NUMERICAL SOLUTIONS OF THE FRACTIONAL KdV-BURGERS-KURAMOTO EQUATION

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1 S5 NUMERICAL SOLUTIONS OF THE FRACTIONAL KdV-BURGERS-KURAMOTO EQUATION by Doga KAYA a*, Sema GULBAHAR a, ad Asif YOKUS b a Departmet of Matematics, Istabul Commerce Uiversity, Uskudar, Istabul, Turkey b Departmet of Actuary, Firat Uiversity, Elazıg, Turkey Origial scietific paper ttps://doi.org/0.98/tsci7068k Itroductio No-liear terms of te time-fractioal KdV-Burgers-Kuramoto equatio are liearized usig by some liearizatio teciques. Numerical solutios of tis equatio are obtaied wit te elp of te fiite differece metods. Numerical solutios ad correspodig aalytical solutios are compared. Te L error orms are computed. Stability of give metod is ivestigated by usig te Vo Neuma stability aalysis. Key words: time-fractioal KdV-Burgers-Kuramoto equatio, fiite differece metod, stability Te birt of fractioal calculus goes as Leibiz ad Newto s differetial calculus. Te otio of fractioal-order derivative of o-iteger order was firstly itroduced by Leibiz: m de m = m e () d were is o-iteger value (cf. []). Later, fractioal PDE drew attetio of may matematicias ad ave also sow a icreasig developmet (cf. [-0]). Time-fractioal KdV-Burgers-Kuramoto equatio is oe of te importat fractioal differece equatio wic is defied by te followig form: wit iitial coditio ad wit boudary coditios 4 u u u u u + u + + λ λ λ = 0 4 t ( ) () u (,0) = f () uat (, )= β, ubt (, )= β, t t, (4) 0 were is te order of te fractioal time derivative ad λ, λ, ad λ 0 are parameters caracterizig istability, dispersio ad dissipatio, respectively []. * Correspodig autor, semaakkus_mat@otmail.com

2 S54 Time-fractioal KdV-Burgers-Kuramoto equatio is a importat model to describe pysical peomea o te move of turbulece ad oter istability process. It ca also be used to defie a log waves o a viscous fluid flowig dow alog a iclied plae [], ustable drift waves i plasma [], turbulet cascade model i a barotropic atmospere [4]. I tis paper, we study te fiite differece metod to obtai umerical solutios of time-fractioal KdV-Burgers-Kuramoto equatio. Te effectiveess of te proposed metod is tested usig a umerical eample ad te stability aalysis is ivestigated. Fiite differece metods I tis sectio, we sall recall some basic facts about fiite differece metods. Firstly, we sall defie a set of grid poits i te domai D to obtai a umerical solutio to eq. () usig fiite differece metods as follows. Let us coose a state step size ( ) = ( b a)/ N(N is a iteger), a time step size t, draw a set of orizotal ad vertical lie across D, ad get all itersectio poits ( j, t ) or simply ( j, ) were j = a+ j, j = 0,,,, N, ad t = t, = 0,,, M. If D= [ ab, ] [ 0, T], te we ca coose t= TM / (M is a iteger) ad t = t, = 0,,, M. Puttig appropriate fiite differece approimatio i eq. (), it ca be see tat solutio of eq. () reduced to solutio problem of algebraic differetial systems of liear ad o-liear equatios cosistig of fiite differece equatio. However, it is kow tat o-liear system of equatios ca ot be solved directly. Terefore, we sall use two liearizatio teciques for o-liear terms eistig i te eq. (). Liearizatio Firstly, puttig te Caputo fractioal derivative approimatio for u/ t (0< ) ad usual fiite differece approimatios at te odal poit ( m+, ) wic are give, respectively, by te followig forms [0, 5]: u Γ t Γ ( t ) k k ( ) ( ) ( ) m m ( ) ( 0 ) m m + u u k + k, u u, =0 U um+ um + um (5) U u u u u ( ) m m m m um+ 4um+ + 6um 4um um U 4 i eq. () ad later applyig te followig liearizatio tecique defied: uu u u m+ m ( um) for te o-liear term uu i eq. () [6], cosequetly we ave te followig system of algebraic equatio:

3 S55 Liearizatio ( ) k k ( ) ( ) + u m um k + k = Γ um+ u m um+ um + u m = ( u) λ m + um+ um+ + um u m um+ 4um+ + 6um 4um u m λ λ 4 Firstly, puttig eq. (5) i eq. () ad later applyig te followig liearizatio tecique defied: um + u m+ uu ( um+ um ) (7) i eq. (), te we ave te followig system of algebraic equatio: Stability aalysis ( ) k k ( ) ( ) + u m um k + k = Γ ( )( ) u m + um+ um+ um um+ um + u m = + λ 4 um+ um+ + um u m um+ 4um+ + 6um 4um + u m λ λ 4 I tis subsectio, stability of te fiite differece metod is ivestigated wit te elp of Vo-Neuma aalysis. Takig ito cosideratio te Vo-Neuma stability aalysis, te growt factor of a typical Fourier mode is defied: im u m (6) (8) = ξ e (9) were i = ad ξ deotes te amplicatio factor. To ivestigate te stability of te umerical sceme, te o-liear term uu i te modified Burgers equatio as bee liearized by makig te quatity u to a local costat. Tus te o-liear term i te equatio coverts ito uu ˆ ad i tat case te eq. () turs ito: we get: 4 u u u u u + uˆ λ + λ + λ = 0 (0) 4 t Substitutig te Fourier mode of eq. (9) ito te recurrece relatiosip of eq. (6), ( ) Γ k im k im ( ξ e ξ e ) ( k ) + k + +

4 S56 Let ξ followig: were im ( + ) im ( ) im ( + ) im im ( ) ξ e ξ e ξ e ξ e + ξ e + uˆ λ + im ( + ) im ( + ) im ( ) im ( ) ξ e ξ e + ξ e ξ e + λ + im ( + ) im ( + ) im im ( ) im ( ) ξ e 4ξ e + 6ξ e 4ξ e + ξ e + λ = = ζξ ( ) ( ) () ad assume tat ζ = ζ( ) is idepedet of time. Te, we obtai te ( ) ( ) i i t k+ k e e k k ξ ξ + + uˆ Γ i i i i i i e + e e e + e e λ λ + + i i i i e 4e + 6 4e + e + λ = 0 4 Te we easily obtai te followig epressio: uˆ λ λ 4λ i + si( ) cos( ) ( ) cos( ) 4 cos + + ξ = k k ( ξ ξ ) ( k + ) k Γ ( ) For more details, we refer to [7]. Hece, we get: X [ ] [ ] = X + ξ ix Y λ 4λ = cos( ) cos ( ) cos( ) uˆ λ X = + si( ) [ ] [ + ] 4 ( ) k k ( ξ ξ ) ( ) Y = k + k Γ Accordig to te Fourier stability, for te give sceme to be stable, te coditio ξ must be satisfied. Tis implies tat if te followig iequality: Y is provided te te scema become ucoditioally stable: ζ () () X + X = (4)

5 S57 Numerical eamples ad results I tis subsectio, umerical results of te equatio ave bee obtaied for te test problem used i te preset study. To sow ow accurate te results, bot te error orm L give: ad te error orm L give: are goig to be computed ad preseted. Test problem were N eact eact = N = j N j J =0 L U U U U L U U U U ( ) ( ) eact eact = N = ma j N j j Te aalytical solutio of te fractioal KdV- Burgers-Kuramoto is give: ut (, )= v ( a+ a ) + ( a + a ), t t0, 0 (5) I our computatios, tree differet liearizatio teciques ave bee applied for te umerical solutio of te test problem. Te error orms L are computed for t = = I tab., te error orms L obtaied usig te liearizatio teciques are compared. I fig., solutio for liearizatio, solutio liearizatio ad eact solutio for = are compared. Coclusio a = ta ( vt b) 4 9, v =, b =. Table. Compariso of error orms L ad L usig te liearizatio teciques at = 0. for = 0.000, 0 Numerical solutio Eact solutio Liearizatio Liearizatio L L Numerical solutios of te fractioal KdV-Burgers-Kuramoto equatio are obtaied by usig fiite differece metods wit two differet liearizatio teciques. Te computatioal efficiecy ad effectiveess of proposed metod are tested o a problem. Te error orms L are computed ad preseted. Te obtaied results sow tat te error orms are sufficietly small durig all computer rus. Cosiderig te tables, it is obvious tat te obtaied results usig liearizatio is better ta obtaied results usig oter liearizatio. It is sow tat te preset metod is a particularly successful umerical sceme to solve te fractioal KdV-Burgers-Kuramoto equatio.

6 S u Lieerizatio Eact solutio Lieerizatio Figure. Compariso of eact solutio ad umerical solutios wic are obtaied wit te elp of liearizatio teciques at = 0. for = 0.000, 0 7. Refereces [] Das, S., Fuctioal Fractioal Calculus, Spriger-Verlag Berli, Heidelberg, Germay, 0 [] Safari, M., et al., Applicatio of He s Variatioal Iteratio Metod ad Adomia s Decompositio Metod to te Fractioal KdV-Burgers-Kuramoto Equatio, Computers&Matematics wit Applicatios, 58 (009), -, pp [] Hasemi, M. S., Baleau, D., Numerical Approimatio of Higer-Order Time-Fractioal Telegrap Equatio by Usig a Combiatio of a Geometric Approac ad Metod of Lie, Joural of Computatioal Pysics, 6 (06), July, pp. 0-0 [4] Hasemi, M. S., et al., A Lie Group Approac to Solve te Fractioal Poisso Equatio, Romaia Joural of Pysics, 60 (05), 9-0, pp [5] Hasemi, M. S., et al., Solvig te Time-Fractioal Diffusio Equatio Usig a Lie Group Itegrator, Termal Sciece, 9 (05), Suppl., pp. S77-S8 [6] Sog, L., Zag, H., Applicatio of Homotopy Aalysis Metod to Fractioal KdV-Burgers-Kuramoto Equatio, Pysics Letter A, 67 (007), -, pp [7] Wei, L., et al., Numerical Algoritm Based o a Implicit Fully Discrete Local Discotiuous Galerki Metod for te Time-Fractioal KdV-Burgers-Kuramoto Equatio, Joural of Applied Matematics ad Mecaics, 9 (0), pp. 4-8 [8] Pasayi, S., et al., Aalytical Lie Group Approac for Solvig Fractioal Itegro-Differetial Equatios, Commuicatios i Noliear Sciece ad Numerical Simulatio, 5 (07), Oct., pp [9] Yokus, A., Numerical Solutio for Space ad Time Fractioal Order Burger Type Equatio, Aleadria Egieerig Joural, O-lie first, ttps://doi.org/0.06/j.aej [0] Yokus, A., Kaya, D., Numerical ad Eact Solutios for Time Fractioal Burgers Equatio, Joural of Noliear Scieces ad Applicatios, 0 (07), 7, pp [] Kawaara, T., Formatio of Saturated Solitos i a Noliear Dispersive System wit Istability ad Dissipatio, Pysical Review Letters, 5 (98), Aug., pp. 8-8 [] Topper, J., Kawaara, T., Approimate Equatios for Log Noliear Waves o a Viscous Fluid, Joural of te Pysical Society of Japa, 44 (978), July, pp [] Coe, B., et al., No-Liear Saturatio of te Dissipative Trapped-Io Mode by Mode Couplig, Nuclear Fusio, 6 (976), 6, pp [4] Huag, F., Liu, S., Pysical Mecaism ad Model of Turbulet Cascades i a Barotropic Atmospere, Advaces i Atmosperic Scieces, (004),, pp [5] Oldam, K. B., Spaier, J., Te Fractioal Calculus, Academic Press, New York, USA, 974 [6] Ucar, Y., et al., Numerical Solutios of te Modified Burgers Equatio by Fiite Differece Metods, Joural of Applied Matematics, Statistics ad Iformatics, (07),, pp. 9-0 [7] Guo, B., et al., Fractioal Partial Differetial Equatios ad teir Numerical Solutios, World Scietific, Sigapure, 05 Paper submitted: Jue, 07 Paper revised: November 9, 07 Paper accepted: November, Society of Termal Egieers of Serbia Publised by te Viča Istitute of Nuclear Scieces, Belgrade, Serbia. Tis is a ope access article distributed uder te CC BY-NC-ND 4.0 terms ad coditios