A Comparison Among Homotopy Perturbation Method And The Decomposition Method With The Variational Iteration Method For Dispersive Equation

Transcription

1 Inernaional Jornal of Basic & Applied Sciences IJBAS-IJENS Vol:9 No: A Comparison Among Homoopy Perrbaion Mehod And The Decomposiion Mehod Wih The Variaional Ieraion Mehod For Dispersive Eqaion Hasan BULUT* and HMehme BASKONUS** *Deparmen of Mahemaics Fira Universiy 9 Elazig-TURKEY ** Deparmen of Mahemaics Hakkari Universiy Hakkari- TURKEY hbl@firaedr hmbaskons@gmailcomr Absrac-- In his aricle we implemen a relaively new nmerical echniqe and we presen a comparaive sdy among Homoopy perrbaion mehod and Adomian decomposiion mehod he variaional ieraional mehod These mehods in applied mahemaics can be an effecive procedre o obain for approimae solions The sdy olines he significan feares of he hree mehods The analysis will be illsraed by invesigaing he homogeneos Dispersive eqaion model problem This paper is pariclarly concerned a nmerical comparison wih he Adomian decomposiion and Homoopy perrbaion mehod he variaional ieraional mehod The nmerical resls demonsrae ha he new mehods are qie accrae and readily implemened Inde Term-- Dispersive Eqaion The Decomposiion Mehod Homoopy perrbaion Mehod The variaional ieraional Mehod I INTRODUCTION Parial differenial eqaions which arise in real-world physical problems are ofen oo complicaed o be solved eacly And even if an eac solion is obainable he reqired calclaions may be oo complicaed o be pracical or i migh be diffcl o inerpre he ocome Very recenly some promising approimae analyical solions are proposed sch as Ep-fncion mehod [ ] Adomian decomposiion mehod [ 7] variaional ieraion mehod [8 ] and Homoopy-perrbaion mehod [ 6] Oher mehods are reviewed in Refs[7 8] HPM is he mos effecive and convenien on efor boh linear and nonlinear eqaions This mehod does no depend on a small parameer Using homoopy echniqe in opology a homoopy is consrced wih an embedding parameer p [] which is considered as a small parameer HPM has been shown o effecively easily and accraely solve a large class of linear and nonlinear problems wih componens converging rapidly o accrae solions HPM was firs proposed by He [] and was sccesflly applied o varios engineering problems [9 ] Recenly VIM is applied for eac solions of Dispersive Eqaion [ 7] The aim of his work is o employ HPM and ADM o obain he eac solions for Dispersive Eqaions and o compare he resls wih hose of VIM Differen from ADM specific algorihms are sally sed o deermine he Adomian polynomials HPM handles linear and nonlinear problems in a simple manner by deforming a diffcl problem ino a simple one Dispersive eqaion has he following form: () wih he bondary and iniial condiions ( ) cos( ) () ( ) cos( are ) given fncions These eqaions appear in sch diverse phenomena as: elasic waves in solid inclding vibraing sring bars membranes sond or acosics elecromagneic waves and nclear reacors II NUMERICAL METHODS Fndamenals of he Homoopy Perrbaion Mehod To illsrae he basic ideas of his mehod we consider he following eqaion []: A () wih bondary condiion f r r B r n () A is a general differenial operaor B a bondary operaor f (r) a known analyical fncion and Γ is he bondary of he domain Ω A can be divided ino wo pars which are L and N L is linear and N is nonlinear Eq () can herefore be rewrien as follows: L N f r r () Homoopy perrbaion srcre is shown as follows: (6) r p: v (7) In Eq (6) p [ ] is an embedding parameer and is he firs approimaion ha saisfies he bondary condiion IJBAS-IJENS December 9 IJENS

2 Inernaional Jornal of Basic & Applied Sciences IJBAS-IJENS Vol:9 No: 6 We can assme ha he solion of Eq (6) can be wrien as a power series in p as following: v v pv p v p v (8) and he bes approimaion for solion is lim v v v v v p (9) The above convergence is discssed in [] Adomian Decomposiion Mehod The decomposiion series mehod does no reqire his discriminaion and resling massive compaion In his work we apply he second mehod o obain analyic and approimae solions of he eqaion from () and sing he decomposiion mehod he eqaion () is approimaed by he operaors in he following form () L f ( ) L L f ( ) L L he operaor respecively Assming he invers of L d eiss and i can convenienly be inegraed wih respec o from o hen applying he inverse operaor L o () yields L f ( ) [ ( )] Therefore i follows ha () ( ) ( ) L L () The zeroh componens is obained by sing he iniial condiion as ( ) n L L n ( n) ( ) cos( ) cos( ) () Which is defined by all erms ha arise from he iniial condiions Ths he nknown fncion ( ) is comped in erms of he componens defined by he decomposiion series given as () n ( ) n( ) n The remaining componens n( ) n can be compleely deermined sch ha each erm is comped by sing he previos erm Since is know (6) L ( L ) L d L ( L ) L d L ( L ) L d L ( L ) L d L L n ( n) () As a resls he series solions is given by ( ) L n n is he previosly given inegraion operaor The solion ( ms saisfy he reqiremens imposed by he iniial condiions The decomposiion mehod provides a reliable echniqe ha reqires less work if compared wih he radiional echniqes Moreover he proposed mehod does no need discriminaion of he problem o obain nmerical resls We can evalae he approimae solion by sing he erm approimaion Tha is (7) k ( Where he componens are prodce as ( k ( ( ( ( ( ( ( ( ( IJBAS-IJENS December 9 IJENS

3 Inernaional Jornal of Basic & Applied Sciences IJBAS-IJENS Vol:9 No: 7 II THE VARIATIONAL ITERATIONAL METHODS Consider he differenial eqaion L N g() (8) Where L is a linear operaor N is a non-linear operaor and g() is a known and Nonlineer analyical fncion Ji Han He has modified he above mehod ino an ieraion mehod 7 in he following way: ( L ( ) N ( ) g( )) d n n n n (9) is a general Lagrange s mlipler which can be ~ is a idenified opimally via he variaional heory and n resriced variaion which means ~ I is obvios now ha he main seps of He s variaional ieraion mehod reqire firs he deerminaion of he Lagrangian mliplier ha will be idenified opimally Having deermined he Lagrangian mliplier he sccessive approimaions n n of he solion will be readily obained pon sing any selecive fncion Conseqenly he solion lim n for ( n ) () In oher words correcion fncional (9) will give several approimaions and herefore he eac solion is obained a he limi of he resling sccessive approimaions To give a clear overview of he mehodology we consider several eamples in he following secion Eample : For comparison prposes we consider a Dispersive eqaion model problem in order o illsrae he echniqe discssed above This problem is as follows; wih in iniial condiions cos( ) () () In order o solve Eq() sing HPM we can consrc a homoopy for his eqaion p [ Y ] p[ Y Y "'] Y py p py py "' () Y Y Y Y "' and p Wih iniial Y Cos sppose he solion approimaion of Eq () has he form: Y Y py p Y p Y p Y n () n p Yn ''' ''' ''' ''' Y py p Y p Y p Y p( Y py p Y p Y ) p( Y py p Y p Y ) ''' ''' ''' ''' Y py p Y p Y p Y py p Y p Y p Y py p Y p Y p Y ''' ''' ''' ''' Y py py py p Y p Y p Y p Y p Y p Y p Y p Y p Y ''' ''' ''' ''' Y p( Y Y Y ) p ( Y Y Y ) p ( Y Y Y ) p ( Y Y Y ) Then sbsiing Eq () ino Eq () and rearranging based on powers of p -erms we obain: p Y : () ''' p : Y Y Y ''' (6) p : Y Y Y (7) ''' p : Y Y Y (8) ''' p : Y Y Y (9) wih solving Eqs ()-(9) Y cos( ) () Y sin( ) () 6 Y cos () 6 9 Y sin () ( ) ( ) Y cos () ( ) IJBAS-IJENS December 9 IJENS

4 Inernaional Jornal of Basic & Applied Sciences IJBAS-IJENS Vol:9 No: 8 he above erms of he series () cold calclaed When we consider he series () wih he erms ()-() and sppose p we obain approimaion solion of Eq () as following: Y Y Y Y Y () ( ) Cos( ) 6 9 cos( ) sin( ) cos( ) sin( ) cos( ) 6 Y Y Y are idenified and As a resl he componens he series solion hs enirely deermined In order o solve his eqaion by sing he Adomian decomposiion mehod we simply ake he eqaion in an operaor form Following he same manner as given by eqaion () o as find he zeroh componen of (6) cos( ) and remaining componens ec We are comped by a recrsive scheme direcly by hand sing () Some of he symbolically comped componens are as follows L ( L ) L cos( d sin[ ] (7) 6 L ( L ) L sin( ) d cos( ) L ( L ) L cos( ) d sin( ) 9 6 L ( L ) L sin( ) d cos( ) L ( L ) In his manner for componen of he decomposiion series were obained of which ( was evalaed o have he following epansion ( ) 6 9 ( ) cos( ) sin( ) cos( ) sin( ) cos( ) 6 As a simply; cos 6 6 ( ) (( ) ( ) ( 6 )sin( )) Conining he epansion o he las erm i may be proved ha he secion of he decomposiion series () as In order o solve () eqaion by sing he variaional ieraion mehod we simply ake he eqaion in an operaor form and n n L N g() ( ) ( ) d (8) ( ) ( ) d n n (9) i( i ) are general Lagrange mlipliers [] and can be idenified opimally via he variaional heory[8-] is an iniial approimaion or rial- fncion wih possible nknowns n resriced variaions [7-] ie ~ Making he correcion fncional (9) saionary noicing ha ~ n( ) n( ) n n n y () ~ are considered as ( ) ( ) ( ) d n( ) n ( ) n( ) d () ' n ( ) n ( ) n( )( ( )) ( ) n( ) " ( ) d yields he following saionary condiions : ' n ( ) n : ( ) n : " ( ) The Lagrange mliplier herefore can be readily idenified ( ) and by he same maniplaion we have () IJBAS-IJENS December 9 IJENS

5 Inernaional Jornal of Basic & Applied Sciences IJBAS-IJENS Vol:9 No: 9 As a resl we obain he following ieraion formlae in y - and - direcions: cos( ) cos( ) ( cos( ) ( ) cos( ) d y n( ) n( ) y n ( n( ( ( ) n d hs we have cos( ) cos( ) y n( ) n( ) ( cos( ) ( ) cos( ) d n ( n( ( ) ( ) n d y y () Now we begin wih an arbiary iniial approimaion: A A and B are consans B in o be deermined by sing he iniial condiions () hs we have cos By he above variaional ieraion formla in -direcion () we can obain following resl: Conining in his manner we obain he firs few componens of n ( cos cos( ) cos( ) ( cos( ) ( ) cos( ) d y ( ) ( ) ( ( ( ) ( ) d y () and so on in he same manner he res of componens of he ieraion formla () were obained sing he Mahemaica Package (=(-6 9 ) cos[ ] (- 8 8 )sin[ ] The Solion ( in a closed form is ( ) Cos( ) Fig The nmerical resls for Y in comparison wih he analyic solion ( when = wih iniial condiion of Eq() by means of HPM The approimaion can also be obained by y-direcion or by alernaing se of -and y-direcions ieraions formla - - app eac app hpm (a): Eac solion (b): Approimaion solion via HPM - - Fig The plos of he nmerical resls for Y in comparison wih he analyic solion ( when = wih iniial condiion of Eq () by means of HPM (a): Eac solion (b): Approimaion solion HPM IJBAS-IJENS December 9 IJENS

6 Inernaional Jornal of Basic & Applied Sciences IJBAS-IJENS Vol:9 No: (a): Eac solion (b): Approimaion solion via ADM Fig The nmerical resls for Y in comparison wih he analyic solion ( when = wih iniial condiion of Eq () by means of ADM (a): Eac solion (b): Approimaion solion via VIM Fig The nmerical resls for Y in comparison wih he analyic solion ( when = wih iniial condiion of Eq () by means of VIM - - app eac - app eac app adm (a): Eac solion (b): Approimaion solion via ADM Fig The plos of he nmerical resls for Y in comparison wih he analyic solion ( when = wih iniial condiion of Eq () by means of ADM - app vı m (a): Eac solion (b): Approimaion solion via VIM Fig The plos of he nmerical resls for Y in comparison wih he analyic solion ( when = wih iniial condiion of Eq () by means of VIM IV COMPARISON AMONG HPM VIM AND ADM I can be seen from he eamples sdied ha: Comparison among HPM VIM and ADM shows ha alhogh he resls of hese mehods when applied o he Dispersive eqaion are he same HPM does no reqire specific algorihms and comple calclaions sch as ADM or consrcion of correcion fncionals sing general IJBAS-IJENS December 9 IJENS

7 Inernaional Jornal of Basic & Applied Sciences IJBAS-IJENS Vol:9 No: Lagrange mlipliers sch as VIM and is mch easier and more convenien han ADM and VIM HPM handles linear and nonlinear problems in a simple manner by deforming a difficl problem ino a simple one B in nonlinear problems we enconer difficlies o calclae he so-called Adomian polynomials when sing ADM Also opimal idenificaion of Lagrange mlipliers via he variaional heory can be difficl in VIM [] [] H Bl Comparing Nmerical Mehods for Bossiness Eqaion Model Problem Nmerical Mehods for Parial Differenial Eqaions () (9) V CONCLUSIONS In his leer we have sccessflly developed HPM and ADM o obain he eac solions of Dispersive eqaion The resls are hen compared wih hose of VIM I is apparenly seen ha hese mehods are very powerfl and efficien echniqes for solving differen kinds of problems arising in varios fields of science and engineering and presen a rapid convergence for he solions The solions obained show ha he resls of hese mehods are in agreemen b HPM is an easy and convenien one REFERENCES [] JH He X-H W Chaos Solions Fracals () (6) 7 [] JH He MA Abdo Chaos Solions Fracals ()(7) [] AM Wazwaz Appl Mah Comp 66 () 6 [] AM Wazwaz A Gorgis Appl Mah Comp 9 () [] AM Wazwaz Appl Mah Comp () [6] D Lesnic Chaos Solions Fracals 8 (6) 776 [7] S Saha Ray Appl Mah Comp 7 (6) 6 [8] JH He In J Non-Linear Mech (999) 699 [9] JH He Appl Mah Comp () [] DD Ganji A Sadighi J Comp Appl Mah 7 ()(7) [] JH He In J Non-Linear Mech ()() 7 [] JH He In J Mod Phys B (8) (6) 6 [] JH He Appl Mah Comp () () 7 [] JH He Phys Le A ( ) (6) 87 [] JH He In J Nonlinear Sci Nmer Siml 6 () () 7 [6] [6] DD Ganji A Sadighi In J Nonlinear Sci Nmer Siml 7 () (6) [7] JH He In J Mod Phys B () (6) [8] JH He Non-perrbaive mehods for srongly nonlinear problems disseraion [9] Verlag im Inerne Berlin 6 [] A Rajabi DD Ganji H Taherian Phys Le A 6 ( ) (7) 7 [] DD Ganji Phys Le A ( ) (6) 7 [] AM Siddiqi R Mahmood QK Ghori Phys Le A (6) [] AM Wazwaz Phys Le A (6) in press [] AD Polyanin Handbook of Linear Parial Differenial Eqaions for Engineers [] and Scieniss Chapman & Hall/CRC Press Boca Raon [6] A Sommerfeld Parial Differenial Eqaions in Physics Academic Press New York 99 [7] N Bellomo and D Sarafyan J Mah Anal Appl [8] DKaya and HBl Fıra Ünv Fen ve Mühendislik Bilimleri Dergisi() [9] D SMomani SAbasad Chaos Solions Fracals 7 (6) 9 [] [8] D J Evans H Bl A new approach o he gas eqaions: An applicaion of he Decomposiion Mehod Iner J Comper Mah 79 () 87 8 [] [9] H Bl D J Evans Oscillaions of solions of iniial vale problems for Parabolic Eqaions by he Decomposiion Mehod Iner J Comper Mah 8 () IJBAS-IJENS December 9 IJENS