A Novel Approach for Solving Burger s Equation

Transcription

1 Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: Vol. 9, Issue (December 4), pp Applicaios ad Applied Mahemaics: A Ieraioal Joural (AAM) A Novel Approach for Solvig Burger s Equaio Amrua Daga ad Vikas Pradha Deparme of Applied Mahemaics & Humaiies S.V. Naioal Isiue of Techology Sura-3957, Idia amrua.a.bhadari@gmail.com; pradha65@yahoo.com Received: Jauary 9, 3; Acceped: Augus 8, 3 Absrac The paper preses a ew aalyical mehod called Variaioal Homoopy Perurbaio Mehod (VHPM), which is a combiaio of he well-kow Variaioal Ieraio mehod (VIM) ad he Homoopy Perurbaio mehod (HPM) for solvig he oe-dimesioal Burger s equaio. Two es problems are preseed o demorae he efficiecy ad he accuracy of he proposed mehod.the umerical soluios obaied are compared wih he exac soluio. Furhermore, his mehod does o require spaial discreizaio or resricive assumpios ad is free from roud-off errors ad herefore reduce he umerical compuaio sigificaly. The resuls reveal ha he Variaioal Homoopy Perurbaio Mehod is very effecive ad coveie o solve oliear parial differeial equaios. Keywords: Burger s equaio; Variaioal Homoopy Perurbaio Mehod (VHPM); Variaioal Ieraio Mehod (VIM); Homoopy Perurbaio Mehod (HPM) MSC No.: 76T99, Iroducio I oe dimesio, Burger s equaio is give by u( x, ) u( x, ) u( x, ) u( x, ), a x b,, x x () where ε > is kiemaic viscosiy. The iiial codiio is give by 54

2 54 Amrua Daga ad Vikas Pradha u( x,) u ( x). Burger used he equaio () i a mahemaical modellig of urbulece (Burger 939; Burger 948). This equaio arises i may siuaios such as he heory of shock waves, urbulece problem ad coiuous sochasic processes. I has bee reaed as a opic of ceral ieres by may auhors for he cocepual udersadig of a class of physical flows ad for esig various umerical mehods. I is well kow ha he exac soluio of Burger s equaio ca be solved oly for resriced values of kiemaic viscosiy. The mahemaical properies of Burger s Equaio have bee sudied by Cole (95). Ozis (996) applied a direc variaioal mehod o geerae a limied form of he soluio of Burger s equaio. Ozis e al. (3) applied a simple fiie-eleme approach wih liear elemes o Burger s equaio reduced by he Hopf-Cole rasformaio. Aksa ad Ozdes (4) have reduced Burger s equaio o he sysem of oliear ordiary differeial equaios by discreizaio i ime ad solved he sysem by he Galerki mehod i each ime sep. Aksa e al. (6) applied he leas square mehod o he soluio of equaio (). Abu ad Solima (5) applied he Variaioal Ieraio Mehod o obai he soluio of equaio (). Noorzad (8) applied Homoopy perurbaio Mehod ad Variaioal Ieraio Mehod ad made compariso bewee boh he mehods. Recely, Deepi Mishra () applied he Homoopy Perurbaio Trasform Mehod o solve he o-liear Burger s equaio. I he prese paper, Burger s equaio is solved by he Variaioal Homoopy Perurbaio Mehod for wo es problems. Numerical ad graphical values are also preseed for ε = corary o he coservaive mehods which require he iiial ad boudary codiios, he VHPM provide a aalyical soluio by usig oly he iiial codiios.. The Variaioal Ieraio ad Homoopy Perurbaio Mehod To illusrae he basic coceps of he VIM ad HPM, we firs cosider he followig oliear differeial equaio Lu Nu g( x), () where L = A liear operaor, N = A oliear operaor, g(x) = A ihomogeeous erm. Accordig o he VIM (He, 999; Baiha, 7) we way cosruc a correcio fucioal as follows:

3 AAM: Ier. J., Vol. 9, Issue (December 4) 543 ( x) u ( x) u ( ) Lu Nu g d, (3) where () is a geeral Lagrage muliplier, which ca be ideified opimally via variaioal heory. The secod erm o he righ had side is called he correcio ad is cosidered as a resriced variaio, i.e., δ u =. By his mehod, firs i is required o deermie he Lagrage muliplier () ha will be ideified opimally. The successive approximaios u (, ) x, of he soluio u( x, ) will be readily obaied upo usig he deermied Lagrage muliplier ad ay selecive fucio u (, ) x. Cosequely, he soluio is give by u( x, ) lim u ( x, ). (4) The esseial idea of HPM is o iroduce a Homoopy parameer (say p), which akes values from o. Whe p =, he sysem of equaios is i sufficiely simplified form which ormally admis a raher simple soluio. As p gradually icreases o, he sysem goes hrough a sequece of deformaio, he soluio of each is close o ha a he previous sage of deformaio. Eveually a p =, he sysem akes he origial form of equaio ad he fial sage of deformaio gives he desired soluio. To illusrae he basic cocep of Homoopy Perurbaio Mehod, we cosider he followig oliear sysem of differeial equaios A(u) f ( r), r, (5) wih he boudary codiios: where u Bu,, r, (6) A = a differeial operaor, B = a boudary operaor, f (r) = a kow aalyic fucio, Γ = he boudary of he domai Ω. Geerally, he operaor A ca be divided io wo pars L ad N, where L is a liear operaor ad N is a oliear operaor. Therefore equaio (6) ca be rewrie as follows: L ( u) N( u) f ( r). (7)

4 544 Amrua Daga ad Vikas Pradha We cosruc a Homoopy v( r, p) : [,] R, which saisfies H( v, p) ( p)[ L( v) L( u )] p[ A( v) f ( r)], p[,], r (8) or H( v, p) L( v) L( u ) pl( u ) p[ N( v) f ( r)], (9) where u is iiial approximaio of equaio (6). I his mehod, usig he Homoopy parameer p, we ca express i erms of power series as v v pv p v. ()... Seig p = yields he approximae soluio of equaio () as below u lim v v v v.... () p The covergece of series () is discussed by Biazar ad Ghazvii (9). 3. The Variaioal Homoopy Perurbaio Mehod I he Homoopy Perurbaio Mehod, he basic assumpio is ha he soluios ca be wrie i erms of power series i p as i i... () i u p u u pu p u To illusrae he cocep of he Variaioal Homoopy Perurbaio Mehod, we cosider he geeral differeial equaio (5). We cosruc he correcio fucioal equaio (6) ad apply he Homoopy Perurbaio Mehod o obai, i () i i (, ) ( ) i,, i p u u x p N p u x g x d (3)

5 AAM: Ier. J., Vol. 9, Issue (December 4) 545 Thus he procedure is formulaed by he couplig of Variaioal Ieraio Mehod ad Homoopy Perurbaio Mehod. A compariso of like powers of p gives soluios of various orders. 4. Saeme of Problem 4.. Problem u u u u, x x (4) u( x,) x( x ). (5) Equaio (4) alog wih he iiial codiio (5) has he exac soluio (Cole, 95) u(x,)= exp( )si = exp( )cos = A x A A x wih Fourier coefficies 4 A exp 8 ( x x ) dx, 4 A exp 8 ( x x ) cos x dx. 4.. Problem Cosider he Burger s equaio (4) wih he iiial codiio ad homogeeous boudary codiio u( x,) si( x), x, u(, ) u(, ),. (6) The exac soluio (Cole, 95) of he Burger s equaio (4) wih iiial ad boudary codiios (6) is obaied as

6 546 Amrua Daga ad Vikas Pradha ae si( x) (, ), a ae cos( x) u x where he Fourier coefficies a ad respecively. a (=,,..) are defied by he followig equaios, exp ( ) cos( ) a x dx, exp ( ) cos( ) cos( ), (,,3, ). a x x dx 5. Mehod of Soluio 5.. Soluio of Problem Accordig o he Variaioal Homoopy Perurbaio Mehod, we cosruc he correcio fucioal for equaio (4) as u u u u x, u x, u d, x x (7) which yields he saioary codiios, ' ( ), ( ). Therefore, he geeral Lagrage muliplier ca be readily ideified as λ =, which yields he followig ieraio formula u u u u x, u x, u d x x (8) Applyig he variaioal Homoopy perurbaio mehod, we ge

7 AAM: Ier. J., Vol. 9, Issue (December 4) 547 ( ) x u pu p u f x p u pu p u u pu p u d p u pu p u d x Comparig he coefficie of like powers of p, we have p : u ( x, ) x( x ), x x p : u ( x, ) u u d u d x( x )( 3 x ) 6 x, p : u ( x, ) x( 7 4x 3 x ), 4 p : u( x, ) u ud u ud u d x x x x( 7 5x 5x 9 x ) x(7 9x 7x 5 x ) 6 x( 5 x ), p : u ( x, ) 3 x( 4 9x x x ). 4 6 Similarly, furher approximaios ca be obaied up o desired accuracy. The soluio becomes u( x, ) x( x )+ x( 7 4x 3 x ) 3 x( 4 9x x x ). (9) 5.. Soluio of Problem Accordig o Variaioal Homoopy Perurbaio Mehod, we cosruc he correcio fucioal for equaio, he Lagrage muliplier ca be deermied as λ =, which yields he followig ieraio formula. u u u u x, u x, u d x x. () Applyig he Variaioal Homoopy Perurbaio Mehod, we ge ( ) x u pu p u f x p u pu p u u pu p u d p u pu p u x.

8 548 Amrua Daga ad Vikas Pradha Comparig he coefficie of like powers of p, we have p : u( x, ) si x, p : u( x, ) u ud u d x x si cos si x xd xd, p : u ( x, ) ( si xcos x) si( x), p : u( x, ) u ud u ud u d x x x ( si x cos x) si( x) ( si x cos x) si( x) x p : u ( x, ) ( cos( x) ( 4cos( x)) si ( si cos ) si( ) d x x x x d x ( si x cos x) si( x) d, x cos( x) ( +cos( x))( cos( x) cos( x)))si( x). Similarly, furher approximaios ca be obaied up o desired accuracy. The soluio becomes u( x, ) si x ( si xcos x) si( x) ( cos( x) ( 4cos( x)) cos( x) ( +cos( x))( cos( x) cos( x)))si( x). () 6. Resuls ad Discussio Here a approximae soluio is obaied for wo problems ad are compared o he exac soluio wih pu emphasis o he accuracy of he prese mehod where he viscosiy value is oe. The abular compariso bewee he VHPM soluios ad he exac soluios a differe imes for specific value of x are summarized i Table for problem. I shows ha he soluios are i good harmoy wih hose of he exac soluio. The soluios obaied for problem ad problem are compared wih he exac soluio a paricular imes show i Table ad Table 3 respecively. The plos of he umerical soluios obaied for various values of ime ad space, cosiderig ε = are show i Figures -3.

9 AAM: Ier. J., Vol. 9, Issue (December 4) 549 Table. Compariso of VHPM soluios of (Problem ) wih Exac soluios a ε = a differe imes. x VHPM Exac Table. Compariso of VHPM soluiosof (Problem-) obaied wih Exac soluios a =. ad differe values of x x Exac VHPM Table 3. Compariso of VHPM soluio of (Problem ) wih exac soluio for ε = ad a differe ime levels =. =. x Exac VHPM Exac VHPM

10 55 Amrua Daga ad Vikas Pradha Figure. The hree dimesioal graph of problem for ε = Figure. Graph of he soluio of problem a ime =. ad =. for ε= Figure 3. Graph of u(x,) Vs x of problem a ime =. ad =. for ε =

11 AAM: Ier. J., Vol. 9, Issue (December 4) Coclusio I his paper, soluio of Burgers equaio is obaied by applyig Variaioal Homoopy Perurbaio Mehod wih specific iiial codiios. The Variaioal Homoopy Perurbaio Mehod is proved o be a effecive approach for solvig he Burger s equaio due o he excelle agreeme bewee he obaied umerical soluio ad he exac soluio. A compariso is made o show ha mehod has small size of compuaio i compariso wih he compuaioal size required i oher umerical mehods ad is rapid covergece shows ha mehod is reliable ad iroduces a sigifica improveme i solvig parial differeial equaio. REFERENCES Abdou, Mohamed ad Solima, Abd El-Maksoud (5). Variaioal Ieraio Mehod for Solvig Burger s ad Coupled Burger s Equaios, Joural of compuaioal ad Applied Mahemaics, Vol. 8, No.. hp:// Aksa, Emie N.ad Ozdes, Ali (4). A Numerical Soluio of Burgers equaio, Applied Mahemaics ad Compuaio, Vol. 56, No.. hp:// Aksa, Emie, Ozdes, Ali adturgu, Ozis (6).A Numerical Soluio of Burgers Equaio based o Leas Squares Approximaio, Applied Mahemaics ad Compuaio, Vol. 76, No.. hp:// Biazar, Jafar ad Ghazvii, Hossei (9). Covergece of he Homoopy Perurbaio Mehod for Parial Differeial equaios, Noliear Aalysis: Real World Applicaios, Vol., No.5. hp:// Chapai, Himashu V., Pradha, Vikas H., ad Meha, Maoj N. ().Numerical Simulaio of Burger s equaio usig Quadraic B-splies, Ieraioal Joural of Applied Mahemaics ad Mechaics (IJAMM), Vol. 8, No.. hp://ijamm.bc.ciyu.edu.hk/ijamm/oubox/yv8np8c pdf Cole, Julia D. (95). O a Quasi liear Parabolic Equaio Occurrig i Aerodyamics. Quarerly Applied Mahemaics, Vol. 9, No.. hp:// He, Ji-Hua (999).Variaioal Ieraio Mehod kid of o-liear Aalyical Techique: Some examples, Ieraioal Joural of No-Liear Mechaics, Vol. 34, No. 4. hp:// He, Ji-Hua (3). Homoopy perurbaio mehod: a New Noliear Aalyical Techique, Applied Mahemaics ad Compuaio, Vol. 35, No.. hp://

12 55 Amrua Daga ad Vikas Pradha Mishra, Deepi D., Pradha, Vikas H. ad Meha, Maoj N. (). Soluio of Burger s Equaio by Homoopy Perurbaio Trasform Mehod, Ieraioal Joural of Maageme, IT ad Egieerig (IJMRA), Vol., No. 7. hp:// Noor, Muhammad Aslam ad Mohyud-Di, Syed Tauseef (8). Variaioal Homoopy Perurbaio Mehod for solvig Higher Dimesioal Iiial Boudary Value Problems, Mahemaical Problems i Egieerig, Vol., No. hp:// Noorzad, Reza, Arash,Tahmasebi Poor ad Omidvar, Mehdi, (8). Variaioal Ieraio Mehod ad Homoopy Perurbaio Mehod for Solvig Burgers Equaio i Fluid Dyamics, Joural of Applied Scieces, Vol. 8, No.. hp:// Ozis, Turgu, Aksa Emie N. ad Ozdes Ali (3). A Fiie Eleme Approach for Soluio of Burgers equaio, Applied Mahemaics ad Compuaio, Vol.39, No.. hp:// Ozis, Turgu ad Ozdes, Ali (996). A direc Variaioal Mehods Applied o Burger s Equaio, Joural of Compuaio ad Applied Mahemaics, Vol. 7, No.. hp://