Journal of Applied Nonlinear Dynamics
Explicit Solutions of Coupled Time Fractional KaupBoussinesq Equation with Weak Dispersion
Journal of Applied Nonlinear Dynamics 8(4) (2019) 585594  DOI:10.5890/JAND.2019.12.006
Hemanta Mandal, B. Bira
Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Chennai603203, India
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Abstract
In the current article, we consider a system of time fractional KaupBoussinesq shallow water equations. Using Lie group analysis, we obtain the symmetry group of transformations which reduces the system of fractional partial differential equations (FPDEs) to system of fractional ordinary differential equations (FODEs). Further, we investigate the exact explicit group invariant solution as well as power series solution of the given system of equations. Next the physical significance of the group invariant solution under the influence of fractional order is studied graphically. Lastly, conserved vectors for the FPDEs are obtained using the conservation theorem.
References

[1]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Elsevier Science B.V: Amsterdam. 

[2]  Hilfer, R. (2000), Applications of fractional calculus in physics, World Scientific: Singapore. 

[3]  Carpinteri, A. and Mainardi, F. (1997), Fractals and Fractional Calculus in Continuum and Mechanics, Springer: New York. 

[4]  Xu, X.Y. and Tan, W.C. (2006), Intermediate processes and critical phenomena: theory, method and progress of fractional operators and their applications to modern mechanics, Sci. Series Ser. G: Phys. Mech. Astron., 49, 257272. 

[5]  Whitham, G.B. (1999), Linear and Nonlinear Waves, J. Wiley and Sons: New York. 

[6]  El, G.A., Grimshaw, R.H.J., and Pavlov, M.V. (2001), Integrable shallowwater equations and undular bores, Stud. Appl. Math., 106, 157186. 

[7]  Ivanov, R.I. (2009), Two component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46, 389396. 

[8]  Dehghan, M., Manafian, J., and Saadatmandi, A. (2010), Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Eq., 26, 448479. 

[9]  Gazizov, R.K. and Kasatkin, A.A. (2013), Construction of exact solutions for fractional order differential equations by invariant subspace method, Comput. Math. Appl., 66, 576584. 

[10]  Bekir, A., Askoy, E., and Cevikel, A.C. (2015), Exact solutions of nonlinear time fractional differential equations by subequation method, Math. Methods Appl. Sci., 38, 27792784. 

[11]  ElSayed, A.M.A. and Gaber, M. (2006), The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys Lett A, 359, 175182. 

[12]  Liu, J. and Hou, G. (2011), Numerical solutions of the spaceand timefractional coupled Burgers equations by generalized differential transform method, Appl. Math. Comput., 217, 70017008. 

[13]  Yasar, E. and Giresunlu, I.B. (2015), Lie symmetry reductions exact solutions and conservation laws of the third order variant Boussinesq system, Appl. Anal., 128, 252255. 

[14]  Singla, K. and Gupta, R.K. (2017), Conservation laws for certain time fractional nonlinear systems of partial differential equations, Commun. Nonlinear Sci. Numer. Simulat., 53, 1021. 

[15]  Hu, J., Ye, Y., Shen, S., and Zhang, J. (2014), Lie symmetry analysis of the time fractional KdVtype equation, Appl. Math. Comput., 233, 439444. 

[16]  Liu, H.Z. (2013), Complete group classifications and symmetry reductions of the fractional fifthorder KdV types of equations, Stud. Appl. Math., 131, 317330. 

[17]  Djordjevic, V.D. and Atanackovic, T.M. (2008), Similarity solutions to nonlinear heat conduction and Burgers/KortewegdeVries fractional equations, J. Comput. Appl. Math., 212, 701714. 

[18]  Buckwar, E. and Luchko Y. (1998), Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, J. Math. Anal. Appl., 227, 8197. 

[19]  Samko, S., Kilbas, A.A., and Marichev, O. (1993), Fractional integrals and derivatives: theory and applications, Gordon and Breach science: Switzerland. 

[20]  Miller, K.S. and Ross, B. (1993), An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons: New York. 

[21]  Oldham, K.B. and Spanier, J. (1974), The fractional calculus, Academic Press: New York. 

[22]  Kamchatnov, A.M., Kraenkel, R.A., and Umarov, B.A. (2003), Asymptotic soliton train solutions of Kaup Boussinesq equations, wave motion, 38, 355365. 

[23]  Olver, P.J. (1986), Applications of Lie groups to Differential groups, Springer: New York. 

[24]  Bluman, G.W. and Kumei, S. (1989), Symmetries and differential equation, Springer: New York. 

[25]  Bluman, G.W. and Anco, S.C. (2002), Symmetries and integration methods for differential equations, Springerverlag: New York. 

[26]  Costa, F.S., MarĂ£o, J.A.P.F., Soares, J.C.A., and de Oliveira, E.C. (2015), Similarity solution to fractional nonlinear spacetime diffusionwave equation, J. Math. Phys., 56, 033507. 

[27]  Plociniczak, L. (2014), Appriximation of the ErdelyiKober operator with application to the timefractional porous medium equation, Siam J. Appl. Math., 74, 12191237. 