Simulation of Singular Fourth- Order Partial Differential Equations Using the Fourier Transform Combined With Variational Iteration Method
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1 Amirkabir Uiversi of Techolog (Tehra Polechic) Vol. 5, No., Sprig 3, pp. - 5 Amirkabir Ieraioal Joral of Sciece & Research (Modelig, Ideificaio, Simlaio & Corol) (AIJ - MISC) Simlaio of Siglar Forh- Order Parial Differeial Eqaios Usig he Forier Trasform Combied Wih Variaioal Ieraio Mehod S. S. Norazar *, H. Tamim, S. Khalili 3, A. Mohammadzadeh - Associae Professor, Deparme of Mechaical Egieerig, Amirkabir Uiversi of Techolog, Tehra, Ira - MSc Sde, Deparme of Mechaical Egieerig, Amirkabir Uiversi of Techolog, Tehra, Ira 3- MSc Sde, Deparme of Mechaical Egieerig, Amirkabir Uiversi of Techolog, Tehra, Ira - MSc Sde, Tehra iversi alms i mechaical egieerig, Tehra, Ira ABSTRACT I his paper, we prese a comparaive sd bewee he modified variaioal ieraio mehod (MVIM) ad a hbrid of Forier rasform ad variaioal ieraio mehod (FTVIM). The sd olies he efficiec ad covergece of he wo mehods. The aalsis is illsraed b ivesigaig for siglar parial differeial eqaios wih variable coefficies. The solio of siglar parial differeial eqaios sall eeds a coordiae rasformaio i order o discard he siglari of he parial differeial eqaio. Mos ofe his rasformaio is o applicable ad eve does o eis. Therefore i his case he solio for he siglar parial differeial eqaio does o eis. I he prese sd he resls of simlaio for he siglar parial differeial eqaios wih variable coefficies sig he Forier rasform variaioal ieraio mehod are compared wih he resls of simlaio sig he modified variaioal ieraio mehod. The compariso shows ha he effeciveess ad accrac of Forier rasform variaioal ieraio mehod is more ha ha of he modified variaioal ieraio mehod for he simlaio of siglar parial differeial eqaios. KEYWORDS Forier Trasformaio Modified Variaioal Ieraio Mehod, Hbrid of Forier Trasform, Variaioal Ieraio Mehod, Siglar Parial Differeial Eqaios Wih Variable Coefficies. Correspodig Ahor, icp@a.ac.ir Vol. 5, No., Sprig 3
2 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) - INTRODUCTION This paper olies a reliable compariso bewee wo powerfl mehods ha were recel developed. The firs is a hbrid of Forier rasform ad variaioal ieraio mehod (FTVIM) developed b S.S. Norazar e al i []. The secod is he modified variaioal ieraio mehod (MVIM) developed b M. A. Noor e al i []. This paper is devoed o he sd of he siglar forh - order parabolic parial differeial eqaio wih variable coefficie. I is well kow i he lierare ha a wide class of problems arisig i mahemaics, phsics ad asrophsics ad egieerig scieces ma be disicivel formlaed as siglar iiial ad bodar vale problems. Siglar forh order parabolic parial differeial eqaios gover he rasverse vibraios of a homogeeos beam. Sch pes of eqaio arise i he mahemaical modelig of viscoelasic ad ielasic flows, deformaio of beams ad plae deflecio heor, see [ - ]. The sdies of sch problems have araced he aeio of ma mahemaicias ad phsiciss. So fidig a mehod wih less amo of compaioal work i compariso wih he previos mehods ma be sefl. The sd of forh - order parial differeial eqaios wih variable coefficies are performed b solvig he vibraio eqaio of beam ad shafs sig he secod - order fiie differece mehod [, 5]. However i heir resls he covergece of he solio wih mesh refieme failed ad herefore he accrac of he resls was limied o a cerai amo of refied mesh. Wazwaz [, 8, 9] sdied he behavior of he forh - order parial differeial eqaios wih variable coefficies. The applicaio of sch eqaios ecoers i he sd of deformaio of beams ad plaes as well as he flow of viscoelasic ad elasic flids. Wazwaz [, 8, 9] solved he goverig eqaios sig he semi - aalical mehod sch as he Adomia decomposiio mehod. However i he Adomia decomposiio mehod he bodar codiios of he goverig eqaio were oall igored. This ma case a iaccrae resls as well as icorporaig eormos amos of erms i series solio. I he prese sd we ied o se he FTVIM developed b Norazar e al [] o solve he siglar forh - order S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh parial differeial eqaios wih variable coefficies ha ma arise i he sd of he vibraio of beams ad plaes as well as he flow of viscoelasic ad elasic (Newoia) flid. These eqaios are solved b sig he modified He s variaioal ieraio mehod (MVIM) []. Firs, a coordiae rasformaio is sed o resolve he siglari problem ad he he MVIM [] is applied o solve he goverig eqaios. However, he advaage of he FTVIM is ha here is o eed o fid a coordiae rasformaio o ge rid of he siglari problem. We show he effeciveess of he FTVIM b solvig for siglar forh - order parabolic differeial eqaios wih variable coefficies as case sdies. The firs case sd problem is he vibraio of a elasic beam wih variable maerial properies alog he ais of beam. I he secod case sd we solve he vibraio of a hi wo dimesioal elasic plae wih variable maerial properies alog he wo dimesios (, ). I he hird case sd we solve he goverig eqaio of a hree dimesioal vibraig plae wih eeral sisoidal forcig fcio. I he forh case sd we cosider he vibraio of a elasic beam wih a sisoidal variable proper alog he beam ais. - NUMERICAL APPLICATION I his secio, we appl he FTVIM proposed b S.S. Norazar e al i [] for solvig he forh order parabolic parial differeial eqaios. For sig FTVIM we cosrc a correcio fcioal b sig he Lagrage mlipliers ha calclaed opimall via variaioal heor. Ad fiall we ca approimae he eac solios. For he sake of compariso, we ake he same eamples as sed i [] ad he merical resls are ver ecoragig ad coclsive. EXAMPLE Cosider he followig siglar forh parabolic parial differeial + + = < <, > () Wih iiial codiios: (,) = < < () 5 (,) = + < < (3) 8 Vol. 5, No., Sprig 3
3 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod Firs we ake Forier rasform from eqaio () ad immediael cosrc a correcio fcioal as is doe i referece [3]. Here û deoes he Forier rasform of. Now for sig FTVIM we cosrc a correcio fcioal as[3]: (, ) (, ) (, ) F F û + ω = û ω + λξ û d ωξ + + ξ ξ ω () For deailed derivaio of cosrcig he correcio fcioal (Eq. ()) oe ma refer o Wazwaz [3]. B sig he variaioal priciple ad iegraig b pars we ma obai he followig. Iegraig b pars ad ake firs variaio, we ge λξ = ξ as follows: δ (, ) (, ) (, ) ω δ ω δ λξ F F d ωξ û + = û + û + + ξ ξ ω û (5) δû + ( ω, ) = δû ( ω, ) + δ λξ û (, ) d ωξ ξ () ξ δû + = δû + δ λξ û λ ( ξ) + λ ( ξ) dξ ξ û û () δ û λ ( ξ ) δ δλ( ξ ) λ + = û + + ( ξ ) δ dξ ξ û û (8) λ ( ξ) = ξ = λξ = λξ = ξ ξ = λ ( ξ) = ξ = (9) Assmig (, ), sig he same mehod as sed i [3], ad sbsiig for he vale of λξ = ξ, io Eq.() he sccessive approimaio û + ( ω,) are obaied as follows: (, ) (, ) (, ) û ω = û ω + ξ û ωξ + F + F dξ ξ ω () () (, ) (, ) (, ) F F û ω = û + ω + ξ û d ωξ + + ξ ξ ω (3) For obaiig (,) firs we calclae followigs: 5 (,) = + () Eq. () is obaied b iegraig he Nema iiial codiio (Eq. (3)) ad cosiderig as (,). ( I πdirac 5 ) 5, ωω + πdirac ( ωω ) I ω û ( ω,) = (5) ω () 5 + = Vol. 5, No., Sprig 3 () 9
4 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh 5 + = I F = πdirac ( ω) ω () (8) (9) () () Usig he Maple package he iverse Forier rasform, (,) is: 5 3 (,) = + () Here, we are akig he iverse Forier rasform from eqaio () sig he Maple package. Afer some simple maiplaios we ge Eq. () For calclaig (,), we se he correcio fcioal of Eq. () sig he vale of û (ɷ,) ad akig he iverse Forier rasform as: ξ û (ω, ξ) = ( I π Dirac(5, ω) ω - + π Dirac (ω) ω - ω I ω 5 (-ξ) (3) 3 I F πdirac ( ω) = ω () 3 Vol. 5, No., Sprig 3
5 ) Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod ω F = I(πDirac(5,ω) ω + 8 πdirac (,ω) ω -πdirac(3,ω)) ω 8I (π Dirac (, ω) ω + π Dirac (3,ω) ω + π Dirac (,ω)) ω 3 + 3I ( π Dirac (3, ω) ω + π Dirac (,ω) ω + π Dirac (,ω)) ω 9I(πDirac(,ω)ω + π Dirac (,ω)ω) I(πDirac(,ω)ω + I) + + ω 5 ω ) - 3 Usig correcio fcioal ad he Maple package, he iverse Forier rasform, (,) is: (,) = + + Here, we are sig he correcio fcioal of Eq. () ad akig he iverse Forier rasform oe ges Eq. (). For 3 (,): 5 ( I πdirac ) 5, ωω + πdirac ( ωω ) I ω û (,) 3 ω = ξ + ξ () ξ ω (5) () 3 5 I F πdirac ( ω) = + ω (8) (9) Usig correcio fcioal ad he Maple package he Forier rasform ad iverse Forier rasform, 3 (,) is: (,) = (3) Ad so o. The Talor series epasio for si () is wrie as: i (i + ) 3 5 si = = + + ( ) (3) ( i + )! 5 i = Ad: (, ) = lim B sbsiig Eq (3) io Eq (3) hs Eq (3) ca limael be redced o: 5 (, ) = lim si = + Which, i is he eac solio. (3) (33) EXAMPLE Cosider he followig siglar forh parabolic parial differeial eqaio i wo space variables: =!! Vol. 5, No., Sprig 3 (3) 3
6 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Wih iiial codiios: (,,) =, (,,) = + +!! Now sig FTVIM we cosrc a correcio fcioal as: S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh (35) (,, ) (,, ) û (,, ) (,,) (,,) () ω ξ ξ F F ξ û ω = û ω λξ dξ + + +!! ξ (3) Iegraig b pars ad akig he firs variaio, we ge λξ = ξ as follows: (,, ) (,, ) û (,, ) (,,) (,,) () ω ξ ξ F F ξ δû ω δ ω δ λξ + = û + + dξ + + +!! ξ (3) û (,, ) (,,) (,,) () ω ξ δû + ω = δû ω + δ λξ dξ ξ (38) û δû = δû + + δ λξ λ ( ξ) û + λ ( ξ) û dξ ξ (39) δ û + = λ ξ δ û + δλ ξ + λ ( ξ ) dξ ξ û λ ( ξ) = λξ = λ ( ξ) = λξ = ξ ξ ξ ξ Assmig û () = = = () (,,) = + +!!, sig he same mehod as sed i [3], ad sbsiig for he vale of λξ = ξ io Eq. (3) he sccessive approimaio û + (ɷ,,) are obaied as follows: () (3) () (5) For obaiig (,,), firs we calclae followigs: + + =!! + + =!! + = + = +! () () (8) 3 Vol. 5, No., Sprig 3
7 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod + = + = +! (9) (, ) πdirac ωω + πdirac ωω πdirac ωω + I I ω I ω F + + =!! ω (5) (5) F Dirac + = π ω + 3 (5) Dirac Dirac Dirac (, ) I I I,,) π ωω + π ωω π ωω + ω ω û ω = + ( ξ ) ξ ( ω πdirac (, ωω ) + πdirac ( ωω ) + I I ω ξ + + πdirac ( ω) d ξ ω 3 (53) Usig he Maple package, he iverse Forier rasform, (,,) is: 3 (,, ) = = + + 3!! (5) For (,,) 3 3 ( (,,)) = + + =!! 3 3 ( (,,)) = + + =!! (55) (5) (5) Usig correcio fcioal ad he Maple package, he Forier rasform ad iverse Forier rasform, ) is: (58) (59) For 3 (,,) () () () Vol. 5, No., Sprig 3 33
8 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh Usig correcio fcioal ad he Maple package, he Forier rasform ad iverse Forier rasform, 3 ) is: (3) () Ad so o. The Talor series epasio for si()is wrie as: (5) Ad (,, ) = lim () B sbsiig Eq. () ad Eq. (5) io Eq. () hs Eq. () ca limael be redced o: (,, ) = lim = + + si!! () Which, i is he eac solio of Eq. (3). EXAMPLE 3 Cosiderig he followig hree dimesioal o - homogeos siglar parial differeial eqaio, we solve his eqaio wih FTVIM. z + + = cos! z!! z z z Wih iiial codiio: z (,,,) = + + (9) z z ( ) (8) Now we cosrc a correcio fcioal as: ( ω, z,,) ( ω, z,,) +! z! ( ω, z,,) û ( ω, z,,) = û + + ( ω, z,,) + λξ () F + dξ! z z ( ω, z,,) cos ( ξ ) + z z ξ () Iegraig b pars ad akig he firs variaio, we ge λξ = ξ as follows: + +! z!! z δ = δ + δ λξ ξ û + û F d () z cos ( ξ ) + z z ξ ! z!! z δ = δ + δ λξ ξ+ δ λξ ξ 3 Vol. 5, No., Sprig 3 û û + F d F d ξ + z cos ( ξ ) () z z 5 5 5
9 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod û δû + + = δû + δ λξ dξ (3) ξ û δû δ δ λξ λ + ( ξ) λ + = û + û + ( ξ) d ξ ξ û d () δ = λ ξ δ + δ λξ + δ λ ξ ξ ξ û û [ ] + + û d û d (5) λ ( ξ) = ξ = λξ = λξ = ξ ξ = λ ( ξ) ξ = = () Assmig z, ) = z, ), sig he same mehod as sed i [3], ad sbsiig for he vale of λξ = ξ, ad sig Eq. () he sccessive approimaio û + (ɷ,, z, ) are obaied as follows: () (8) (9) (8) For obaiig û (ɷ,, z, ), firs we calclae followig erms: (, z,, ) = (, z,, ) z = 5 (, z,, ) = 5 z (, z,, ) = z 5 ( ω, z,,) = 5! z (8) (8) Usig he Maple package, he iverse Forier Trasform, (,,,) z is: cos cos cos cos cos z cos, z,, = ( z) z z z z,,, = + z cos z z z z z (, z,, ) = cos (83) z z For (, z,,) Vol. 5, No., Sprig 3 35
10 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh z,,, cos = z z z z z z (,,, ) = + + cos z + + cos z z (8) z cos 8 8cos 8, z,, = cos 8 8cos 8, z,, = (85) cos 8 8cos 8, z,, = z z z z z cos cos 35, z,, = ! z z z z cos cos 35, z,, = ! cos cos 35, z,, = ! z z z z z (8) Usig he Maple package, he iverse Forier Trasform, (,,,) z is: For 3 (, z,,) (8) (88) 3 Vol. 5, No., Sprig 3
11 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod (89) (9) Usig he Maple package, he iverse Forier Trasform, (,,,) 3 z is: cos( z + z + z ) ( z + z + z ) + 5 ( z + z + z ) ( z + z + ) 98( z + z + ) ( + + ) 395 ( z + z + ) ( z + z + z ) + 98cos( z + z + ) ( z + z + z ) + cos( z + z + z ) (, z,,) = z z z ( z + z + z ) cos + +!!! (, z,, ) = ( z + z + ) cos + + z!!! cos( z + z + z ) z 3(, z,, ) 35 cos = z z z z z z z z !!! z z z (9) Ad so o. The Talor series epasio for cos() is wrie as: i ( ) i = ( i ) i cos( ) = = + +!!!! (9) Ad (,) = lim (,) (93) B sbsiig Eq. (9) ad Eq. (9) io Eq. (93) hs Eq. (9) ca limael be redced o Eq (9): z 3(, z,, ) = cos z z z z z z z z cos z z z (9) Therefore, he eac solio is give as: z (,) = lim (,) = + + cos z (95) Vol. 5, No., Sprig 3 3
12 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) EXAMPLE Cosider he followig siglar forh parabolic parial differeial. + = si( ) S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh (9) Wih iiial codiios: (,) = si( ) < < (,) = [ si] < < (98) Now sig FTVIM we cosrc a correcio fcioal as: (9) (99) Iegraig b pars ad ake firs variaio, we ge λξ = ξ as follows: δ δ δ λξ F d ξ si( ) ξ + = + + û + û û () û δû + + = δû + δ λξ dξ ξ () û δû + + = δû + δλξ λ ( ξ) û + λ ( ξ) d ξ ξ û d () û δû + + = δû + δλξ λ ( ξ) û + λ ( ξ) d ξ ξ û d (3) λ ( ξ) = ξ = λξ = λξ = ξ ξ = λ ( ξ) ξ = = () Assmig (,) = ( - si ()) (-), sig he same mehod as sed i [3], ad sbsiig for he vale of λξ = ξ, io Eq. (99) he sccessive approimaios, (,) ω, are obaied as follows: û + + û ( ωξ, ) (, ξ) ( ω,) = û ( ( ω,) + λξ () ( û ( + F dξ ξ si( ) û ( ωξ, ) (, ξ) û (( ω,) = û (( ( ω,) + λξ () + F dξ ξ si( ) (5) () û ( ωξ, ) (, ξ) û 3 ( ( ω,) = û ( ( ω,) + λξ () ( + F dξ ξ si( ) () ( ωξ, ) (, ξ) + ( ω,) = ( ω,) + λξ () + F dξ ξ si( ) û + û û (8) For obaiig û (ɷ, ), firs we calclae followigs: si( ) = si ( ) ( ωπ (, ω) + ) I Dirac I F {( si( ) ) ( )} = I πdirac ( ω + ) I πdirac ( ω ) ( ) ( ω + z )( ω ) ω (9) () 38 Vol. 5, No., Sprig 3
13 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod Usig Eq. (5) we obai û (ɷ, ) as: () Usig he Maple package, he iverse Forier Trasform, (,) is: 3 (, ) ( si ) = + () For (,) 3 ( si ) = + si( ) Usig correcio fcioal ad he Maple package, he iverse Forier rasform, (,) is: 3 5 (, ) ( si ) = + + (3) () For 3 (,) 3 5 ( si ) = + + si( ) (5) Usig correcio fcioal ad he Maple package, he iverse Forier rasform, (,) 3 is: 3 5 (, ) 3 ( si ) = () Ad so o. The Talor series epasio for e is wrie as: e i i 3 5 ( )... () = = i! 5 i = Ad: (,) = lim (,) (8) B sbsiig Eq. () ad Eq. () io Eq. (8) hs Eq. () ca limael be redced o: ( ) (, ) = lim (, ) = si e (9) Which, i is he eac solio of Eq. (9). RESULTS I he followig ables ad figres, we show he red of covergece of o 3 of he FTVIM ad MVIM solio a hree differe locaios ad a hree differe imes. The red of rapid covergece of he FTVIM owards he eac solio is clearl show. Usig he ew mehod, FTVIM, idicaes ha he amo of compaioal work is mch less ha ha of he MVIM. Followig ables below ehibis he relaive errors obaied b he FTVIM, MVIM ad he eac solio. Vol. 5, No., Sprig 3 39
14 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh Table idicaes he relaive errors of he resls of he o 3 of he FTVIM ad MVIM for eample relaive error = =.5 = FTVIM MVIM FTVIM MVIM FTVIM MVIM =. =. =. (, ) (, ) (, ) e - 3 (, ) e e - e e e (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Table idicaes he relaive errors of he resls of he o 3 of he FTVIM ad MVIM for eample relaive error (, ) =(, ) (, ) =(.5,.5) (, ) =(, ) FTVIM MVIM FTVIM MVIM FTVIM MVIM =. =. =. ) ) ) e - 3 ) e e - e e e ) ) ) ) ) ) ) ) Vol. 5, No., Sprig 3
15 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod Table 3 idicaes he relaive errors of he resls of he o 3 of he FTVIM ad MVIM for eample 3 relaive error z) =(,, ) z) =(.5,.5,.5) z) =(,, ) FTVIM MVIM FTVIM MVIM FTVIM MVIM =. =. =. z, ) z, ) z, ) 3 z, ) z, ) z, ) z, ) 3 z, ) z, ) z, ) z, ) 3 z, ) e e e e e Table idicaes he relaive errors obaied b he FTVIM, MVIM ad he eac solio for eample relaive error = =.5 = FTVIM MVIM FTVIM MVIM FTVIM MVIM =. =. (, ) (, ) (, ) (, ) e e - 3 e (, ) (, ) (, ) (, ) e e - 8 e (, ) =. (, ) (, ) (, ) Vol. 5, No., Sprig 3
16 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh The efficiec ad rapid covergece of FTVIM are show i he followig figres: Figre : eac solio ad firs approima of (, ) for = i eample Figre : eac solio ad secod approima of (, ) for = i eample Figre 3: eac solio ad hird approima of (, ) for = i eample Figre : eac solio ad forh approima of 3(, ) for = i eample Figre 5: eac solio ad firs approima of ) for =, = i eample Figre : eac solio ad secod approima of ) for =, = i eample Vol. 5, No., Sprig 3
17 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod Figre : eac solio ad hird approima of ) for =, = i eample Figre 8: eac solio ad forh approima of 3 ) for =, = i eample Figre 9: eac solio ad firs approima of z, ) for =, =, z= i eample 3 Figre : eac solio ad secod approima of z, ) for =, =, z= i eample 3 Figre : eac solio ad hird approima of z, ) for =, =, z= i eample 3 Figre : eac solio ad forh approima of 3 z, ) for =, =, z= i eample 3 Vol. 5, No., Sprig 3 3
18 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) S. S. Norazar, H. Tamim, S. Khalili, A. Mohammadzadeh Figre 3: eac solio ad firs approima of (, ) for = i eample Figre : eac solio ad secod approima of (, ) for = i eample Figre 5: eac solio ad hird approima of (, ) for = i eample Figre : eac solio ad forh approima of 3(, ) for = i eample 3- CONCLUSIONS A ew effecive modificaio o he variaioal ieraio mehod, he Forier rasform variaioal ieraio mehod (FTVIM), is preseed i his paper. The resls obaied b FTVIM are compared wih MVIM. The validi ad effeciveess of he ew mehod, FTVIM is show b solvig for siglar differeial eqaios wih variable coefficies ad he ver rapid approach o he eac solios is show schemaicall. The ver rapid approach owards he eac solios of he ew mehod, FTVIM, idicaes ha he amo of compaioal work is mch less ha hose reqired for ha of MVIM. Moreover, he deficiec of he MVIM cased b saisfied bodar codiios is overcome b he ew mehod, FTVIM, where, he solio is show o be valid i he eire rage of problem domai. I is coclded ha he FTVIM is a powerfl ad efficie ool i obaiig he accrae solios as well as oher effecive merical mehods. - REFERENCES [] S.S. Norazar, A. Nazari - Golsha, M. Norazar, O he closed form solio of liear ad oliear Cach Reacio - Diffsio Vol. 5, No., Sprig 3
19 Amirkabir Ieraioal Joral of Sciece& Research (Modelig, Ideificaio, Simlaio & Corol) eqaios sig he hbrid of Forier Trasform ad Variaioal Ieraio Mehod. Phsics Ieraioal,, pp. 8 -,. [] M.A. Noor, K.I. Noor, S.T. Mohd - Di, Modified variaioal ieraio echiqe for solvig siglar forh - order parabolic parial differeial eqaios, Noliear Aalsis,, pp. 3 -, 9. [3] A.M. Wazwaz, Parial Differeial Eqaios ad Soliar Waves Theor, s Ed, Higher Edcaio Press, Beijig, Spriger Dordrech Heidelberg Lodo New York, ISBN: [] D.J. Evas, W.S. Yosef, A oe o solvig he forh - order parabolic eqaio b a AGE mehod, I. J. Comp. Mah,, pp. 93-9, 99. [5] D.J. Gorma, Free Vibraios Aalsis of Beams ad Shafs, Wile, New York, 95. [] A.M. Wazwaz, The variaioal ieraio mehod for solvig oliear siglar bodar vale Simlaio of siglar forh- order parial differeial eqaios sig he Forier rasform combied wih variaioal ieraio mehod problems arisig i varios phsical models, Comm Noliear Sci Nmer Simla,, pp ,. [] A.Q.M. Khaliq, E.H. Twizell, A famil of secod - order mehods for variable coefficie forh - order parabolic parial differeial eqaios, I. J. Comp. Mah. 3, pp. 3 -, 98. [8] A.M. Wazwaz, O he solio of he forh - order parabolic eqaios b he decomposiio mehod, I. J. Comp. Mah., 5, pp., 3 -, 995. [9] A.M. Wazwaz, Aalic reame for variable coefficie forh - order parabolic parial differeial eqaios, Appl. Mah. Comp., 3, pp. 9 -,. [] X.W. Zho, L.Yao, The variaioal ieraio mehod for Cach problems, Comp. Mah. Appl.,, pp. 5 -,. [] J.H. He, X.W, Variaioal ieraio mehod: New developme ad applicaios, Comp. Mah. Appl., 5, pp ,. Vol. 5, No., Sprig 3 5
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