Maeaical and Copaional Applicaions, Vol., No., pp. 8-89,. Associaion for Scienific Researc ANALYTICAL ASPECT OF FOURTH-ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS Najeeb Ala Kan, Asa Ara, Maad Afzal and Aza Kan Deparen of Maeaics, Universi of Karaci, Karaci-77, Pakisan. Deparen of Maeaics & Basic Sciences, NEDUET, Karaci-77, Pakisan. Absrac- In is work, e ooop analsis eod (HAM) is applied o solve e for-order parabolic parial differenial eqaions. Tis eqaion pracicall arises in e ransverse vibraion probles. Te proposed ieraive scee finds e solion wio an discriizaion, linearizaion, or resricive asspions. Soe applicaions are given o verif e reliabili and efficienc of e eod. Te convergence conrol paraeer in e HAM solions as provided a convenien wa of conrolling e convergence region of series solions. I is also sown a e solions a are obained b Adoian decoposiion eod (ADM) and variaional ieraion eod (VIM) are special cases of e solions obained b HAM. Kewords- For-order parabolic parial differenial eqaions, Hooop analsis eod, Convergence conrol paraeer, ransverse vibraions.. INTRODUCTION Liao [] eploed e basic idea of e ooop in opolog o propose eod for nonlinear probles, nael, ooop analsis eod (HAM) [-]. Tis eod as an advanages over e classical eods; ainl, i is independen of an sall or large qaniies. So, e HAM can be applied no aer if governing eqaions and bondar/iniial condiions conain sall or large qaniies or no. Te HAM also avoids discreizaion and provides an efficien nerical solion wi ig accrac, inial calclaion, and avoidance of psicall nrealisic asspions. Frerore, e HAM alwas provides s wi a fail of solion epressions in e ailiar paraeer; e convergence region and rae of eac solion ig be deerined convenienl b e ailiar paraeer. Varios ecniqes for seeking analical solions o e nonlinear parial differenial eqaions are proposed, Adoian Decoposiion Meod (ADM) [-], variaion ieraion eod (VIM) [7] and ooop perrbaion eod (HPM) [8]. Te HAM is a general analic approac o ge series solion of varios pes of nonlinear eqaions, inclding algebraic eqaion, ordinar and parial differenial eqaions. In is paper, e for-order parabolic parial differenial eqaions wi variable coefficiens will be approaced analicall. Evans e al [9] as sdied for-order parabolic eqaion wi consan coefficiens via AGE eod. Evans [] sdied e second order parabolic eqaions via finie difference eod. Wazwaz [-] and Biazar e al [7] sdied for-order parabolic parial differenial eqaions wi variable coefficiens b ADM and VIM. Le s asse e following nonlinear differenial eqaion in for
8 N. A. Kan, A. Ara, M. Afzal and A. Kan [ τ ] Ν () were N is a nonlinear operaor,τ is an independen variable and ( τ) is e solion of eqaion. We define e fncion, φ( τ, p) as follows: Li p ( τ, p) ( τ) φ () were, p [, ] and ( τ) condiions is e iniial gess wic saisfies e iniial or bondar Li φ p ( τ, p) ( τ) () and b sing e generalized ooop eod, Liao s so-called zero-order deforaion () will be: ( p) L[ φ( τ, p) ( τ) ] ph ( τ) N[ φ( τ, p) ] () paraeer, e ailiar fncion H ( τ ), e iniial gess ( τ) and e ailiar linear operaor L. Tis freedo plas an iporan role in esablising e kesone of validi and fleibili of HAM as sown in is paper. Ts, wen p increases fro τ, p τ τ. o e solion φ canges beween e iniial gess and e solion Te Talor series epansion of φ ( τ, p) wi respec o p is ( τ p) ( τ) p, τ φ () and ( ) ( τ ; p) φ τ () p p Were ( ) ( τ) for brevi is called e order of deforaion derivaion wic reads: ( ) ( τ ; p) φ τ (7) p p I's clear a if e ailiar paraeer will becoe: ( p) L[ ( τ, p) ( τ) ] p N[ φ( τ, p) ] H τ, en () ailiar fncion φ (8)
Analical Aspec of For-Order Parabolic Parial Differenial Eqaions 8 Tis saeen is coonl sed in HPM procedre. Indeed, in HPM we solve e nonlinear differenial eqaion b separaing an Talor epansion er. Now, we define e vecor of v v v v v v (9) {,,,..., } According o e definiion in (7), e governing eqaion and e corresponding iniial condiions can be dedced fro zero-order deforaion eqaion (). Differeniaing () ies wi respec o e ebedding paraeer p and seing p and finall dividing b, we will ave e so-called order deforaion eqaion in e for: [ ( Ω) ( Ω) ] R[ ] L were r χ () R r ( ) [ φ( Ω; p) ] N p p () and χ > B appling inverse linear operaor o bo sides of eqaion, (), we can easil solve e eqaion and cope e solions b copaional sofware MATHEMATICA. Nerical resls reveal a e HAM is eas o ipleen and redces e copaional work o a angible level wile sill ainaining a ver iger level of accrac. For e sake of coparison, we ake e sae eaples as sed in [-,7].. EXEMPLIFICATION Consider e following case of one-diensional eqaion (,) (,), < <, > () sbjec o e iniial and bondar condiions:. (, ), (, ),, S in,, S in (,) S in,, S in () To solve is eaple b HAM, we ake
8 N. A. Kan, A. Ara, M. Afzal and A. Kan (,) () (,; p) Φ(,; p) Φ N( Φ ) () and fro (), we ave R v ( N( )) (,) (,) () B appling () wi (), we ave (7) (8) ( ) 7 (9) ( ) ( ) Fig.. Te -crve of e and 8 (Tick, Dased) Fig.. Coparison beween HAM ( (ADM,VIM), order approiaion for Eaple a.9,. 9. 9,. 88 and eac), solion a. 9 for Eaple afer -ieraions (Red, Tick, Aoaic, Dased) Te res of e coponens of e HAM solion can be obained. Te for er approiae solion is given b 7 () Wen, e solion obained b [,7].
Analical Aspec of For-Order Parabolic Parial Differenial Eqaions 8 Consider e following case of one-diensional eqaion (,) (,) Sin, < <, > () sbjec o e iniial and bondar condiions: (, ) Sin, (, ) Sin, (,), (, ). (,) e ( Sin), (,) e Sin () To solve is proble b HAM, we ake (,) ( )( Sin) () (,; p) Φ(,; p) Φ N( Φ ) () Sin and fro (), we ave R v ( N( )) (,) (,) Sin () B appling (), we ave ( Sin) () ( ) ( ) ( Sin) (7) ( ) ( ) ( ) ( ) 7 ( Sin) (8)
8 N. A. Kan, A. Ara, M. Afzal and A. Kan Fig.. Te -crve of e and 8 (Tick, Dased) Fig.. Coparison beween HAM ( (ADM, VIM) order approiaion for Eaple. a.9,. 9.,. and eac), solions a. 9 for Eaple afer -ieraions (Red, Tick, Aoaic, Dased) Te res of e coponens of e HAM solion can be obained. Te for er approiae solion is given b - ( ) ( ) ( ) 7 7 ( Sin) (9) Wen, e solion obained b [,7]. Consider e following case of wo-diensional eqaion (,,) (,,) (,,), <, <, > () sbjec o e iniial and bondar condiions: (,, ), (,, ).,,, S in, (,,) S in,.,, S in,.,, S in () (,,) S in, (,,) S in To solve is eaple b HAM, we ake
Analical Aspec of For-Order Parabolic Parial Differenial Eqaions 87, () p,;, p,;, p,;, N Φ Φ Φ Φ () and fro (), we ave,,,,,, N R v () B appling (), we ave () () 7 (7) Te res of e coponens of e HAM solion can be obained. Te for er approiae solion is given b Fig.. Te -crve of e and 8 (Tick, Dased) Fig.. Coparison beween HAM ( (ADM, VIM) order approiaion for Eaple. a 9..8, 8 9.,. and eac), solions 8..9, for Eaple afer -ieraions (Red, Tick, Aoaic, Dased) Te res of e coponens of e HAM solion can be obained. Te for er approiae solion is given b
88 N. A. Kan, A. Ara, M. Afzal and A. Kan 7 (8) Wen, e solion obained b [,7].. FINAL REMARKS In is paper, e HAM sed o obain e analical solions of for-order parabolic parial differenial eqaions wi variable coefficiens. Te copaions associaed wi e eaples in is work were perfored b sing MATHEMATICA. HAM provides s wi a convenien wa o conrol e convergence of approiaion series b adaping wic is a fndaenal qaliaive difference in analsis beween HAM and oer eods. Te proposed eod is sccessfll ipleened. Tere are wo iporan poins o ake ere. Firs, e eod was sed in a direc wa wio sing linearizaion, perrbaion or resricive asspion. Second, e HAM avoids e cbersoe of e copaional eods wile sill ainaining e iger level of accrac. Te coparison beween e HAM, ADM and VIM ade and i is fond a HAM is ore effecive an ADM and VIM, a leas for ose pariclar eaples. Te sggesed algori is qie efficien and is pracicall well sied for se in ese probles, wic are arising in ransverse of a vibraing bea [].. REFERENCES. S. J. Liao, Te proposed ooop analsis ecniqes for e solion of nonlinear probles. P.D. disseraion, Sangai Jiao Tong Universi, Sangai, 99 (in Englis).. S. J. Liao, Beond perrbaion: Inrodcion o e ooop analsis eod. Boca Raon: CRC Press, Capan and Hall,.. S. Abbasband, Te applicaion of ooop analsis eod o nonlinear eqaions arising in ea ransfer, Ps Le A, 9,.. A. M. Wazwaz, Analical reaen of for variable coefficiens for order parabolic parial differenial eqaions, Appl. Ma. Cop., 9-7,.. A. M. Wazwaz, Eac solions of variable coefficiens for order parabolic parial differenial eqaions in iger diension spaces, Appl. Ma. Cop.,, -.. H. Haddadpor, An eac solion for variable coefficiens for-order wave eqaion sing e Adoian eod, Ma. and Cop. Model. (-), -,. 7. J. Biazar, H. Gazvini, He s variaional eod for for-order parabolic eqaion, Cop. Ma. Appl.,7-, 7. 8. N. A. Kan, A. Ara, A. Maood, S. A. Ali, Analical sd of Navier-Sokes eqaion wi fracional orders sing He s ooop perrbaion and variaional ieraion eods, In. J. of Nonlinear Sc. and N. Sil., 7-, 9. 9. D. J. Evans, A sable eplici eod for e finie difference solion of for order parabolic parial differenial eqaions, Cop. J. 8, 8-87, 9.
Analical Aspec of For-Order Parabolic Parial Differenial Eqaions 89. D. J. Evans, W. S. Yosef, A noe on solving e for order parabolic eqaion b e Age eod. In J. Cop. Ma., 9-97, 99.. D. J. Goran, Free vibraions analsis of Bea and safs, Wile, New York 97.