International Journal of Pure and Applied Sciences and Technology
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1 It. J. Pure Al. Sci. Tecol Iteratioal Joural o Pure ad Alied Scieces ad Tecoloy ISSN 9-67 Aailable olie at Researc Paer Few Numerical Aloritms or Miimizatio Ucostraied No iear Fuctios K. Kartikeya * ad V. M. Cadrasekara Scool o Adaced Scieces VIT Uiersity Vellore-64 Idia. *Corresodi autor kkartikeya67@yaoo.co.i Receied: 7-8- ; Acceted: -- Abstract: I tis aer we roose ew umerical aloritms or miimizatio o ucostraied o-liear uctios by usi Modiied omotoy erturbatio metod ad te aoter ew aloritm based o Modiied Adomia decomositio metod. Te comaratie study amo te see ew aloritms ad Newto s aloritm is establised by meas o eamles. Keywords: No liear uctios Newto s metod Modiied omotoy Perturbatio Metod Modiied Adomia decomositio metod.. Itroductio Otimizatio roblems wit or witout costraits arise i arious ields suc as sciece eieeri ecoomics maaemet scieces etc. were umerical iormatio is rocessed. I recet times may roblems i busiess situatios ad eieeri desis ae bee modeled as a otimizatio roblem or taki otimal decisios. I act umerical otimizatio teciques ae made dee i to almost all braces o eieeri ad matematics. Seeral metods [578] are aailable or soli ucostraied miimizatio roblems. Tese metods ca be classiied i to two cateories as o radiet ad radiet metods. Te o radiet metods require oly te objectie uctio alues but ot te deriaties o te uctio i idi miimum. Te radiet metods require i additio to te uctio alues te irst ad i some cases te secod deriaties o te objectie uctio. Sice more iormatio about te uctio bei miimized is used trou te use o deriaties radiet metods are eerally more eiciet ta o radiet metods. All te ucostraied miimizatio metods are iteratie i ature ad ece tey start rom a iitial trial solutio ad roceed towards te miimum oit i a sequetial maer.
2 It. J. Pure Al. Sci. Tecol omotoy teciques were alied to id all roots o a o liear equatios[ 4] ad te omotoy teory becomes a owerul matematical tool we it is successully couled wit erturbatio teory [ ]. Sice te beii o te 98 s te Adomia decomositio metod as bee alied to a wide class uctioal equatios [4 5]. Adomia ies te solutio as a iiite series usually coeri to a accurate solutio. Abbaoui et. al. [] alied to te stadard Adomia decomositio o simle iteratio metod to sole te equatio were is a oliear uctio ad roed te coerece o te series solutio. Babolia et. al. [6] modiied te stadard Adomia metod wic roosed i []. Recetly S.Abbasbady itroduced [] modiied omotoy erturbatio metod or o liear equatios ad comariso wit Adomia decomositio metod. Viay Kawar et al. [9] itroduced ew aloritms called eteral touc tecique ad ortooal itersectio tecique or soli te o liear equatios. I tis aer we itroduce ew ew aloritms or miimizatio o o liear uctios by usi Modiied omotoy erturbatio metod ad comared wit Modiied Adomia decomositio metod. Te comaratie study is establised amo te see ew aloritms wit Newto s aloritm by meas o eamles.. New Aloritms Cosider te oliear otimizatio roblem: Miimize { R : R R } were is a o-liear twice dieretiable uctio. Cosider te uctio G were. ere is te uctio to be miimized. G is deied aroud te critical oit * o i ad is ie by G. I we assume tat we ae G i. Cosider te equatio wose oe or more roots are to be oud. y reresets te ra o te uctio ad assume tat a iitial estimate is kow or te desired root o te equatio. Cosider te equatio were α be a root o it ad is a C uctio o a iteral cotaii α ad we suose tat α >. omotoy teciques were alied to id all te roots o. [4]. Also te omotoy erturbatio metod or soli. was alied i [7 9 ]. I tis aer we will use te omotoy erturbatio metod o aoter equatio wic ca be aroimately te solutio o.. By usi te Taylor s easio ear o For a small we ae wic ies Taylor easio o to a ier order is O ad we are looki or suc as
3 It. J. Pure Al. Sci. Tecol or A N c were c is a costat is a liear uctio ad N is o liear uctio ad or aroimati we ca aly te omotoy erturbatio metod. By te omotoy tecique[ 4] a omotoy R R ] [ wic satisies c c A ] [ ] [ ] [ c N were is a imbeddi arameter is a iitial aroimatio o. is costructed. ece we ae c A ad te cai rocess o rom to is just tat o rom to A c. I tooloy tis is called deormatio ad A c are called omotoic. Alyi te erturbatio tecique [6] due to te act tat ca be cosidered as small arameter we ca assume tat te solutio o. or.4 ca be eressed as a series i we teds to. or.4 corresod to. or.5 becomes te aroimatio o..... lim series.6 is coeret or most cases ad also te rate o coeret deeds o A. [8]. Substituti.5 i to.4 ad equati te terms wit idetical owers o we ae : : : F c N were F. We set. ece we ae / / F F
4 It. J. Pure Al. Sci. Tecol et m m... deotes te m term aroimatio o. Sice te series coeres ery raidly te sum m i m i ca sere as a ractical solutio i eac iteratio. For m c α wic is te Newto s metod ad te same as Adomia decomositio metod or m []. For m α Aloritm wic is te ouseolder s iteratio [] ad te same as Adomia decomositio metod or m [ ] New Aloritm - I Sice te aloritm. becomes For m
5 It. J. Pure Al. Sci. Tecol α ad te same as Adomia decomositio metod or m [] Aloritm New Aloritm - II Sice te aloritm. becomes i Order Iteratios Uder some coditios o reularity o ad its deriatie we obtai ood teciques. For eamle i we cosider 6 4 O Te or m ad or m Aloritm as te same Adomia decomositio metod []. New Aloritm - III Sice te Aloritm. becomes
6 It. J. Pure Al. Sci. Tecol Coerece Aalysis o New Aloritms I II III Te coerece aalysis o New Aloritm-I.8 is same as te coerece aalysis o ouseolder s iteratio [] ad te same as Adomia decomositio metod [] or m te coerece aalysis o New Aloritm-II. is same as Adomia decomositio metod [] or m ad coerece aalysis o New Aloritm-III. is same as Adomia decomositio metod [] or m i te i order iteratios.. Numerical Illustratios Eamle.: Cosider te uctio 5 R. Te miimizi oit o te uctio is equal to wic is obtaied i iteratios by Newto metod i iteratios by New aloritm I New aloritm-ii ad New Aloritm-III or te iitial alue ad also see te ariatios i iteratios or te iitial alue o ad. Iterati Newto s New Aloritm- I New Aloritm-II New Aloritm-III os Aloritm Iterati Newto s New Aloritm-I New Aloritm-II New Aloritm-III os Aloritm Iterati Newto s New Aloritm-I New Aloritm-II New Aloritm-III os Aloritm Eamle.: Cosider te uctio R. Te miimizi oit o te uctio is equal to.6996 wic is obtaied i 4 iteratios by Newto s Aloritm
7 It. J. Pure Al. Sci. Tecol i iteratios by New Aloritm-I New aloritm-ii ad New Aloritm-III or te iitial alue ad also see te ariatio o iteratios or te iitial alue o ad. Iteratios Newto s New Aloritm-I New Aloritm-II New Aloritm-III aloritm Iteratios Newto s New Aloritm-I New Aloritm-II New Aloritm-III aloritm Iteratios Newto s New Aloritm-I New Aloritm-II New Aloritm-III aloritm Eamle.: Cosider te uctio e R. Te miimizi oit o te uctio is equal to - wic is obtaied i 7 iteratios by Newto s aloritm 5 iteratios by New aloritm I i iteratios by New aloritm II i 5 iteratios by New Aloritm III or te iitial alue ad also see te ariatios i iteratio or te iitial alue o ad. Iteratios Newto s New Aloritm- I New Aloritm-II New Aloritm-III aloritm Iteratios Newto s New Aloritm-I New Aloritm-II New Aloritm-III aloritm....
8 It. J. Pure Al. Sci. Tecol Iteratios Newto s New Aloritm-I New Aloritm-II New Aloritm-III aloritm Coclusio I tis aer we ae itroduced ew umerical aloritms amely New Aloritm I New Aloritm II New Aloritm III. From te aboe illustratios it is clear tat te rate o coerece o tese ew aloritms is aster ta Newto s Aloritm. I real lie roblems te ariables caot be cose arbitrarily rater tey ae to satisy certai seciied coditios called costraits. Suc roblems are kow as costraied otimizatio roblems. I ear uture we ae a la to eted te roosed ew aloritms to costraied otimizatio roblems. Reereces [] K. Abbaoui ad Y. Cerruault Coerece o adomia s metod alied to o liear equatios Mat. Comut. Model [] S. Abbasbady Imroi ewto-raso metod or o liear equatios by modiied adomia decomositio metod Al. Mat. Comut [] S. Abbasbady Modiied omotoy erturbatio metod or oliear equatios ad comariso wit adomia decomositio metod Al. Mat. Comut [4] G. Adomia Soli Frotier Problems o Pysics: Te Decomositio Metod Kluwer Academic Publisers Dordrect 994. [5] G. Adomia ad R. Rac O te solutios o alebraic equatios by te decomositio metod Mat. Aal. Al [6] E. Babolia ad J. Biazar Solutio o oliear equatios by modiied adomia decomositio metod Al. Mat. Comut [7] J.. e Newto-like iteratio metod or soli alebraic equatios Commu. Noliear Sci. Numer. Simul
9 It. J. Pure Al. Sci. Tecol [8] J.. e omotoy erturbatio tecique Comut. Metods Al. Mec. E. 78/ [9] J.. e A couli metod o omotoy tecique ad erturbatio tecique or oliear roblems It. J. Noliear Mec [] J.. e Comariso o omotoy erturbatio metod ad omotoy aalysis metod Al. Mat. Comut [] J.. e Te omotoy erturbatio metod or oliear oscillators wit discotiuities Al. Mat. Comut [] A.S. ouseolder Te Numerical Treatmet o a Sile Noliear Equatio McGraw ill New york 97. [] S.J. iao Boudary elemet metod or eeral oliear dieretial oerators E. Aal. Boudary Elemet [4] S.J. iao A aroimate solutio tecique ot deedi o small arameters: a secial eamle It. J. No iear Mec [5] C. Moa Josi ad M. Kaa Moudalya Otimizatio Teory ad Practice Narosa Publicatio ouse New Deli 4. [6] A.. Naye Problems i Perturbatio J. Wiley New York 985. [7] S.S. Rao Eieeri Otimizatio Teory ad Practice New Ae Iteratioal imited New Deli 996. [8] G.V. Reklaitis A. Raidra ad K.M. Rasdell Eieeri Otimizatio Metods ad Alicatios Jo Wiley ad Sos New York 98. [9] Viay Kawar Sukjit Si J.R. Sarma ad Mamta New umerical teciques or soli o-liear equatios Idia Joural o Pure ad Alied Matematics [] Bezad Gabari Medi Golami Porsokoui ad Bija Raimi New iteratie metod o solutio o te model o zabotiski-belouso reactio ad comariso wit rue kutta metod It. J. Pure Al. Sci. Tecol
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