Research Article Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations

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1 Applied Maheaics Volue 23, Aricle ID 57956, pages hp://ddoiorg/55/23/57956 Research Aricle Couple of he Variaional Ieraion Mehod and Legendre Waveles for Nonlinear Parial Differenial Equaions Fukang Yin, Junqiang Song, Xiaoqun Cao, and Fengshun Lu 2 College of Copuer, Naional Universiy of Defense Technology, Changsha 473, China 2 China Aerodynaics Research and Developen Cener, Mianyang, Sichuan 62, China Correspondence should be addressed o Fukang Yin; yinfukang@nudeducn Received 7 Ocober 22; Revised 27 Deceber 22; Acceped 27 Deceber 22 Acadeic Edior: Francisco Chiclana Copyrigh 23 Fukang Yin e al This is an open access aricle disribued under he Creaive Coons Aribuion License, which peris unresriced use, disribuion, and reproducion in any ediu, provided he original work is properly cied This paper develops a odified variaional ieraion ehod coupled wih he Legendre waveles, which can be used for he efficien nuerical soluion of nonlinear parial differenial equaions (PDEs) The approiae soluions of PDEs are calculaed in he for of a series whose coponens are copued by applying a recursive relaion Block pulse funcions are used o calculae he Legendre waveles coefficien arices of he nonlinear ers The ain advanage of he new ehod is ha i can avoid solving he nonlinear algebraic syse and sybolic copuaion Furherore, he developed vecor-ari for akes i copuaionally efficien The resuls show ha he proposed ehod is very effecive and easy o ipleen Inroducion Nonlinear phenoena are of fundaenal iporance in applied aheaics and physics and hus have araced uch aenion I is well known ha os engineering probles are nonlinear, and i is very difficul o achieve he soluion analyically or nuerically The analyical ehods coonly used o solve he are very resriced, while he nuerical echniques involving discreizaion of he variables on he oher hand give rise o rounding off errors Considerable aenion has been paid o developing an efficien and fas convergen ehod Recenly, several approiae ehods areinroducedofindhenuericalsoluionsofnonlinear PDEs, such as Adoian s decoposiion ehod (ADM) 6], hooopy perurbaion ehod (HPM) 7 2], hooopy analysis ehod (HAM) 3, 4], variaional ieraion ehod (VIM) 5 23], and waveles ehod 24 29] The variaional ieraion ehod (VIM) proposed by He 5 23] hasbeenshownobeveryefficienforhandling a wide class of physical probles 6 8, 3 4] If he eac soluion of he nonlinear PDEs eiss, he VIM gives rapidly convergen successive approiaions; oherwise, a few approiaions can be used for nuerical purposes In order o iprove he efficiency of hese algorihs, several odificaions, such as variaional ieraion ehod using He s Polynoials 42 48] or using Adoian s Polynoials 49 54], have been developed and successfully applied o various engineering probles However, since he variaional ieraion ehod provides he soluion as a sequence of ieraes, is successive ieraions ay be very cople, so ha he resuling inegraions in is ieraive relaion ay be ipossible o perfor analyically In recen years, waveles have found heir way ino any differen fields of science and engineering Various waveles 24 29] havebeenusedforsudyingprobleswihgreaer copuaional copleiy and proved o be powerful ools o eplore a new direcion in solving differenial equaions Unlike he variaional ieraion ehod ha requires sybolic copuaions, he waveles ehod convers he PDE ino algebraic equaions by he operaional arices, which canbesolvedbyanieraiveprocedureiisworhyo enion here ha he ehod based on operaional arices of an orhogonal funcion for solving differenial equaions is copuer oriened The proble wih his approach is ha he algebraic equaions ay be singular and nonlinear Recenly, soe efficien odificaions of ADM (using 55, 56]) and VIM or HAM 57] (using Legendre polynoials) are presened o approiae nonhoogeneous ers in

2 2 Applied Maheaics nonlinear differenial equaions Moivaed and inspired by he ongoing research in hese areas, we ipleen Legendre waveles wihin he fraework of VIM o faciliae he copuaional work of he ehod while sill keeping he accuracy The reainder of he paper is organized as follows Secion 2 inroduces he VIM In Secion 3, we describe he basic forulaion of Legendre waveles and he operaional ari required for our subsequen developen In Secion 4, weproposeanewvariaionalieraionehod using Legendre waveles (VIMLW) In order o deonsrae he validiy and applicabiliy of VIMLW, four eaples are given in Secion5 Finally, a brief suary is presened 2 Variaional Ieraion Mehod This secion inroduces he basic ideas of variaional ieraion ehod (VIM) Here a descripion of he ehod 5 23] is givenohandlehegeneralnonlinearproble: L (u) +N(u) =g(), () polynoials, and is he noralized ie They are defined on he inerval, ) as follows: ψ n () = { +/22 (k/2) L (2 k n), for n n+, { 2 k 2 k {, oherwise, (4) where =,,2,,M, n=,2,,2 k Thecoefficien +/2is for orhonoraliy, he dilaion paraeer is a= 2 k, and he ranslaion paraeer b= n2 k Here,L () are he well-known Legendre polynoials of order defined on he inerval, ] Afuncionf() defined over, ) ay be epanded by Legendre wavele series as + + f () = c n ψ n (), (5) n= = where L is a linear operaor, N is a nonlinear operaor, and g() is a known analyic funcion According o He s VIM, we can consruc a correcion funcional as follows: wih c n = f(),ψ n (), (6) u n+ () =u n () + λ (τ) {L (u n (τ))+n( u n (τ)) g(τ)}, n, (2) where λ is a general Lagrange uliplier which can be opially idenified via variaional heory and u n is a resriced variaion which eans δ u n = Therefore, he Lagrange uliplier λ should be firs deerined via inegraion by pars The successive approiaion u n () (n ) of he soluion u() will be readily obained by using he obained Lagrange uliplier and any selecive funcion u The zeroh approiaion u ay selec any funcion ha jusees,aleas,heiniialandboundarycondiions Wih λ deerined, several approiaions u n (), n, follow iediaely Consequenly, he eac soluion ay be obained as u () = li n u n () (3) The VIM depends on he proper selecion of he iniial approiaion u () Finally, we approiae he soluion of he iniial value proble () byhenh-order er u n () I hasbeenvalidaedhavimiscapableofeffecively,easily, and accuraely solving a large class of nonlinear probles 3 Legendre Waveles 3 Legendre Waveles Legendre waveles ψ n () = ψ(k, n,, ) have four arguens: k is any posiive ineger, n = 2n (n =,2,3,,2 k ), is he order for Legendre in (6);, denoes he inner produc If he infinie series in (5) isruncaed,henicanbe wrien as f () = 2 k M n= = c n ψ n () =C T Ψ (), (7) where C and Ψ() are 2 k M arices given by C () = c,c,,c M,c 2,,c 2M,, c 2 k,,c 2 k M ]T, Ψ () = ψ (),ψ (),,ψ M (),, ψ 2 k (),,ψ 2 k M ()] T A wo-diensional funcion f(, ) defined over, ), ) ay be epanded by Legendre wavele series as wih f (, ) = 2 k M c ij = 2 k M j= (8) (9) c ij ψ i () ψ j () =Ψ T () CΨ (), () f (, ) ψ i () d f (, ) ψ j () d () Equaion () can be wrien ino he discree for (in ari for) by f (, ) =Ψ T () CΨ (), (2)

3 Applied Maheaics 3 where C is a 2 k M 2 k M ari given by c, c, c,2 k M c, c, c,2 C= k M (3) ] c 2 k M, c 2 k M, c 2 k M,2 k M] The inegraion and derivaive operaion arices of he Legendre waveles have been derived in 58, 59] The inegraion of he vecor Ψ() defined in (9) canbe obained as Ψ (s) ds PΨ (), (4) where P is a 2 k M 2 k M ari given by 58] The derivaive of he vecor Ψ() can be epressed by dψ () d =DΨ(), (5) where D is he 2 k M 2 k M operaional ari of derivaive given by 59] The inegraion of u(, ) = Ψ T ()CΨ() wih respec o variable canbeepressedas u (, τ) dτ = (Ψ T () CΨ (τ))dτ =Ψ T () C( Ψ (τ) dτ) = Ψ T () CPΨ () (6) Siilarly, he inegraion of u(, ) = Ψ T ()CΨ() wih respec o variable canbeepressedas u (τ, ) dτ = (Ψ T (τ) CΨ ())dτ = ( Ψ T (τ) dτ) CΨ () =Ψ T () P T CΨ () (7) The derivaive of u(, ) = Ψ T ()CΨ() wih respec o variable canbeepressedas u (, ) = (ΨT () CΨ ()) =Ψ T () CDΨ () =Ψ T Ψ () () C (8) Siilarly, he derivaive of u(, ) = Ψ T ()CΨ() wih respec o variable canbeepressedas u (, ) = (ΨT () CΨ ()) =Ψ T () D T CΨ () = ΨT () CΨ () (9) 32 Block Pulse Funcions The block pulse funcions (BPFs) for a coplee se of orhogonal funcions ha are defined on he inerval, b) by b i () = { i, { b < i b {, elsewhere (2) for i =,2,, I is also known ha for arbirary absoluely inegrable funcion f() on, b) can be epanded in block pulse funcions: in which f () ξ T B (), (2) ξ T =f,f 2,,f ], B () =b (),b 2 (),,b ()], (22) where f i are he coefficiens of he block pulse funcion given by f i = b b f () b i () d = (i/)b b ((i )/)b f () b i () d The eleenary properies of BPFs are as follows where (23) () Disjoinness: he BPFs are disjoined wih each oher in he inerval,t): for i,j=,2,, b i () b j () =δ ij b i () (24) (2) Orhogonaliy: he BPFs are orhogonal wih each oher in he inerval,t): for i,j=,2,, T b i () b j () d = hδ ij (25) (3) Copleeness: he BPFs se is coplee when approaches infiniy This eans ha for every f L 2 (, T)), when approaches o he infiniy, Parseval s ideniy holds: T f 2 () d = f 2 i b i () 2, (26) f i = T h f () b i () d (27) Definiion Le A and B be wo arices of,hena B=(a ij b ij )

4 4 Applied Maheaics Lea 2 Assuing ha f() and g() are wo absoluely inegrable funcions, which can be epanded in block pulse funcion as f() = FB(), andg() = GB() respecively, hen one has where H=F G f () g () =FB() B T () G T =HB(), (28) Proof According o he disjoinness propery of BPFS in (6), we have FB () B T () G T =f g φ () f 2 g 2 φ 2 () f g φ 2 ()] =HB() (29) Lea 3 Le f(,) and g(, ) be wo absoluely inegrable funcions, which can be epanded in block pulse funcion as f(, ) = B T ()FB() and g(, ) = B T ()GB(), respecively, one has where H=F G f (, ) g (, ) =B T () HB (), (3) 33 Nonlinear Ter Approiaion The Legendre waveles can be epanded ino -se of block pulse funcions as Taking he collocaion poins as follow, Ψ () =Φ B () (3) i = i /2 2 k M,,2,,2k M (32) The -square Legendre ari Φ is defined as Φ Ψ( ) Ψ( 2 ) Ψ( 2 k M )] (33) The operaional ari of produc of Legendre waveles canbeobainedbyusingheproperiesofbpfslef(, ) and g(, ) be wo absoluely inegrable funcions, which can be epanded in Legendre waveles as f(, ) = Ψ T ()FΨ() and g(, ) = Ψ T ()GΨ(),respecively Fro (3), we have f (, ) =Ψ T () FΨ () =B T () Φ T FΦ B (), g (, ) =Ψ T () GΨ () =B T () Φ T GΦ B (), and le F b =Φ T FΦ, G b =Φ T GΦ, H b =F b G b By eploying Lea3,we ge f (, ) g (, ) =B T () H b B () =B T () Φ T inv (ΦT )H binv (Φ ) Φ B () =Ψ T () HΨ (), where H=inv(Φ T )H binv(φ ) (34) (35) 4 Variaional Ieraion Mehod Using Legendre Waveles In his secion, we presen a new odificaion of variaional ieraion ehod using Legendre waveles (called VIMLW) This algorih can be ipleened for solving nonlinear PDEs effecively To deduce he basic relaions of our proposed algorih, consider he following fors of iniial value probles: L u (, )] +Nu (, )] =g(, ),, ], >, (36) where L and N are linear operaor and nonlinear operaor, respecively, and g(, ) is a known analyic funcion, subjec o he iniial condiion u(, ) I should be noed here ha Lu(, )] conains he er u/,where is a posiive ineger According o he radiional VIM, we can consruc he correcion funcional for (36)as u k+ (, ) =u k (, ) + λl(u k (, τ))+n(u k (, τ)) g (, τ)]dτ (37) The Lagrange uliplier of (37)is λ (, τ) = ( τ) ( )! = ( ) (τ ) (38) ( )! In order o iprove he perforance of VIM, we inroduce Legendre waveles o approiae u k (, ) and he nonhoogeneous er g(, ) as u k (, ) =Ψ T () C k Ψ (), g(, ) =Ψ T () GΨ () (39) Now for he nonlinear par, by nonlinear er approiaion described in Secion 33,we have N u k (, )] =Ψ T () N k Ψ (), (4) where N is ari of order 2 k M 2 k M For he linear par, we have Lu k (, )] =Ψ T () L k Ψ (), (4) where L is a ari of order 2 k M 2 k M Then he ieraion forula (37)can be consruced as Ψ T () C k+ Ψ ()=Ψ T () C k Ψ () If λ is consan, we have + λψ T () L k +N k G]Ψ(τ) dτ (42) C k+ =C k +λl k +N k G] P (43)

5 Applied Maheaics 5 Table : Nuerical values when = 25, 5, 75, and for (52) VIM VIMLW Eac VIM VIMLW Eac When λ is a funcion of τ,helegendrewavelesareused o approiae λ(τ) as Subsiuing (44)ino(42), we have Ψ T () C k+ Ψ () Since =Ψ T () C k Ψ () λ (, τ) =Ψ T () SΨ (τ) (44) + Ψ T () L k +N k G]Ψ(τ) Ψ T (τ) S T Ψ () dτ (45) φ i B i (τ) φ φ 2 φ B (τ) φ 2 φ 22 φ 2 B 2 (τ) φ 2i B i (τ) Ψ (τ) = ] =, ] φ φ 2 φ ] B (τ) ] ] φ i B i (τ) ] (46) we ge Ψ (τ) Ψ T (τ) = = φ i B i () φ 2i B i () ] φ i B i () ] φ i B i () j= ( 2 φ ji B i ()) φ 2i B i () φ i B i ()] (47) According o he propery of block pulse funcions, we obain Ψ (τ) Ψ T (τ) where = φ 2 ji B i () j= φ 2 φ 2 2 φ 2 φ 2 2 φ 2 22 φ 2 B () 2 B 2 () = ] ] φ 2 φ2 2 ] φ2 B ()] ] φ 2 φ 2 2 φ 2 φ 2 2 φ 2 22 φ 2 2 = inv (Φ )Ψ() ] φ 2 φ2 2 ] φ2 ] =HΨ() ], (48) φ 2 φ 2 2 φ 2 φ 2 2 φ 2 22 φ 2 2 H= inv (Φ ) (49) φ 2 φ2 2 ] φ2 ] Subsiuing (48)ino(45), we have Ψ T () C k+ Ψ () =Ψ T () C k Ψ ()+ Ψ T () L k +N k G] HΨ(τ) ]S T Ψ () dτ

6 6 Applied Maheaics Table 2: Nuerical values when = 25, 5, 75, and for (54) VIM VIMLW Eac VIM VIMLW Eac Eac soluion 6 4 VIMLW soluion Figure : Eac soluion and VIMLW approiae soluion of Eaple 4 =Ψ T () C k Ψ () +Ψ T () L k +N k G] HPΨ() ]S T Ψ () =Ψ T () C k Ψ () +Ψ T () L k +N k G] HPΨ() Ψ T () S ] T =Ψ T () C k Ψ () +Ψ T () L k +N k G] HPHΨ() s, where s= ] S ] T Finally, we ge he ieraion forula as follows: (5) 5 Nuerical Eaples To deonsrae he effeciveness and good accuracy of he VIMLW, four differen eaples will be eained Eaple 4 Consider he regularized long-wave (RLW) equaion 39]: u u +( u2 2 ) =, <<, > (52) C k+ =C k +L k +N k G] HPH ] S ] T (5) wih he iniial condiion u(, ) = and he eac soluion is u(, ) = (/( + ))

7 Applied Maheaics Eac soluion VIMLW soluion Figure 2: Eac soluion and VIMLW approiae soluion of Eaple Eac soluion 5 VIMLW soluion Figure 3: Eac soluion and VIMLW approiae soluion of Eaple 6 By assuing u k (, ) = Ψ T ()C k Ψ() and fro (52), we have Lu k ]= u k 3 u k 2 =ΨT () L k Ψ (), Nu k ]=u k u k =ΨT () N k Ψ (), where L k =C k D (D T ) 2 C k D, N k =(D T C k ) C k (53) We uilize he ehods presened in his paper o solve (52) wihm=6and k= Table shows he approiae soluions for (52) obained for differen poins using he variaional ieraion and VIMLW ehod Figure presens heeacsoluionandvimlwapproiaesoluionof Eaple 4 Noe ha only he fifh-order er of heir soluions is used in evaluaing he approiae soluions for Eaple 4 We can see ha he approiae soluion obained wih VIMLW gives alos he sae resuls as ha

8 8 Applied Maheaics Eac soluion 5 VIMLW soluion Figure 4: Eac soluion and VIMLW approiae soluion of Eaple 7 wih VIM I indicaes ha he approiae soluion is quie closeoheeacone Eaple 5 Consider he following equaion 39]: u + 2 uu =u, <<, > (54) wih he iniial condiions u(, ) = 2, and he eac soluion is u(, ) = ( 2 /( + )) By assuing u k (, ) = Ψ T ()C k Ψ() and fro (54), we have L k u k ]= u k 3 u k 3 =ΨT () L k Ψ (), N k u k ]= u k 2 2 u k 2 =ΨT () N k Ψ (), (55) where L k =(D T ) 3 C k D, N k = ((/2)(D T ) 2 C k ) C k We eploy he ehods presened in his paper o solve (54) wihm = 6 and k = Thenuericalresuls are presened in Table 2 and shown in Figure 2 Iisobe noed ha only he fifh-order ers are used in evaluaing he approiae soluions The resuls obained using he VIMLW are in good agreeen wih he resuls of VIM Eaple 6 We consider he following equaion 4]: u (, ) +u(, ) u (, ) =g(, ), >, R, <α (56) wih he iniial condiions u(, ) = and he eac soluion is u(, ) =,whereg(,)=+ 2 By assuing u k (, ) = Ψ T ()C k Ψ(), g(, ) = Ψ T () GΨ(),wehave L k u k ]=u (, ) g(, ) =Ψ T () L k Ψ (), N k u k ]=u k u k =ΨT () N k Ψ (), where L k =C k D G, N k =C k (D T C k ) (57) Table 3 shows he approiae soluions for (56) wih M=6and k=using he VIM and he VIMLW ehods and he resuls are ploed in Figure 3 Iisobenoedha only he fourh-order ers of VIM and VIMLW are used in evaluaing he approiae soluions in Table 3We observe ha he approiae soluion of (56) wihvimlwgives analogous resuls o ha obained by VIM, which shows ha he approiae soluion reains closed for o he eac one Eaple 7 Consider he following Burgers-Poisson (BP) equaion of he for 4]: u u +u +uu =(3u u +uu ), <<, > (58) wih he iniial condiions u(, ) =, and he eac soluion is u(, ) = ( + )/( + )

9 Applied Maheaics 9 Table 3: Nuerical values when = 25, 5, 75, and for (56) VIM VIMLW Eac VIM VIMLW Eac Table 4: Nuerical values when = 25, 5, 75, and for (58) VIM VIMLW Eac VIM VIMLW Eac By assuing u k (, ) = Ψ T ()C k Ψ(),wehave L u k ] = u k 3 u k 2 + u k =ΨT () L k Ψ (), (59) where L k =C k D (D T ) 2 C k D+D T C k And u Nu k ]=u k k 3 u k 2 u k 2 u 3 u k k 3 =ΨT () N k Ψ (), (6) where N k = C k (D T C k ) 3(D T C k ) (D T ) 2 C k ] C k (D T ) 3 C k ] Table 4 shows he approiae soluions o (58) wih M = 6 and k = wih VIM and VIMLW, and Figure 4 presens he Eac soluion and VIMLW approiae soluion of Eaple 7 Only he fourh-order ers are used in evaluaingheapproiaesoluionsintable 4FroTable 4 and Figure 4 he approiae soluion of he given Eaple 7 by using VIMLW is in good agreeen wih he resuls of VIM and i clearly appears ha he approiae soluion reains closed for o eac soluion 6 Conclusion A new odificaion of variaional ieraion ehod using Legendre waveles is proposed and eployed o solve a nuber of nonlinear parial differenial equaions The proposed ehod can give approiaions of higher accuracy and closed for soluions if eised There are four iporan poins o ake here Firs, unlike he VIM, he VIMLW can easily overcoe he difficuly arising in he evaluaion inegraion and he derivaive of nonlinear ers and does no need sybolic copuaion Second, by using he properies of BPFs, operaional arices of produc of Legendre waveles are derived and uilized o deal wih nonlinear ers Third, copared wih Legendre waveles ehod, he VIMLW only needs a few ieraions insead of solving a syse of nonlinear algebraic equaions Fourh and os iporan, VIMLW is copuer oriened and can use eising fas algorihs o reduce he copuaion cos Acknowledgens This work is suppored by he Naional Naural Science Foundaion of China (Gran no 4563) The auhors are very graeful o he reviewers for carefully reading he paper and for hier coens and suggesions which have iproved he paper References ] G Adoian, Solving Fronier Probles of Physics: The Decoposiion Mehod, vol6offundaenal Theories of Physics, Kluwer Acadeic Publishers, Dordrech, The Neherlands, 994 2] G Adoian, A review of he decoposiion ehod in applied aheaics, Maheaical Analysis and Applicaions, vol 35, no 2, pp 5 544, 988

10 Applied Maheaics 3] G Adoian, Soluions of nonlinear P D E, Applied Maheaics Leers, vol, no 3, pp 2 23, 998 4]QEsaili,ARaiar,EAlizadeh,andDDGanji, An approiaion of he analyical soluion of he Jeffery-Hael flow by decoposiion ehod, Physics Leers A,vol372,no 9, pp , 28 5] A-M Wazwaz, A new algorih for calculaing Adoian polynoials for nonlinear operaors, Applied Maheaics and Copuaion, vol, no, pp 53 69, 2 6] S Moani and Z Odiba, Analyical soluion of a iefracional Navier-Sokes equaion by Adoian decoposiion ehod, Applied Maheaics and Copuaion, vol 77, no2, pp , 26 7] J-H He, Hooopy perurbaion echnique, Copuer Mehods in Applied Mechanics and Engineering,vol78,no3-4,pp , 999 8] J-H He, A coupling ehod of a hooopy echnique and a perurbaion echnique for non-linear probles, Inernaional Non-Linear Mechanics,vol35,no,pp37 43,2 9] J-H He, The hooopy perurbaion ehod nonlinear oscillaors wih disconinuiies, Applied Maheaics and Copuaion,vol5,no,pp ,24 ] S T Mohyud-Din and M A Noor, Hooopy perurbaion ehod for solving fourh-order boundary value probles, Maheaical Probles in Engineering, vol27,aricleid 9862, 5 pages, 27 ] S T Mohyud-Din and M A Noor, Hooopy perurbaion ehod for solving parial differenial equaions, Zeischrif für Naurforschung A,vol64,no3-4,pp57 7,29 2] S T Mohyud-Din and M A Noor, Hooopy perurbaion ehod and Padé approians for solving Flierl-Peviashivili equaion, Applicaions and Applied Maheaics, vol3,no2, pp ,28 3] S J Liao, An approiae soluion echnique no depending on sall paraeers: a special eaple, Inernaional Journal of Non-Linear Mechanics,vol3,no3,pp37 38,995 4] S J Liao, Boundary eleen ehod for general nonlinear differenial operaors, Engineering Analysis wih Boundary Eleens,vol2,no2,pp9 99,997 5] J H He, Variaional ieraion ehod a kind of non-linear analyical echnique: soe eaples, Inernaional Non-Linear Mechanics, vol 34, no 4, pp , 999 6] J-H He, Variaional ieraion ehod for auonoous ordinary differenial syses, Applied Maheaics and Copuaion,vol4,no2-3,pp5 23,2 7] J-H He, Variaional principles for soe nonlinear parial differenial equaions wih variable coefficiens, Chaos, Solions &Fracals,vol9,no4,pp847 85,24 8] J-H He and X-H Wu, Consrucion of soliary soluion and copacon-like soluion by variaional ieraion ehod, Chaos, Solions & Fracals,vol29,no,pp8 3,26 9] J-H He and X-H Wu, Variaional ieraion ehod: new developen and applicaions, Copuers & Maheaics wih Applicaions, vol 54, no 7-8, pp , 27 2] J-H He, Variaional ieraion ehod soe recen resuls and new inerpreaions, Copuaional and Applied Maheaics,vol27,no,pp3 7,27 2] J H He, G C Wu, and F Ausin, The variaional ieraional ehod which should be follow, Nonlinear Science Leers A, vol,no,pp 3,2 22] J-H He, Soe asypoic ehods for srongly nonlinear equaions, Inernaional Modern Physics B, vol2, no,pp4 99,26 23] J H He, Asypoic ehods for soliary soluions and copacons, Absrac and Applied Analysis, vol 22, Aricle ID 96793, 3 pages, 22 24] G Hariharan, K Kannan, and K R Shara, Haar wavele ehod for solving Fisher s equaion, Applied Maheaics and Copuaion,vol2,no2,pp ,29 25] Ü Lepik, Nuerical soluion of evoluion equaions by he Haar wavele ehod, Applied Maheaics and Copuaion, vol85,no,pp695 74,27 26] S G Venkaesh, S K Ayyasway, and S Raja Balachandar, The Legendre wavele ehod for solving iniial value probles of Brau-ype, Copuers & Maheaics wih Applicaions, vol 63, no 8, pp , 22 27] S A Yousefi, Legendre waveles ehod for solving differenial equaions of Lane-Eden ype, Applied Maheaics and Copuaion,vol8,no2,pp47 422,26 28] RKPandey,NKuar,ABhardwaj,andGDua, Soluionof Lane-Eden ype equaions using Legendre operaional ari of differeniaion, Applied Maheaics and Copuaion, vol 28, no 4, pp , 22 29] F Yin, J Song, F Lu, and H Leng, A coupled ehod of Laplace ransfor and legendre waveles for Lane-Eden-ype differenial equaions, JournalofAppliedMaheaics,vol 22,AricleID6382,6pages,22 3] L M B Assas, Variaional ieraion ehod for solving coupled-kdv equaions, Chaos, Solions and Fracals, vol38, no 4, pp , 28 3] Z M Odiba, Consrucion of soliary soluions for nonlinear dispersive equaions by variaional ieraion ehod, Physics Leers A,vol372,no22,pp ,28 32] E M Abulwafa, M A Abdou, and A A Mahoud, The soluion of nonlinear coagulaion proble wih ass loss, Chaos, Solions & Fracals,vol29,no2,pp33 33,26 33] E M Abulwafa, M A Abdou, and A A Mahoud, Nonlinear fluid flows in pipe-like doain proble using variaionalieraion ehod, Chaos, Solions & Fracals,vol32,no4,pp , 27 34] N Bildik and A Konuralp, The use of variaional ieraion ehod, differenial ransfor ehod and adoian decoposiion ehod for solving differen ypes of nonlinear parial differenial equaions, Inernaional Nonlinear Sciences and Nuerical Siulaion,vol7,no,pp65 7,26 35] S Moani and S Abuasad, Applicaion of He s variaional ieraion ehod o Helholz equaion, Chaos, Solions & Fracals,vol27,no5,pp9 23,26 36] S Moani and Z Odiba, Nuerical coparison of ehods for solving linear differenial equaions of fracional order, Chaos, Solions & Fracals,vol3,no5,pp ,27 37] S T Mohyud-Din, M A Noor, K I Noor, and M M Hosseini, Soluion of singular equaion by He s variaional ieraion ehod, Inernaional Nonlinear Sciences and Nuerical Siulaion,vol,no2,pp8 86,2 38] N H Sweila and M M Khader, Variaional ieraion ehod for one diensional nonlinear heroelasiciy, Chaos, Solions &Fracals,vol32,no,pp45 49,27 39] D D Ganji, H Tari, and M Bakhshi Jooybari, Variaional ieraion ehod and hooopy perurbaion ehod for nonlinear evoluion equaions, Copuers & Maheaics wih Applicaions,vol54,no7-8,pp8 27,27

11 Applied Maheaics 4] Z Odiba and S Moani, Nuerical ehods for nonlinear parial differenial equaions of fracional order, Applied Maheaical Modelling,vol32,no,pp28 39,28 4] E Hizel and S K Küçükarslan, A nuerical analysis of he Burgers-Poisson (BP) equaion using variaional ieraion ehod, in Proceedings of he 3rd WSEAS Inernaional Conference on Applied and Theoreical Mechanics, Tenerife, Spain, Deceber 27 42] S T Mohyud-Din and M A Noor, Modified variaional ieraion ehod for solving Fisher s equaions, Applied Maheaics and Copuing, vol3,no-2,pp295 38, 29 43] M A Noor and S T Mohyud-Din, Variaional ieraion ehod for solving higher-order nonlinear boundary value probles using He s polynoials, Inernaional Nonlinear Sciences and Nuerical Siulaion, vol9,no2,pp 4 56, 28 44] M A Noor and S T Mohyud-Din, Modified variaional ieraion ehod for hea and wave-like equaions, Aca Applicandae Maheaicae,vol4,no3,pp ,28 45] M A Noor and S T Mohyud-Din, Modified variaional ieraion ehod for solving Helholz equaions, Copuaional Maheaics and Modeling,vol2,no,pp4 5,29 46] M A Noor and S T Mohyud-Din, Variaional ieraion ehod for fifh-order boundary value probles using He s polynoials, Maheaical Probles in Engineering,vol28, Aricle ID , 2 pages, 28 47] M A Noor and S T Mohyud-Din, Modified variaional ieraion ehod for solving fourh-order boundary value probles, Applied Maheaics and Copuing, vol 29, no -2, pp 8 94, 29 48] M A Noor and S T Mohyud-Din, Modified variaional ieraion ehod for Goursa and Laplace probles, World Applied Sciences Journal,vol4,no4,pp ,28 49] S Abbasbandy, A new applicaion of He s variaional ieraion ehod for quadraic Riccai differenial equaion by using Adoian s polynoials, Copuaional and Applied Maheaics,vol27,no,pp59 63,27 5] S Abbasbandy, Nuerical soluion of non-linear Klein- Gordon equaions by variaional ieraion ehod, Inernaional Journal for Nuerical Mehods in Engineering,vol7,no 7,pp876 88,27 5] S T Mohyud-Din and M A Noor, Solving Schrödinger equaions by odified variaional ieraion ehod, World Applied Sciences Journal,vol5,no3,pp ,28 52] S T Mohyud-Din, M A Noor, and K I Noor, Modified variaional ieraion ehod for solving Sine Gordon equaions, World Applied Sciences Journal,vol5,no3,pp ,28 53] M A Noor and S T Mohyud-Din, Soluion of singular and nonsingular iniial and boundary value probles by odified variaional ieraion ehod, Maheaical Probles in Engineering,vol28,AricleID9747,23pages,28 54] M A Noor and S T Mohyud-Din, Variaional ieraion decoposiion ehod for solving eighh-order boundary value probles, Differenial Equaions and Nonlinear Mechanics,vol27,AricleID9529,6pages,27 55] M M Hosseini, Adoian decoposiion ehod wih Chebyshev polynoials, Applied Maheaics and Copuaion,vol 75, no 2, pp , 26 56] W C Tien and C K Chen, Adoian decoposiion ehod by Legendre polynoials, Chaos, Solions and Fracals, vol 39, no 5, pp 293 2, 29 57] Z Odiba, On Legendre polynoial approiaion wih he VIM or HAM for nuerical reaen of nonlinear fracional differenial equaions, Copuaional and Applied Maheaics,vol235,no9,pp ,2 58] M Razzaghi and S Yousefi, The Legendre waveles operaional ari of inegraion, Inernaional Syses Science, vol32,no4,pp495 52,2 59] F Mohaadi and M M Hosseini, A new Legendre wavele operaional ari of derivaive and is applicaions in solving he singular ordinary differenial equaions, he Franklin Insiue,vol348,no8,pp ,2

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