Time-fractional Klein-Gordon equation: formulation and solution using variational methods

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1 ime-fracional Klein-Gordon equaion: formulaion and soluion using variaional mehods YOUWEI ZHANG Hei Universiy School of Mahemaics and Saisics Beihuan oad 846 Zhangye CHINA Absrac: his paper presens he formulaion of ime-fracional Klein-Gordon equaion using he Euler-Lagrange variaional echnique in he iesz derivaive sense and derives an approimae soliary wave soluion. Our resuls winess ha He s variaional ieraion mehod was very efficien and powerful echnique in finding he soluion of he proposed equaion. he basic idea described in his paper is efficien and powerful in solving wide classes of nonlinear fracional high order evoluion equaions. Key Words: iesz fracional derivaive; Euler-Lagrange equaion; Klein-Gordon equaion; He s variaional ieraion mehod; Soliary wave Inroducion Our aim is o apply he reliable reamen of He s variaional ieraion mehod [ 4] o obain he soluion of he iniial value problem of he ime-fracional Klein- Gordon equaion of he form wih he iniial condiions Dα u a(u + b(u =, u(, = φ (, u (, = φ (, ( where < α <, a, b >, u = u(, is a field variable, Ω (Ω is a space coordinae in he propagaion direcion of he field and (= [, ] ( > is he ime, Dα is he iesz fracional derivaive, φ ( and φ ( are given funcions. he subscrips denoe he parial differeniaion of he funcion u wih respec o he parameer and. Fracional calculus have araced much aenion during recen years due o heir numerous applicaions in physics and engineering [5 8], such as, he nonlinear oscillaion of earhquake can be modeled wih fracional derivaives [9], and he fluid dynamic raffic model wih fracional derivaives can eliminae he deficiency arising from he assumpion of coninuum raffic flow []. Finding of a new mahemaical algorihm o consruc eac soluion of nonlinear fracional order evoluion equaions is imporan and migh have significan impac on fuure research, we noice ha he Lagrangian of conservaive sysem is consruced using fracional derivaives, he resuling equaions of moion can be nonconservaive. herefore, in many cases, he real physical processes could be modeled in a reliable manner using fracional-order differenial equaions []. Based on he sochasic embedding heory, Cresson [] defined he fracional embedding of differenial operaors and provided a fracional Euler-Lagrange equaion for Lagrangian sysems, hen invesigaed a fracional Noeher heorem and a fracional Hamilonian formulaion of fracional Lagrangian sysems. Herzallah and Baleanu [] presened he necessary and sufficien opimaliy condiions for he Euler-Lagrange fracional equaions of fracional variaional problems wih deermining in which spaces he funcional mus eis. Malinowska [4] proposed he Euler-Lagrange equaions for fracional variaional problems wih muliple inegrals and proved he fracional Noeher-ype heorem for conservaive and nonconservaive generalized physical sysems. iewe [5, 6] formulaed a version of he Euler-Lagrange equaion for problems of calculus of variaion wih fracional derivaives. Wu and Baleanu [7] developed some new variaional ieraion formulae o find approimae soluions of fracional differenial equaions and deermined he Lagrange muliplier in a more accurae way. For generalized fracional Euler-Lagrange equaions we can refer o he works by Odzijewicz [8]. Oher he known resuls we can see Baleanu e al [9] and Inokui e al []. hanks o he mos of physical phenomena may be considered as nonconservaive, hen hey can be described using fracional-order differenial equa- E-ISSN: Volume 5, 6

2 ions. Klein-Gordon equaion describes relaivisic elecrons, and correcly describes he spinless pion, he equaion was firs considered as a quanum wave e- quaion by Schrödinger in his search for an equaion describing de Broglie waves, and plays a significan role in mahemaical physics and many scienific applicaions such as solid-sae physics, nonlinear opics, and quanum field heory [, ]. he equaion has araced much aenion in sudying solions and condensed maer physics [ 5], in invesigaing he ineracion of solions in a collisionless plasma, he recurrence of iniial saes, and in eamining he nonlinear wave equaions [6]. ecenly, several mehods have been used o solve fracional Klein-Gordon equaion using echniques of nonlinear analysis, such as homoopy perurbaion mehod [7], homoopy analysis mehod [8], iccai epansion mehod [9], new ieraive mehod [,], he firs inegral mehod [] and series epansion mehod [] e al. I was menioned ha he variaional ieraion mehod has been used successfully o solve fracional KdV equaion [4]. he objecive of his paper is o eend he applicaion of he fracional variaional ieraion mehod o formulae a high-order ime-fracional Klein-Gordon equaion and derive an approimae soliary wave soluion. his paper is organized as follows: Secion s- aes some background maerial from fracional calculus. Secion presens he principle of He s variaional ieraion mehod. Secion 4 is devoed o describe he formulaion of he ime-fracional Klein-Gordon equaion using he Euler-Lagrange variaional echnique and o derive an approimae soliary wave soluion. Secion 5 makes some analysis for he obained graphs and figures and discusses he presen work. Preliminaries We recall he necessary definiions for he fracional calculus (see [5 7] which is used hroughou he remaining secions of his paper. Definiion A real mulivariable funcion f(,, > is said o be in he space C γ, γ wih respec o if here eiss a real number p(> γ, such ha f(, = p f (,, where f (, C(Ω. Obviously, C γ C δ if δ γ. Definiion he lef-hand side iemann-liouville fracional inegral of a funcion f C γ, (γ is defined by I α f(, = Γ(α I f(, = f(,. ( τ α f(, τdτ, α >, Definiion he iemann-liouville fracional derivaives of he order n α < n of a funcion f C γ, (γ are defined as D α f(, = Γ(n α n D α f(, n n = Γ(n α n ( τ nα f(, τdτ, (τ nα f(, τdτ. Lemma 4 he inegraion of iemann-liouville fracional derivaive of he order < α < of he funcions f, g, D α f(,, D α g(, C(Ω by pars is given by he rule f(, D α g(, d = g(, D α f(, d. Definiion 5 he iesz fracional inegral of he order n α < n of a funcion f C γ, (γ is defined as I α f(, = (I α f(, + I α f(, = Γ(α τ α f(, τdτ, where I α and I α are respecively he lef- and righhand side iemann-liouville fracional inegral operaors. Definiion 6 he iesz fracional derivaive of he order n α < n of a funcion f C γ, (γ is defined by D α f(, = (D α f(, + ( n D α f(, d n = Γ(n α d n τ nα f(, τdτ, where D α and D α are respecively he lef- and righ-hand side iemann-liouville fracional differenial operaors. E-ISSN: Volume 5, 6

3 Lemma 7 Le α > and β > be such ha n < α < n, m < β < m and α + β < n, and le f L (Ω and I mα f AC m (Ω. hen we have he following inde rule: D α ( D β f(, = D α+β f(, m D βi αi f(, = Γ( α i. i= emark 8 One can epress he iesz fracional differenial operaor Dα of he order < α < as he iesz fracional inegral operaor Iα τ, i.e. D α f(, = I α f(,,. Variaional-ieraion mehod he variaional ieraion mehod provides an effecive procedure for eplici and soliary wave soluions of a wide and general class of differenial sysems represening real physical problems. Moreover, he variaional ieraion mehod can overcome he foregoing resricions and limiaions of approimae echniques so ha i provides us wih a possibiliy o analyze srongly nonlinear evoluion equaions. herefore, we eend his mehod o solve he ime-fracional Klein- Gordon-ype equaion. he basic feaures of he variaional ieraion mehod oulined as follows. Considering a nonlinear evoluion equaion consiss of a linear par Lu, nonlinear par N u, and a free erm g(= g(, represened as Lu + N u = g. ( According o he variaional ieraion mehod, he n+-h approimae soluion of ( can be read using ieraion correcion funcional as u n+ = u n + λ(τ ( Lũ + N ũ g dτ, ( where λ(τ is a Lagrangian muliplier and ũ = ũ(, is considered as a resriced variaion funcion, i.e., δũ =. Ereming he variaion of he correcion funcional ( leads o he Lagrangian muliplier λ(τ. he iniial ieraion u can be used as he iniial value u(,. As n ends o infiniy, he ieraion leads o he soliary wave soluion of (, i.e. u = lim n u n. 4 ime-fracional Klein-Gordon e- quaion he Klein-Gordon equaion is given as u a(u + b(u =. (4 Employing a poenial funcion v on he field variable u, se u = v yields he poenial equaion of he Klein-Gordon equaion (4 in he form, v a(v + b(v =. (5 he Lagrangian of his Klein-Gordon equaion (4 can be defined using he semi-inverse mehod [8, 9] as follows. he funcional of he poenial equaion (5 can be represened as J(v = d d ( c vv ac v(v Ω + bc v(v, (6 wih c i (i =,, is unknown consan o be deermined laer. Inegraing (6 by pars and aking v = v = v Ω = (v Ω = (v Ω = (7 yield J(v = d d ( c v v ac (vv Ω + bc (v v. (8 he consans c i (i =,, can be deermined aking he variaion of he funcional (8 o make i opimal. By applying he variaion of he funcional, inegraing each erm by pars, and making use of he variaion opimum condiion of he funcional J(v, i yields he following represenaion c v ac (v + bc (v =. (9 We noice ha he obained resul (9 is equivalen o (5, so one has ha he consans c i (i =,,..., 6 are respecively c =, c = c =. hen he funcional becomes J(v = d d ( v v a(vv + b(v v. Ω E-ISSN: Volume 5, 6

4 In addiion, he funcional epression given by (8 obains direcly he Lagrangian form of he Klein- Gordon equaion, N(v, v, v, v = v v av v + bv + bv v v. he Lagrangian of he ime-fracional version of he Klein-Gordon equaion ( could be read as F ( D α v, v, v, v = D α vv av v + bv + bv v v, α ], ]. ( hen he funcional of he ime-fracional Klein- Gordon equaion will ake he represenaion J(v = d F ( D α v, v, v, v d, ( Ω where he ime-fracional Lagrangian F ( D α v, v, v, v is given by (. Following Agrawal s mehod [4 4], he variaion of funcional ( wih respec o v leads o ( F δj(v = d d Ω D αv δ( D α v + F δv v + F δv + F δ(v. ( v v By Lemma 4, upon inegraing he righ-hand side of (, one has ( C δj(v = d d D α v δv + ( v D α δv Ω v D α δv D α v δv ( av v + bv v δv + ( av + bv v + 6bv ( bv v δv, δv noing ha δv = δv Ω = δv Ω = δv Ω = δv Ω = δv =. v = in (7 implies C D α v = D α v. Obviously, opimizing he variaion of he funcional J(v, i.e., δj(v =, yields he Euler- Lagrange equaion for ime-fracional Klein-Gordon equaion in he following represenaion D α v D α v + ( av v bv v ( av bv v 6bv ( bv v =. ( Subsiuing he Lagrangian of he ime-fracional Klein-Gordon equaion ( ino Euler-Lagrange formula ( obains D α v D α v + 6bv + 6bv v =. Once again, subsiuing he poenial funcion v for u, yields he ime-fracional Klein-Gordon equaion for he sae funcion u as D α u D α u + 6bu + 6bu u =. (4 According o he iesz fracional derivaive Dα u, he ime-fracional Klein-Gordon equaion represened in (4 can wrie as D α u 6bu 6bu u =. (5 Acing from lef-hand side by he iesz fracional op- on (5 leads o eraor Dα u D j j= + D α αj u = Γ(α j ( 6b ( u u u 6b =, (6 from Lemma 7. In view of he variaional ieraion mehod, combining wih (6, he n + -h approimae soluion of (5 can be read using ieraion correcion funcional as ( u n+ (, = u n (, λ(τ τ u n(, τ τ α τ u n(, τ τ= Γ(α τ α4 + u n (, Γ(α ( ( 6b D α ũn(, τ + ũn(, τ ũn(, τ dτ, (7 E-ISSN: Volume 5, 6

5 where he funcion ũ n is considered as a resriced variaion funcion, i.e., δũ n =. he ereme of he variaion of (7 subjec o he resriced variaion funcion sraighforwardly yields δu n+ = δu n + λ(τδ ( τ u n(, τ dτ = δu n + λ(τδ ( τ u n(, τ τ= τ λ(τδu n(, τ τ= + δu n (, τ τ λ(τdτ = ( τ λ(τ δu n (, τ τ= + λ(τδ ( τ u n(, τ τ= + δu n (, τ λ(τdτ =. τ his epression reduces he following saionary condiions τ λ(τ τ= =, λ(τ τ= =, λ(τ =, τ which convered o he Lagrangian muliplier a λ(τ = τ. herefore, he correcion funcional (7 akes he following form ( u n+ (, = u n (, (τ τ u n(, τ Dα τ α τ u n(, τ τ= Γ(α τ α4 + u n (, Γ(α ( ( 6b Dτ α u n(, s + u n(, s u n(, s dτ, (8 since α <, he fracional derivaive operaor reduces o fracional inegral operaor Iα by emark 8. In view of he righ-hand side iemann-liouville fracional derivaive is inerpreed as a fuure sae of he process in physics. For his reason, he righderivaive is usually negleced in applicaions, when he presen sae of he process does no depend on he resuls of he fuure developmen, and so he righderivaive is used equal o zero in he following calculaions. he zero order soliary wave soluion can be aken as he iniial value of he sae variable from (, which is aken in his case as u (, = 4c ( a cosh 4 c b sinh ( b + ξ b + ξ, where c, ξ are he given consans. Subsiuing his zero order approimae soliary wave soluion ino (8 and using he Definiion 6 leads o he firs order approimae soliary wave soluion of he problem ( u (, = 4c ( + α a Γ(α c b ( α Γ(α ac 6 α+ b (α + Γ(α cosh ( + c5 b α+ 6bΓ(α + sinh ( c 4 α 6bΓ(α + cosh ( cosh ( a 4 b + ξ sinh ( a b + ξ b + ξ b + ξ b + ξ. Subsiuing firs order approimae soliary wave soluion ino (8, using he Definiion 6 hen leads o he second order approimae soliary wave soluion of he problem ( as follows u (, = 4c ( + α a Γ(α c b ( α Γ(α cosh ( 4 sinh ( ac 6 α+ 6b (α + Γ(α cosh ( b + ξ b + ξ b + ξ + c5 b α+ bγ(α + sinh ( b + ξ c 4 α bγ(α + cosh ( b + ξ c4 6b cosh ( b + ξ( α Γ(α E-ISSN: 4-88 Volume 5, 6

6 Γ(α α4 + Γ (α Γ(α + α Γ(α + ac6 4b cosh ( b + ξ( α+ (α + Γ(α Γ(α α + Γ (αγ(α α a4 c Γ(α 4b 5 cosh ( b + ξ Γ(α + 5 α+4 (α + Γ (αγ(α + 5 a c 6b 4 cosh ( b + ξ Γ(α + α+ Γ (α + Γ(α + a c 8 6b cosh ( b + ξ Γ(α + α Γ (α + Γ(α + +. u(, 4 6 (A α=. Making use of he Definiion 6 and some calculaion, we obain u, u 4 and so on, subsiuing n order approimae soliary wave soluion ino (8, here leads o he n order approimae soliary wave soluion. As n ends o infiniy, he ieraion leads o he soliary wave soluion of he ime-fracional Klein- Gordon equaion ( u(, = 4c a cosh ( 4 b ( c + ξ. u(, 4 6 (A α=.98 5 Discussion he arge of presen work is o eplore he effec of he fracional order derivaive on he srucure and propagaion of he resuling soliary waves obained from ime-fracional Klein-Gordon equaion. We derive he Lagrangian of he Klein-Gordon equaion by he semi-inverse mehod, hen ake a similar form of Lagrangian o he ime-fracional Klein-Gordon e- quaion. Using he Euler-Lagrange variaional echnique, we coninue our calculaions unil he hreeorder ieraion. During his period, our approimae calculaions are carried ou concerning he soluion of he ime-fracional Klein-Gordon equaion aking ino accoun he values of he coefficiens and some meaningful values namely, α =.98,.95,.9 and.85. he soliary wave soluion of ime-fracional Klein-Gordon equaion is obained. In addiion, - dimensional represenaion of he soluion u for he ime-fracional Klein-Gordon equaion wih space and ime for differen values of he order α is presened respecively in Figure, he soluion u is sill a single solion wave soluion for all values of he order α. I shows ha he balancing scenario beween nonlineariy and dispersion is sill valid. Figure presens he change of ampliude and widh of he solion due u(, u(, (A α=.95 (A4 α=.9 Figure he surfaces of he approimae soluions u(, E-ISSN: 4-88 Volume 5, 6

7 u(,α α o he variaion of he order α, - and -dimensional graphs depiced he behavior of he soluion u a ime =.5 corresponding o differen values of he order α. his behavior indicaes ha he order α can be used o modify he shape of he soliary wave wihou change of he nonlineariy and he dispersion effecs in he medium. Figure devoed o sudy he epression beween he ampliude of he solion and he fracional order a differen ime values. hese figures show ha a he same ime, he change of he fracional α decreases he ampliude of he soliary wave. (B u(, α=.98 α=.95 α=.9 α=.85 Acknowledgemen he auhors would like o hank he anonymous referees for heir careful reading of his manuscrip and for suggesing useful changes (B Figure he funcion u as a funcion of space a ime =.5 for order α: (B -dimensions graph, (B -dimensions graph u(,α u(=, α α=.98 α=.95 α=.9 α=.85 (C (C Figure he ampliude of he funcion u as a funcion of ime a space = for order α: (C -dimensions graph, (C -dimensions graph eferences: [] J. He, A new approach o nonlinear parial differenial equaions, Commun. Nonlinear Sci. Numer. Simul. (4, 997, pp. 5. [] J. He, Approimae analyical soluion for seepage flow wih fracional derivaives in porous media, Compu. Mehods Appl. Mech. Eng. 67, 998, pp []. Y. Molliq, M. S. M. Noorani, I. Hashim, Variaional ieraion mehod for fracional heaand wave-like equaions, Nonlinear Anal. eal World Appl., 9, pp [4] S. Momani, Z. Odiba, A. Alawnah, Variaional ieraion mehod for solving he space- and ime-fracional KdV equaion, Numer. Mehods Parial Differenial Equaions 4(, 8, pp [5]. Gorenflo, F. Mainardi, E. Scalas, M. abero, Fracional calculus and coninuous-ime finance. III. he diffusion limi. Mahemaical finance (Konsanz,, rends Mah., pp [6]. Hilfer, Applicaions of Fracional Calculus in Physics, World Scienific, Singapore,. [7]. Mezler, J. Klafer, he resauran a he end of he random walk: recen developmens in he descripion of anomalous ranspor by fracional dynamics, J. Phys. A 7, 4, pp [8] Y. A. ossikhin, M. V. Shiikova, Applicaion of fracional derivaives o he analysis of damped vibraions of viscoelasic single mass sysems, Aca Mech., 997, pp [9] J. He, Some applicaions of nonlinear fracional differenial equaions and heir approimaions, Bull. Sci. ech. Soc. 5(, 999, pp E-ISSN: 4-88 Volume 5, 6

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9 [7] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fracional Inegrals and Derivaives: heory and Applicaions, Gordon and Breach, New York 99. [8] J. He, Semi-inverse mehod of esablishing generalized variaional principles for fluid mechanics wih emphasis on urbo-machinery aerodynamics, In. J. urbo Je-Engines 4(, 997, pp. 8. [9] J. He, Variaional principles for some nonlinear parial differenial equaions wih variable coefficiens, Chaos Solions Fracals 9, 4, pp [4] O. P. Agrawal, Formulaion of Euler-Lagrange equaions for fracional variaional problems, J. Mah. Anal. Appl. 7(,, pp [4] O. P. Agrawal, A general formulaion and soluion scheme for fracional opimal conrol problems, Nonlinear Dyn. 8(4, 4, pp. 7. [4] O. P. Agrawal, Fracional variaional calculus in erms of iesz fracional derivaives, J. Phys. A 4, 7, pp E-ISSN: Volume 5, 6