Modified Kawahara equation within a fractional derivative with non-singular kernel

The article addresses a time-fractional modified Kawahara equation through a fractional derivative with exponential kernel. The Kawahara equation describes the generation of non-linear water-waves in the long-wavelength regime. The numerical solution of the fractional model of modified version of Kawahara equation is derived with the help of iterative scheme and the stability of applied technique is established. In order to demonstrate the usability and effectiveness of the new fractional derivative to describe water-waves in the long-wavelength regime, numerical results are presented graphically.

[1]  Devendra Kumar,et al.  Numerical solution of time- and space-fractional coupled Burgers’ equations via homotopy algorithm , 2016 .

[2]  Dumitru Baleanu,et al.  Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel , 2016 .

[3]  Sirendaoreji.  New exact travelling wave solutions for the Kawahara and modified Kawahara equations , 2004 .

[4]  Numerical simulation of a fractional model of temperature distribution and heat flux in the semi infinite solid , 2016 .

[5]  S. Arabia,et al.  Properties of a New Fractional Derivative without Singular Kernel , 2015 .

[6]  Devendra Kumar,et al.  Numerical computation of nonlinear shock wave equation of fractional order , 2015 .

[7]  Abdon Atangana,et al.  On the new fractional derivative and application to nonlinear Baggs and Freedman model , 2016 .

[8]  V. E. Tarasov.  Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity , 2016 .

[9]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[10]  Takuji Kawahara,et al.  Oscillatory Solitary Waves in Dispersive Media , 1972 .

[11]  B. Rhoades,et al.  T-Stability of Picard Iteration in Metric Spaces , 2008 .

[12]  J. A. Tenreiro Machado,et al.  New Trends in Nanotechnology and Fractional Calculus Applications , 2010 .

[13]  J. Hristov.  STEADY-STATE HEAT CONDUCTION IN A MEDIUM WITH SPATIAL NON-SINGULAR FADING MEMORY: Derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey’s kernel and analytical solutions , 2016 .

[14]  H. M. Baskonus,et al.  On the Solution of Nonlinear Time-Fractional Generalized Burgers Equation by Homotopy Analysis Method and Modified Trial Equation Method , 2014 .

[15]  J. Machado,et al.  Analytical Solution of Fractional Order Diffusivity Equation With Wellbore Storage and Skin Effects , 2016 .

[16]  Maysaa Mohamed Al Qurashi,et al.  Analytical Solutions of the Electrical RLC Circuit via Liouville-Caputo Operators with Local and Non-Local Kernels , 2016, Entropy.

[17]  M. Kurulay.  Approximate analytic solutions of the modified Kawahara equation with homotopy analysis method , 2012 .

[18]  Dumitru Baleanu,et al.  Numerical Computation of a Fractional Model of Differential-Difference Equation , 2016 .

[19]  Devendra Kumar,et al.  A Reliable Algorithm for a Local Fractional Tricomi Equation Arising in Fractal Transonic Flow , 2016, Entropy.

[20]  J. Hristov.  Steady-state heat conduction in a medium with spatial non-singular fading memory: Derivation of Caputo-Fabrizio space-fractional derivative from Cattaneo concept with Jeffrey`s Kernel and analytical solutions , 2017 .

[21]  Abdon Atangana,et al.  On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation , 2016, Appl. Math. Comput..

[22]  Klaus Kirchgässner,et al.  A theory of solitary water-waves in the presence of surface tension , 1989 .

[23]  Carlo Cattani,et al.  Wavelets method for solving fractional optimal control problems , 2016, Appl. Math. Comput..

[24]  S. Momani,et al.  Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order , 2006 .

[25]  Hiroaki Ono,et al.  Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma , 1969 .

[26]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[27]  Jordan Hristov,et al.  Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey’s Kernel to the Caputo-Fabrizio time-fractional derivative , 2016 .