New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves
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摘要
The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.
The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.
The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.
引文
Abdeljawad,T.,2015.On conformable fractional calculus,Journal of Computational and Applied Mathematics,279,57-66.
Aminikhah,H.,Sheikhani,A.H.R.and Rezazadeh,H.,2016.Subequation method for the fractional regularized long-wave equations with conformable fractional derivatives,Scientia Iranica,23(3),1048-1054.
Atangana,A.,Baleanu,D.and Alsaedi,A.,2015.New properties of conformable derivative,Open Mathematics,13(1),2391-5455.
Batarfi,H.,Losada,J.,Nieto,J.J.and Shammakh,W.,2015.Threepoint boundary value problems for conformable fractional differential equations,Journal of Function Spaces,2015,706383.
Biswas,A.,Al-Amr,M.O.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.,Zhou,Q.,Moshokoa,S.P.and Belic,M.,2018a.Resonant optical solitons with dual-power law nonlinearity and fractional temporal evolution,Optik,165,233-239.
Biswas,A.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.,Ekici,M.,Zhou,Q.,Moshokoa,S.P.and Belic,M.,2018b.Optical soliton perturbation with Fokas-Lenells equation using three exotic and efficient integration schemes,Optik,165,288-294.
Biswas,A.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.,Zhou,Q.,Moshokoa,S.P.and Belic,M.,2018c.Optical solitons having weak non-local nonlinearity by two integration schemes,Optik,164,380-384.
Boussinesq,J.,1872.Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal,en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond,Journal de Mathématiques Pures et Appliquées,17,55-108.(in French)
Eslami,M.and Mirzazadeh,M.,2013.Topological 1-soliton solution of nonlinear Schr?dinger equation with dual-power law nonlinearity in nonlinear optical fibers,The European Physical Journal Plus,128(11),140.
Eslami,M.,2016.Trial solution technique to chiral nonlinear Schrodinger’s equation in(1+2)-dimensions,Nonlinear Dynamics,85(2),813-816.
Eslami,M.and Mirzazadeh,M.,2016a.Optical solitons with BiswasMilovic equation for power law and dual-power law nonlinearities,Nonlinear Dynamics,83(1-2),731-738.
Eslami,M.and Rezazadeh,H.,2016b.The first integral method for Wu-Zhang system with conformable time-fractional derivative,Calcolo,53(3),475-485.
Eslami,M.,Rezazadeh,H.,Rezazadeh,M.and Mosavi,S.S.,2017.Exact solutions to the space-time fractional Schr?dinger-Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation,Optical and Quantum Electronics,49(8),279.
Iyiola,O.S.and Nwaeze,E.R.,2016.Some new results on the new conformable fractional calculus with application using D’Alambert approach,Progress in Fractional Differentiation and Applications,2(2),115-122.
Khalil,R.,Al Horani,M.,Yousef,A.and Sababheh,M.,2014.A new definition of fractional derivative,Journal of Computational and Applied Mathematics,264,65-70.
Khodadad,F.S.,Nazari,F.,Eslami,M.and Rezazadeh,H.,2017.Soliton solutions of the conformable fractional Zakharov-Kuznetsov equation with dual-power law nonlinearity,Optical and Quantum Electronics,49(11),384.
Malfliet,W.,1992.Solitary wave solutions of nonlinear wave equations,American Journal of Physics,60(7),650-654.
Osman,M.S.,Korkmaz,A.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.and Zhou,Q.,2018.The unified method for conformable time fractional Schr?dinger equation with perturbation terms,Chinese Journal of Physics,56(5),2500-2506.
Rezazadeh,H.,2018.New solitons solutions of the complex GinzburgLandau equation with Kerr law nonlinearity,Optik,167,218-227.
Rezazadeh,H.,Mirhosseini-Alizamini,S.M.,Eslami,M.,Rezazadeh,M.,Mirzazadeh,M.and Abbagari,S.,2018a.New optical solitons of nonlinear conformable fractional Schr?dinger-Hirota equation,Optik,172,545-553.
Rezazadeh,H.,Mirzazadeh,M.,Mirhosseini-Alizamini,S.M.,Neirameh,A.,Eslami,M.and Zhou,Q.,2018b.Optical solitons of Lakshmanan-Porsezian-Daniel model with a couple of nonlinearities,Optik,164,414-423.
Rezazadeh,H.,Osman,M.S.,Eslami,M.,Ekici,M.,Sonmezoglu,A.,Asma,M.,Othman,W.A.M.,Wong,B.R.,Mirzazadeh,M.,Zhou,Q.,Biswas,A.and Belic,M.,2018c.Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity,Optik,164,84-92.
Sabatier,J.,Agrawal,O.P.and Tenreiro MacHado,J.A.,2007.Advances in Fractional Calculus:Theoretical Developments and Applications in Physics and Engineering,Springer,Dordrecht.
Sachs,R.L.,1988.On the integrable variant of the boussinesq system:Painlevéproperty rational solutions,a related many-body system,and equivalence with the AKNS hierarchy,Physica D:Nonlinear Phenomena,30(1-2),1-27.
Sirendaoreji,and Jiong,S.,2003.Auxiliary equation method for solving nonlinear partial differential equations,Physics Letters A,309(5-6),387-396.
Tariq,K.U.,Younis,M.,Rezazadeh,H.,Rizvi,S.T.R.and Osman,M.S.,2018.Optical solitons with quadratic-cubic nonlinearity and fractional temporal evolution,Modern Physics Letters B,32(26),1850317.
Tasbozan,O.,?enol,M.,Kurt,A.and?zkan,O.,2018.New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves,Ocean Engineering,161,62-68.
Whitham,G.B.,1965.A general approach to linear and non-linear dispersive waves using a Lagrangian,Journal of Fluid Mechanics,22(2),273-283.
Whitham,G.B.and Lighthill,M.J.,1967.Variational methods and applications to water waves,Proceedings of the Royal Society of London.Series A.Mathematical and Physical Sciences,299(1456),6-25.
Yomba,E.,2008.A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations,Physics Letters A,372(7),1048-1060.
Zhang,S.and Xia,T.C.,2007.A generalized new auxiliary equation method and its applications to nonlinear partial differential equations,Physics Letters A,363(5-6),356-360.
Aminikhah,H.,Sheikhani,A.H.R.and Rezazadeh,H.,2016.Subequation method for the fractional regularized long-wave equations with conformable fractional derivatives,Scientia Iranica,23(3),1048-1054.
Atangana,A.,Baleanu,D.and Alsaedi,A.,2015.New properties of conformable derivative,Open Mathematics,13(1),2391-5455.
Batarfi,H.,Losada,J.,Nieto,J.J.and Shammakh,W.,2015.Threepoint boundary value problems for conformable fractional differential equations,Journal of Function Spaces,2015,706383.
Biswas,A.,Al-Amr,M.O.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.,Zhou,Q.,Moshokoa,S.P.and Belic,M.,2018a.Resonant optical solitons with dual-power law nonlinearity and fractional temporal evolution,Optik,165,233-239.
Biswas,A.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.,Ekici,M.,Zhou,Q.,Moshokoa,S.P.and Belic,M.,2018b.Optical soliton perturbation with Fokas-Lenells equation using three exotic and efficient integration schemes,Optik,165,288-294.
Biswas,A.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.,Zhou,Q.,Moshokoa,S.P.and Belic,M.,2018c.Optical solitons having weak non-local nonlinearity by two integration schemes,Optik,164,380-384.
Boussinesq,J.,1872.Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal,en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond,Journal de Mathématiques Pures et Appliquées,17,55-108.(in French)
Eslami,M.and Mirzazadeh,M.,2013.Topological 1-soliton solution of nonlinear Schr?dinger equation with dual-power law nonlinearity in nonlinear optical fibers,The European Physical Journal Plus,128(11),140.
Eslami,M.,2016.Trial solution technique to chiral nonlinear Schrodinger’s equation in(1+2)-dimensions,Nonlinear Dynamics,85(2),813-816.
Eslami,M.and Mirzazadeh,M.,2016a.Optical solitons with BiswasMilovic equation for power law and dual-power law nonlinearities,Nonlinear Dynamics,83(1-2),731-738.
Eslami,M.and Rezazadeh,H.,2016b.The first integral method for Wu-Zhang system with conformable time-fractional derivative,Calcolo,53(3),475-485.
Eslami,M.,Rezazadeh,H.,Rezazadeh,M.and Mosavi,S.S.,2017.Exact solutions to the space-time fractional Schr?dinger-Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation,Optical and Quantum Electronics,49(8),279.
Iyiola,O.S.and Nwaeze,E.R.,2016.Some new results on the new conformable fractional calculus with application using D’Alambert approach,Progress in Fractional Differentiation and Applications,2(2),115-122.
Khalil,R.,Al Horani,M.,Yousef,A.and Sababheh,M.,2014.A new definition of fractional derivative,Journal of Computational and Applied Mathematics,264,65-70.
Khodadad,F.S.,Nazari,F.,Eslami,M.and Rezazadeh,H.,2017.Soliton solutions of the conformable fractional Zakharov-Kuznetsov equation with dual-power law nonlinearity,Optical and Quantum Electronics,49(11),384.
Malfliet,W.,1992.Solitary wave solutions of nonlinear wave equations,American Journal of Physics,60(7),650-654.
Osman,M.S.,Korkmaz,A.,Rezazadeh,H.,Mirzazadeh,M.,Eslami,M.and Zhou,Q.,2018.The unified method for conformable time fractional Schr?dinger equation with perturbation terms,Chinese Journal of Physics,56(5),2500-2506.
Rezazadeh,H.,2018.New solitons solutions of the complex GinzburgLandau equation with Kerr law nonlinearity,Optik,167,218-227.
Rezazadeh,H.,Mirhosseini-Alizamini,S.M.,Eslami,M.,Rezazadeh,M.,Mirzazadeh,M.and Abbagari,S.,2018a.New optical solitons of nonlinear conformable fractional Schr?dinger-Hirota equation,Optik,172,545-553.
Rezazadeh,H.,Mirzazadeh,M.,Mirhosseini-Alizamini,S.M.,Neirameh,A.,Eslami,M.and Zhou,Q.,2018b.Optical solitons of Lakshmanan-Porsezian-Daniel model with a couple of nonlinearities,Optik,164,414-423.
Rezazadeh,H.,Osman,M.S.,Eslami,M.,Ekici,M.,Sonmezoglu,A.,Asma,M.,Othman,W.A.M.,Wong,B.R.,Mirzazadeh,M.,Zhou,Q.,Biswas,A.and Belic,M.,2018c.Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity,Optik,164,84-92.
Sabatier,J.,Agrawal,O.P.and Tenreiro MacHado,J.A.,2007.Advances in Fractional Calculus:Theoretical Developments and Applications in Physics and Engineering,Springer,Dordrecht.
Sachs,R.L.,1988.On the integrable variant of the boussinesq system:Painlevéproperty rational solutions,a related many-body system,and equivalence with the AKNS hierarchy,Physica D:Nonlinear Phenomena,30(1-2),1-27.
Sirendaoreji,and Jiong,S.,2003.Auxiliary equation method for solving nonlinear partial differential equations,Physics Letters A,309(5-6),387-396.
Tariq,K.U.,Younis,M.,Rezazadeh,H.,Rizvi,S.T.R.and Osman,M.S.,2018.Optical solitons with quadratic-cubic nonlinearity and fractional temporal evolution,Modern Physics Letters B,32(26),1850317.
Tasbozan,O.,?enol,M.,Kurt,A.and?zkan,O.,2018.New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves,Ocean Engineering,161,62-68.
Whitham,G.B.,1965.A general approach to linear and non-linear dispersive waves using a Lagrangian,Journal of Fluid Mechanics,22(2),273-283.
Whitham,G.B.and Lighthill,M.J.,1967.Variational methods and applications to water waves,Proceedings of the Royal Society of London.Series A.Mathematical and Physical Sciences,299(1456),6-25.
Yomba,E.,2008.A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations,Physics Letters A,372(7),1048-1060.
Zhang,S.and Xia,T.C.,2007.A generalized new auxiliary equation method and its applications to nonlinear partial differential equations,Physics Letters A,363(5-6),356-360.