# MODIFIED KAWAHARA EQUATION WITHIN A FRACTIONAL DERIVATIVE WITH NON-SINGULAR KERNEL

## Main Article Content

## Abstract

The article addresses a time-fractional modified Kawahara equation through a fractional derivative with exponential kernel. The Kawahara equation describes the generation of nonlinear 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.

## Article Details

**Thermal Science**, [S.l.], mar. 2017. ISSN 2334-7163. Available at: <http://thermal-science.tech/journal/index.php/thsci/article/view/2403>. Date accessed: 20 feb. 2018. doi: https://doi.org/10.2298/TSCI160826008K.

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Accepted 2017-03-14

Published 2017-03-14

## References

[2] Losada, J., Nieto, J.J., Properties of the new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), pp. 87-92.

[3] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.

[4] Tarasov, V.E., Three-dimensional lattice models with long-range interactions of Grünwald- Letnikov type for fractional generalization of gradient elasticity, Meccanica 51 (2016), 1, pp. 125- 138.

[5] Odibat, Z.M., Momani, S., Application of variational iteration method to nonlinear differential equation of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 1, pp. 27-34.

[6] Baleanu, D., Guvenc, Z.B., Machado, J.A.T. (Ed.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer Dordrecht Heidelberg, London New York, 2010.

[7] Atangana, A., Koca, I., On the new fractional derivative and application to nonlinear Baggs and Freedman Model, J. Nonlinear Sci. Appl., 9 (2016), pp. 2467-2480.

[8] Baskonus, H.M., Bulut, H., Pandir, Y., On The Solution of Nonlinear Time-Fractional Generalized Burgers Equation by Homotopy Analysis Method and Modified Trial Equation Method, International Journal of Modeling and Optimization, 4 (2014), 4, pp. 305-309.

[9] Razminia, K., Razminia, A., Machado, J.A.T., Analytical Solution of Fractional Order diffusivity equation with wellbore storage and skin effects, J. Comput. Nonlinear Dynam., 11 (2016), 1, doi: 10.1115/1.4030534.

[10] Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction- diffusion equation, Applied Mathematics and Computation, 273 (2016), pp. 948-956.

[11] Kumar, D., Singh, J., Kumar, S., Sushila , Singh, B.P., Numerical Computation of Nonlinear Shock Wave Equation of Fractional Order, Ain Shams Engineering Journal, 6 (2015), 2, pp. 605- 611.

[12] Bulut, H., Belgacem, F.B.M., Baskonus, H.M., Some New Analytical Solutions for the Nonlinear Time-Fractional KdV-Burgers-Kuramoto Equation, Advances in Mathematics and Statistical Sciences, (2015), pp. 118-129.

[13] Kumar, D., Singh, J., Baleanu, D, Numerical computation of a fractional model of differential- difference equation, Journal of Computational and Nonlinear Dynamics, 11 (2016), doi: 10.1115/1.4033899.

[14] Singh, J., Kumar, D., Nieto, J.J., A reliable algorithm for local fractional Tricomi equation arising in fractal transonic flow, Entropy, 18 (2016), 6, doi: 10.3390/e18060206.

[15] Choudhary, A., Kumar, D., Singh, J., Numerical Simulation of a fractional model of temperature distribution and heat flux in the semi infinite solid, Alexandria Engineering Journal, 55 (2016), 1, 87–91.

[16] Singh, J., Kumar, D., Swroop, R., Numerical solution of time- and space-fractional coupled Burgers equations via homotopy algorithm, Alexandria Engineering Journal, (2016), doi:10.1016/j.aej.2016.03.028.

[17] Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M.M., Cattani, C., Wavelets method for solving fractional optimal control problems, Applied Mathematics and Computation, 286 (2016), pp. 139-154.

[18] Atangana A., Baleanu D., New fractional derivatives with nonlocal and non-singular kernel, Theory and application to heat transfer model, Thermal Science, (2016), 20(2), pp. 763-769.

[19] Gomez-Aguilar J.F., Morales-Delgado V.F., Taneco-Hernandez M.A., Baleanu D., Escobar-Jimenez, R.F., Al Qurashi M.M., Analytical solutions of the electrical RLC circuit via Liouville-Caputo operators with local and non-local kernels, Entropy, 2016, 18(8), Article Number: 402.

[20] Coronel-Escamilla A., Gomez-Aguilar J.F., Baleanu D., Escobar-Jimenez R.F., Olivares- Peregrino V.H., Abundez-Pliego A., Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel, Advances in Difference Equations, 2016, Article Number: 283, doi: 10.1186/s13662-016-1001-5.

[21] Hristov, J., Transient Heat Diffusion with aNon-Singular Fading Memory: From the Cattaneo Constitutive Equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 2, pp.765-770.

[22] Hristov, J., 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, in press; Thermal Science, (2016) OnLine-First (00):115-115; DOI:10.2298/TSCI160229115H.

[23] Kawahara, T., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), pp. 260–264.

[24] Amick C.J., Kirchgӓssner, K., A theory of solitary water-waves in the presence of surface tension, Arch. Rational Mech. Anal., 105 (1989), pp. 1–49.

[25] Amick, C.J., McLeod, J.B., A singular perturbation problem in water waves, Stab. Appl. Anal. of Cont. Media, (1992), pp. 127-148.

[26] Kakutani, T., Ono, H., Weak non-linear hydromagnetic waves in a cold collision free plasma, J. Phys. Soc. Japan, 26 (1969), pp. 1305–1318.

[27] Hasimoto, H., Water waves, Kagaku, 40 (1970), pp. 401–408 (Japanese).

[28] Daoreji, S., New exact travelling wave solutions for the Kawahara and modified Kawahara equations, Chaos Solitons Fractals, 19 (2004), pp. 147-150.

[29] Kurulay, M., Approximate analytic solutions of the modified Kawahara equation with homotopy analysis method, Advances in Difference Equations, 2012(2012), p. 178.

[30] Qing, Y., Rhoades, B.E., T-stability of Picard iteration in metric spaces, Fixed Point Theory and Applications, (2008), Article ID 418971, 4 pages.