Hindawi Publishing Corporaion Inernaional Journal of Differenial Equaions Volume, Aricle ID 954674, pages doi:.55//954674 Research Aricle Soliary Wave Soluions for a Time-Fracion Generalized Hiroa-Sasuma Coupled KdV Equaion by a New Analyical Technique Majid Shaeri and D. D. Ganji Deparmen of Mechanical Engineering, Babol Universiy of Technology, P.O. Bo 484, 4748 767 Babol, Iran Correspondence should be addressed o D. D. Ganji, ddg davood@yahoo.com Received 7 May 9; Acceped 7 July 9 Academic Edior: Shaher Momani Copyrigh q M. Shaeri and D. D. Ganji. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. A new ieraive echnique is employed o solve a sysem of nonlinear fracional parial differenial equaions. This new approach requires neiher Lagrange muliplier like variaional ieraion mehod VIM nor polynomials like Adomian s decomposiion mehod ADM so ha can be more easily and effecively esablished for solving nonlinear fracional differenial equaions, and will overcome he limiaions of hese mehods. The obained numerical resuls show good agreemen wih hose of analyical soluions. The fracional derivaives are described in Capuo sense.. Inroducion In recen years, i has been urned ou ha fracional differenial equaions can be used successfully o model many phenomena in various fields such as fluid mechanics, viscoelasiciy, physics, chemisry, and engineering. For insance, he fluid dynamics raffic model wih fracional derivaives is able o eliminae he deficiency arising from he assumpion of coninuum raffic flow, and he nonlinear oscillaion of earhquakes can be modeled by fracional derivaives. Fracional differeniaion and inegraion operaors can also be used for eending he diffusion and wave equaions. Mos of fracional differenial equaions do no have eac analyical soluions, hence considerable heed has been focused on he approimae and numerical soluions of hese equaions. Alhough variaional ieraion mehod 4 8 and Adomian s decomposiion mehod 9 4 are approaches ha have been uilized eensively o provide analyical approimaions of linear and nonlinear problems, hey have limiaions due o complicaed algorihms of calculaing Adomian polynomials for nonlinear fracional problems, and an inheren inaccuracy in deermining
Inernaional Journal of Differenial Equaions he Lagrange muliplier for fracional equaions. In his sudy, a new alernaive procedure ha needs no Lagrange muliplier or Adomian polynomials is used o obain an analyical approimae soluion of a sysem of nonlinear fracional parial differenial equaions. o illusrae he effeciveness, accuracy, and convenience of his mehod. In his work, we consider he soluion of generalized Hiroa Sasuma coupled KdV of ime fracional order which is presened by a sysem of nonlinear parial differenial equaions, of he form: D α u u uu vw, D α v v uv, <α<,. D α w w uw, subjec o he following iniial condiions: u, β k k anh k, ( v, 4k c ) β k 4k( β k ) anh k, c c. w, c c anh k, where k, c,c /, and β are arbirary consans. The Hiroa Sasuma sysem of equaions 5 was inroduced o describe he ineracion of wo long waves wih differen dispersion relaions. The case of α insysem. was solved by Wu e al. 6.Ifc β, hen in one case he u, β k / k anh k c, v, 4k c β k /c 4k β k /c anh k c and w, c c anh k c are ravelling wave soluions of sysem. when α.. Basic Definiions In his secion, here are some basic definiions and properies of he fracional calculus heory which are used in his paper. Definiion.. A real funcion f, >, is menioned o be in he space C μ,μ R if here eiss a real number p > μ such ha f p f, where f C,, and i is said o be in he space C m μ if f m C μ,m N. Definiion.. The lef sided Riemann Liouville fracional inegral operaor of order α, of a funcion f C μ,μ is defined as D α f Γ α α f d, D f f. α >, >,.
Inernaional Journal of Differenial Equaions The properies of he operaor D α can be found in, 7, and we only menion he following in his case, f C μ,μ, α,β andγ> : D α D β f D α β f, D α D β f D β D α f, D α γ Γ ( γ ) Γ ( α γ )α γ.. The Riemann Liouville derivaive has cerain disadvanages, in rying way o model real world phenomena wih fracional differenial equaions. Therefore, we will employ a modificaion of fracional differenial operaor D α proposed by Capuo, in his work 8 on he heory of viscoelasiciy. Definiion.. The lef sided Capuo fracional derivaive of f is defined as D α f D α m D m f Γ m α m α f m d,. for m <α m, m N, >, f C m Also, we need wo of is basic properies. Lemma.4. If m <α m, m N, and f C m μ,μ, hen one has: D α D α f f, m D α D α f f f k k.4 k!, >. k The Capuo fracional derivaive is considered here, because i allows radiional iniial and boundary condiions o be included in he formulaion of he problem. In his work, we consider he one dimensional linear nonhomogeneous fracional parial differenial equaions in fluid mechanics, where he unknown funcion u, is assumed o be a causal funcion of ime, ha is, vanishing for <. Definiion.5. For m as he smalles ineger ha eceeds α, he Capuo ime fracional derivaive operaor of order α> is defined as D α u, α u, α Γ m α τ m α m u, τ τ m dτ, m <α<m, m u, m, α m N. For more informaion on he mahemaical properies of fracional derivaives and inegrals, one can consul he menioned references..5
4 Inernaional Journal of Differenial Equaions. u,.6 v,.6 w,.6...4 h Figure : h curves of u, : dash, v, :dashdo,andw, : solid when k., α, β.5, c.5, c. a poin,.6.. Basic Ideas of Fracional Ieraion Mehod (FIM) As poined in 9, o illusrae fracional ieraion mehod, we consider he following nonlinear fracional differenial equaion more general form can be considered wihou loss of generaliy : D α y f (, y ), y k a k, k,,...,m,. where he fracional differenial operaor D α is dened as in., m <α m, m N, f is a nonlinear funcion of y, andy is an unknown funcion o be deermined laer. We wan o find a soluion y of. having he form y lim n y n.. Le H / denoe he so called auiliary funcion. Muliplying. by H and hen applying D γ, he Riemann Liouville fracional inegral operaor, of order γ defined by., on boh sides of he resuled erm yields D γ( H [ D α y f (, y )]).. Le h / denoe he so called auiliary parameer. Muliplying. by h and hen adding y, he soluion of., on boh sides of he resuled erm yields y y hd γ( H [ D α y f (, y )])..4
Inernaional Journal of Differenial Equaions 5.5.5.58.58.5.5.498.498.49 4 4 4.49 4 4 4 a b Figure : The soliary wave soluion of u,, FIMresul a and eac soluion b, whenk., α, β.5, c.5, c...8.9... 4 4 4.8.9... 4 4 4 a b Figure : The soliary wave soluion of v,, FIMresul a and eac soluion b, whenk., α, β.5, c.5, c.. However.4 can be solved ieraively as follows: y n y n hd γ( H [ D α y n f (, y n )])..5 In.5, he subscrip n denoes he nh ieraion, and provided ha he righ hand of i, ha is, y hd γ H D α y f, y, is a conracive mapping. The convergence of.5 is ensured by Banach s fied poin heorem, as is shown in 9. Now, we inroduce a new convenien echnique for conrolling he convergence region and rae of soluion series for his mehod. Assume ha we gain a family of soluion series in
6 Inernaional Journal of Differenial Equaions.6.6.55.55.5.5.45.4 4 4 4.45.4 4 4 4 a b Figure 4: The soliary wave soluion of w,, FIMresul a and eac soluion b, whenk., α, β.5, c.5, c.. Table : Numerical values when α.5,.75,. and k., β.5, c.5, c. foru,. α.5 α.75 α. u FIM u HPM u FIM u HPM u FIM u HPM u Eac.49555.495.499749.495.4956.495.4955.5.49597977.494649.494546.494648.49978.499758.49975..5.497898.495556.49599.4948575.4946789.494687.4946789.75.498465.496658.4964956.49588946.49558.49559.49557.49997578.4974688.497885.49766.4966756.496677.4966756.49685.4945.495.4945.49456.4945.494569.5.4988948.49596.495966.494954686.4947797.49477598.494774.4.5.49999.49668.49747.49697847.4957947.495744.4957945.75.494995646.4978898.4985999.4974777.496945.4969446.49694499.4948668.4996476.494559.49985897.498675.49879949.49867.498557.49495.4964568.49495.494944586.49495.49494465.5.499988.49648749.497699.49674.49595555.49597485.49595586.6.5.494589.4979796.499776.49768.49766.4977.497675.75.494647.49986.4947648.4995968.498686.498778.49868.49457679.49494.49459494.4945.49486.49448589.49486 he auiliary parameer h by he means of fracional ieraion mehod. Like HAM, by ploing he amoun of he funcion or one of is derivaives a a paricular poin wih respec o he auiliary parameer h which is he so called h curve, we can obain a proper value of h ha ensures he convergence of he soluion series. This proper value of h corresponds o he curve segmen nearly parallel o he horizonal ais in he h curve plo. Therefore, if we se h any value in his region, which is so called he valid region of h, wearequiesurehahe corresponding soluion series converge. Having freedom for choosing he auiliary funcion H, he auiliary parameer h, he iniial approimaion y, and he fracional inegral order γ, ha is, fundamenal o
Inernaional Journal of Differenial Equaions 7 Table : Numerical values when α.5,.75,. and k., β.5, c.5, c. forv,. α.5 α.75 α. v FIM v HPM v FIM v HPM v FIM v HPM v Eac.485885.99596.8696.485.968.96.968.5.999878.4948.57595.64659.8978.8964.89789..5.9949979.99997784.86.4576.9767.9588.9766.75.989995777.99495.99557.996474487.998975.99895586.9989748.985854.9968.997746.9957.99978.99979.9997.4.6.9987745.58758.5447.49.7945.79.794475.5.9977494.99658458.998569.999477.97784.967.9776.5.98887896.99647.99677.99469.9979449.99797984.997948.75.984995.98667646.988686777.9898999.9998946.99969899.999894.9798857.9877597.98879.9844799.98858789.98845.98858768.994875.994888.997767879.9978555.98965.88.98766.5.989968.9894884.9986759.9985784.99694889.99689977.99694897.5.9849775.98445468.9879868.987987.999969.999488.9999654.75.97958475.979595.98496.9899.9877844.9879.987784.9747646.97466796.978778.978544.986.98544.984 Table : Numerical values when α.5,.75,. and k., β.5, c.5, c. forw,. α.5 α.75 α. w FIM w HPM w FIM w HPM w FIM w HPM w Eac.5759.5499747.548744.54989.5999.5.5999.5.5999674.5748486.57698.567878.5549446.5549557.5549446..5.5457.5996964.598479.5985.5798979.5798864.5798977.75.54976.54449.54.5684858.54658.546458.54658.5799.5489988.547685.544594.59759.5987.59758.55758.59454.587585.5778856.5599798.56.55998.5.5574.5675.564855.5747.5847957.5848679.58479588.4.5.55459985.5447.5874.5746974.595584.5969.5955847.75.5787585.5655.555596.5544.54857.545547.54858.57576.5898588.579785.57644.5586484.558767.5586485.587.5755768.5444.5884.5897568.59.58975778.5.55647.55746.549664.5489.5449479.54764.544957.6.5.577.5767944.5594464.559686.59955.599.59945.75.575.59958.58588.5878.565785.56757.565875.54688.55596.5749779.57865.5877457.58797467.587746 he validiy and fleibiliy of he FIM, we can suppose ha all of hem are properly chosen, herefore, he ieraive scheme.5 will converge o he eac soluion. Accordingly, he successive approimaions y n, n, of he soluion y will be obained by choosing y ha a leas saises he iniial and/or boundary condiions. Consequenly, he eac soluion may be obained by using y lim n y n. 4. Applicaions In his secion, we implemen fracional ieraion mehod o generalized Hiroa Sasuma coupled KdV of ime fracional order when <α. For convenience in applying FIM mehod, we choose he iniial condiions given in. as he iniial approimaions:
8 Inernaional Journal of Differenial Equaions u, β k k anh k, ( v, 4k c ) β k 4k( β k ) anh k, c c w, c c anh k. 4. Choosing γ α and H,, we can consruc he ieraive scheme.5 for invesigaion of he raveling wave soluion of. as follows: α u n, u n, hd [ D α u n, u n, u n, u n, ] v n, w n,, α v n, v n, hd [ D α v n, v n, u n, ] v n,, α w n, w n, hd [ D α w n, w n, u n, ] w n,. 4. Subsiuing he iniial approimaions, 4., ino 4., for he case α, yields ( ( u.49. anh. h.6..anh. ) anh..8 anh. (..anh. ). (.49. anh. ) ( anh...anh. ) (.4.4 anh. ) (.5.5anh..4.4 anh..5.5 anh. ) ), ( ( v.4.4 anh. h.684..anh. ).568 anh. (..anh. ) (.49. anh. ) (.4.4 anh. ) ), ( ( w.5.5anh. h...anh. ).6 anh. (..anh. ) (.49. anh. ) (.5.5 anh. ) ). 4.
Inernaional Journal of Differenial Equaions 9 5. Resul and Discussion In his secion, four figures are presened corresponding o FIM resuls and eac soluions for he soliary wave soluions u,,v,, and w, wih he iniial condiions., when k., α, β.5, c.5, c.. Furhermore, numerical values for he case α.5,.75,., and k., β.5,c.5,c. are obained for u,,v,, and w,. Demonsraing he eacness of FIM, he numerical resuls are presened and only few ieraions are required o achieve accurae soluions. The convergence of FIM for he generalized fracional order Hiroa Sasuma coupled KdV equaion is conrollable, using he so called h curves presened in Figure which are obained based on he fourh order FIM approimae soluions. In general, by he means of he so called h curve, i is sraigh forward o choose a proper value of h which ensures ha he soluion series is convergen. This proper value of h corresponds o he curve segmen nearly parallel o he horizonal ais. Boh eac resuls and approimae soluions obained for he firs four approimaions are ploed in Figures,, and4. There are no visible differences in wo soluions of each pair of diagrams. Tables,, and show he numerical values by FIM when α.5,.75,. and k., β.5,c.5,c. for u,,v,, and w, respecively. 6. Conclusion In his paper, he fracional ieraion mehod FIM has been successfully applied o sudy Hiroa Sasuma coupled KdV of ime fracional order equaion. FIM resuls are compared wih he eac soluions and hose obained by Homoopy perurbaion mehod. The resuls show ha fracional ieraion mehod is a powerful and efficien echnique in finding eac and approimae soluions for nonlinear parial differenial equaions of fracional order. The mehod provides he user wih more realisic series soluions ha converge very rapidly in real physical problems. Compared wih he ADM and VIM, he FIM has following advanages, 9. The auiliary parameer h provides us wih a convenien way o modify and conrol he convergence region of he soluion. The soluion of a given nonlinear problem can be epressed by an infinie number of soluion series and hus can be more efficienly approimaed by a beer selecion of he auiliary parameer values. Unlike he ADM, he FIM mehod is free from he need o use Adomian polynomials. 4 This mehod has no need for he Lagrange muliplier, correcion funcional, saionary condiions, he variaional heory, and so forh, which eliminaes he complicaions ha eis in he VIM. 5 The fracional ieraion mehod can be easily comprehended wih only a basic knowledge of fracional calculus. 6 Compared o he ADM and VIM, he presened mehod proves simpler in is principles and more convenien for compuer algorihms. In his work, we used Maple Package o calculae he series obained by fracional ieraion mehod.
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