. 4. Thn w apply h opraors concrn wih h paricular jump poin. 5. Th abov sps ar rpad and h iraion sop unil w ncounr wih h dsird valus ha has o b found l assum s whr s >. Noic ha w only hav h approima valus of h funcion a s. 5. Numrical Eampls Eampl 1 Considr h IDE givn in [] :

8 6 Nor Shamsidah B Amir Hamzah al d d f and. Δ I f Th impulsiv opraors ac a 1. 1 and. ar givn as follows: I 1 I Hr w wish o approima h valu of s.. W applid h algorihm by using h Eulr mhod i 1 i hf i 5. whr i Z is h ind of iraion and h is h sp siz of ach iraion. Hr h sp siz h.1. Thn w compard h rsuls obaind by using h analyical prssion ha is h soluion of IDE [ [ [ [ Th numrical valus of soluion ar obaind by using h Malab programming and h rsuls of h Eulr mhod as wll as h analyical prssion ar compard in Tabl 1. Absolu rrors ar also givn by aing h approima numrical valus of s. L assum s..

9 Impulsiv diffrnial quaions 7 Tabl 1 : Th approima valus a rror a s Eulr Analyical Eulr Analyical

10 8 Nor Shamsidah B Amir Hamzah al Figur 1 Th approima valus of 1 vrsus im bwn h ulr and analyical mhod for Eampl Figur Th approima valus of vrsus im bwn h ulr and analyical mhod for Eampl 1

11 Impulsiv diffrnial quaions 9 Eampl Considr h IDE f d d... 1 Δ I 5.5 and f A I. Hr w wish o drmin h approima valu a 1. For ha purpos w apply h Eulr mhod 5.. Thn w compard h rsuls obaind by using h analyical prssion ha is h soluion of IDE ] ]

12 Nor Shamsidah B Amir Hamzah al Tabl : Th approima valus a Eulr Analyical Eulr Analyical Figur Th approima valus of 1 vrsus im bwn h ulr and analyical mhod for Eampl

13 Impulsiv diffrnial quaions Figur 4 Th approima valus of vrsus im bwn h ulr and analyical mhod for Eampl Hr h impulsiv opraors I ha acs a h momns also influnc h rror. This is du o our calculaion of h approima valus of h jumps only. 6. Concluding rmars Th accuracy of h rsuls can b improvd by invsigaing h soluions of h ohr numrical mhods. W proposd a gnral numrical procdur for raing h impulsiv diffrnial quaions a fid momns. W inrprd h numrical algorihm following h hory of impulsiv diffrnial quaions and sard wih h Eulr mhod. Alhough i is no h mos accura mhods w will sudy i is by far h simpls and analyzing Eulr s mhod in dail will hopfully carris ovr o h ohr mhods wih highr accuracy wihou a lo of difficuly. Solving h impulsiv diffrnial quaions numrically has no bn don by many rsarchrs. Thrfor many sudis hav o b don in ordr o nhanc and vrify h ising rsuls. In his papr w hav shown br rsuls wih diagrams o h convrgnc and h bhavior of h soluions. Acnowldgmns. Th auhors would li o han h Minisry of Highr Educaion Malaysia for supporing his rsarch undr h Fundamnal Rsarch Gran Schm FRGS.

14 Nor Shamsidah B Amir Hamzah al Rfrncs [1] A. M. Samailno N.A. Prsyu Impulsiv Diffrnial Equaions World Scinific Singapor [] B. M. Randlovic L. V. Sfanovic B. M. Danovic Numrical soluion of impulsiv diffrnial quaions Faca Univ. Sr Mah. Inform [] D. D. Bainov G. Kulv Applicaion of Lyapunov s funcions o h invsigaion of global sabiliy of soluions of sysms wih impulss Appl. Anal [4] D. D. Bainov P.S. Simnov Sysms wih Impuls Effc Sabiliy Thory and Applicaions. Ellis Horwood Limid Chichsr [5] D. D. Bainov S. I. Kosadinov N. van Minh P. P. Zabrio A opological classificaion of diffrnial quaions wih impuls ffc Tamang J. Mah [6] G. Kulv D. D. Bainov On h sabiliy of sysms wih impulsiv by dirc mhod of Lyapunov J. Mah. Anal. Appl [7] M. U. Ahmov A. Zafr Succssiv approimaion mhod for quasilinar impulsiv diffrnial quaions wih conrol Appl. Mah. L [8] S. Jianhua Nw maimum principls for firs ordr impulsiv boundary valu problms Appl. Mah. L [9] V. D.Mil man A. D. Myshis On h sabiliy of moion in h prsnc of impulss Sib. Mah J [1] V. Lashmianham D. D. Bainov P. S. Simonov Thory of Impulsiv Diffrnial Equaions World Scinific Singapor Nw Jrsy London Hong Kong [11] X. J. Ran M. Z. Liu Q. Y. Zhu Numrical mhods for impulsiv diffrnial quaion Mahmaical and Compur Modlling Rcivd: Jun 1