An Application of He's Variational Iteration Method for Solving Duffing - Van Der Pol Equation

Transcription

1 Raf. J. of Comp. & Mah. Vol. No. 3 A Applicaio of He' Variaioal Ieraio Mehod for Solvig Dffig - Va Der Pol Eqaio A J. Al-Saoor Mera A. Aziz College of Comper Sciece ad Mahemaic Uiveriy of Mol Received o: 3// Acceped o: 9/4/3 Dffig-Vav الملخص تم في هذا البحث تطبیق طریقتین ال VIM و ال ADM لا یجاد الحل التقریبي لمعادلة.Der Pol في طریقة ال VIM تعتمد دالة التصحیح بصورة عامة على مضروب لاكرانج الذي یتم إیجاده باستخدام نظریة التغایر. المقارنة العددیة للحلول المتمثلة بشكل متسلسلة تمت مقارنتها مع طریقة RK45 القیاسیة وقد أثبتت طریقة ال VIM كفاءتها في حل هذه النوعیة من المعادلات. هذا البحث أیضا تضمن برهان التقارب لطریقة VIM المستخدمة لحل معادلة.Dffig Va Der Pol ABSTRACT I hi paper e apply He' variaioal ieraio mehod VIM ad he Adomia decompoiio mehod ADM o approximae he olio of Dffig-Va Der Pol eqaio DVP. I VIM a correcio fcioal i corced by a geeral Lagrage mliplier hich ca be ideified via a variaioal heory. The VIM yield a approximae olio i he form of a qickly coverge erie. Compario of he o erie olio ih he claical Rge-Ka order for RK45 mehod ho ha he VIM i a poerfl mehod for he olio of oliear eqaio. The coverge of He' variaioal ieraio mehod o hi eqaio i alo coidered.. Irodcio Chaoic yem have received a flrry of reearch effor i he pa fe decade. Sch yem hich are oliear by are ca occr i vario aral ad mamade yem ad are characerized by a grea eiiviy o iiial codiio [6]. The Dffig Va der Pol eqaio provide a impora mahemaical model for dyamic yem havig a igle able fixed poi alog ih a igle able limi cycle ad i govered by he o-liear differeial eqaio 3 ih he iiial codiio here he over do repree he derivaive ih repec o ime ad hich are poiive coefficie. I geerae he limi cycle for mall vale of μ developig io relaxaio ocillaio he μ become large hich ca be evalaed hrogh he Lided perrbaio mehod [6]. Example of ch pheomea arie i all of he aral ad egieerig ciece [78] i may phyical problem a ell [8]. Mo cieific problem i olid mechaic are iherely o-liear. Excep a limied mber of hee problem mo of hem do o have aalyical olio. Some of hem are olved by ig merical echiqe ad ome are doe o he aalyical perrbaio mehod [9]. 53

2 A J. Al-Saoor & Mera A. Aziz Recely irodced variaioal ieraio mehod by He [7-5] hich give rapidly coverge cceive approximaio of he exac olio if ch a olio exi ha proved ccefl i derivig aalyical ad approximae olio of liear ad oliear differeial eqaio. Thi mehod i preferable over merical mehod a i i free from rodig off error ad eiher reqire large comper poer/memory. He [34] ha applied hi mehod for olvig aalyical olio of aoomo ordiary differeial eqaio o-liear parial differeial eqaio ih variable coefficie ad iegro - differeial eqaio. The variaioal ieraio mehod a cceflly applied o Brger ad copled Brger eqaio [] o Schrodiger- KdV geeralized KdV ad hallo aer eqaio [] o liear Helmholz parial differeial eqaio [9] o eveh order Saada - Koera eqaio [] o Va der Pol Dffig Ocillaor [] Liear ad oliear ave eqaio KdV K Brger ad cbic Boieq eqaio have bee olved by Wazaz [34] by ig he variaioal ieraio mehod. I he pree paper e employ VIM mehod for olvig Dffig Va der Pol eqaio. Frher e compare he rel ih he give olio by ig Adomia Decompoiio Mehod [34 ] ad e prove he covergece of he mehod.. Adomia Decompoiio Mehod for Solvig Dffig Va der Pol Eqaio To olve eq. ADM i employed. We rerie i i he folloig form A 3 i a real Hilber pace H here A H H i eiher a liear or a oliear operaor. The priciple of he ADM i baed o he decompoiio of he o-liear operaor A i he folloig form: A L R N ih L R 3 N Where L+R i liear N o-liear L iverible ih L - a ivere defied by L L L z dzd R z z dzd N [ z z 3 z] dzd A al i ADM he olio of Eq. 3 ca be coidered o be a he m of he folloig ifiie erie 4 From Eq. e have: L L L R L N 5 here L L L R L N 6 54

3 A Applicaio of He' Variaioal Ieraio Mehod for From hich e defie he folloig cheme L R L A [ z z] dzd A z dzd... 7 Where A are called Adomia Polyomial [345].. Algorihm Compig Adomia Polyomial Ip: The Eqaio F F Se N m M k K ; he ip of Adomia Polyomial i eeded. Op: Aj ; he Adomia Polyomial Sep : e j Sep : hile j do ep 3 ad 4 Sep 3: F F j Sep 4: F F Sep 5: expaio of F.r.. f Sep 6: hile j k ad hile j m Aj f j D f j Sep 7: op Aj he Adomia Polyomial Sep 8: ed.. Compig Adomia Polyomial for Eqaio Compig Adomia Polyomial by Algorihm. yield o A N 3 A 3 4 A A No biig 8 i 7 yield :

4 A J. Al-Saoor & Mera A. Aziz ad o o The five erm of he approximaio o he olio are coidered a 3 4 for he covergece of he mehod e refer he reader o [5] i hich he problem of covergece ha bee diced briefly. 3. He' Variaioal Ieraio Mehod for Solvig Dffig Va der Pol Eqaio To explai he baic idea of He' variaioal ieraio mehod VIM e coider a geeral oliear ocillaor ih pecified iiial codiio a follo more geeral form ca be coidered iho he lo of geeraliy F : f g ad for he Eq. e have F : here f ad g are coio oliear operaor ih repec o heir argme ad i a ko variable. We fir coider Eq. a L [ ] N[ ] ad for Eq. e have: L[ ] ad N[ ] f g Where L ih he propery Lf he f deoe he liear operaor ih repec o ad N i a o-liear operaor ih repec o.we he corc a correcio fcioal for Eq. a [] N[ ~ ] d 3 here i he iiial ge ad he bcrip deoe he -h ieraio deoe he Lagrage mliplier hich ca be ideified efficiely via he variaioal heory ad ~ i coidered a a rericed variaio i.e. ~ Takig he variaio ih repec o he idepede variable e oice ha. Aferard e make he correcio fcioal aioary ad e obai ; herefore e have N ~ [ ] d 56

5 A Applicaio of He' Variaioal Ieraio Mehod for 57 d 4 d A a rel e have he folloig aioary codiio: The Lagrage mliplier ca be readily ideified a co i 8 Moreover e have he folloig variaioal ieraio formla: d F d ] [ i 9 Accordigly he cceive approximaio of VIM ill be readily obaied by chooig all he above parameer a follo ]] co ][ co ] co [ i ] co [ co [ ] co [ ] [ i ] co [ i d d ad o o 4. Nmerical Rel The merical approach for 9 ad i comped by ig Malab. We coider he folloig for cae Cae : 75. ad he iiial codiio

6 A J. Al-Saoor & Mera A. Aziz Cae : ad he iiial codiio Cae 3:. 75 ad he iiial codiio Cae 4: ad he iiial codiio Fig. Compario of he Solio of Eq. a ime for cae ig claical ADM VIM ad RK45. Fig. Compario of he Solio of Eq. a ime for cae ig claical ADM VIM ad RK45. 58

7 A Applicaio of He' Variaioal Ieraio Mehod for Fig. 3 Compario of he Solio of Eq. a ime for cae 3 ig claical ADM VIM ad RK45. Fig. 4 Compario of he Solio of Eq. a ime for cae 4 ig claical ADM VIM ad RK Covergece Aalyi: The He' variaioal ieraio formla make a recrrece eqece. Obvioly he limi of he eqece ill be he olio of eq. if he eqece i coverge. I hi ecio e give a e proof of covergece of He' variaioal ieraio mehod i deail by irodcig a e ieraive formlaio of hi procedre. 59

8 A J. Al-Saoor & Mera A. Aziz Here C [ T] deoe he cla of all real valed fcio defied o [T] hich have coio h order derivaive. Lemma If for ay C [ T ] he he He' variaioal ieraio formla Eq.9 i eqivale o he folloig ieraive relaio L[ ] [ ] Where L i a oed i. Proof d Sppoe ad aify he variaioal ieraio formla 9. Applyig d boh ide of 9 rel i d d [ ] i [ ] d i - [ ] d i [ ] d No ig he codiio 5-7 ad i e ill ge d [ ] [ ] [ ] d From he defiiio of L e obai L[ ] [ ] 3 Coverely ppoe ad aify. I vie of he defiiio L ad i. Mliplyig eq. by i ad he iegraig from boh ide of he reled erm from o yield i [ ] d i [ ] d i [ ] d 4 Uig iegraio by par he expreio 4 become i [ ] i [ ] [ i i ][ ] d - i [ ] d 5 Which exacly rel i 9 po he codiio 5-7i.e. o 6

9 A Applicaio of He' Variaioal Ieraio Mehod for i [ ] d 6 ad hi ed he proof. Theorem : If he eqece lim coverge here i prodced by he variaioal ieraio formla of Eq. 9 he i m be he olio of he eqaio Proof: If he eqece coverge e ca rie v lim 7 ad i hold v lim 8 Uig he expreio 7 ad 8 ad he defiiio of L i e ca eaily gai lim L[ ] L lim[ ] 9 From 9 ad accordig o he lemma e obai L lim[ ] lim[ ] 3 Which give lim[ ] 3 From Eq.3 ad coiiy of f ad g operaor i hold lim[ ] lim[ f g ] 3 lim f lim lim glim lim lim lim v f v v g v v v v From he eqaio 3 ad 3 e have v f v v g v v v v 33 O he oher had ig he pecified iiial codiio ad he defiiio of he iiial ge e have v lim ice 34 v lim ice 35 Therefore accordig o he above hree expreio 3334 ad 35 v m be he olio of he Eq..Thi ed he proof. 6. Coclio I hi ork e have give a e proof of covergece of He' Variaioal Ieraio Mehod by preeig a e formlaio of He' mehod. We have compared hi mehod ih ADM ad RK45 ad e ca coclde ha he mai propery of hi mehod i i i flexibiliy ad abiliy o olve Dffig Va Der Pol accraely ad coveiely iho decompoig he o-liear erm hich are very complex. Thi echiqe give a accrae ad eay compable olio by mea of a rcaed erie hoe covergece i fa. 6

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