## THERMAL SCIENCE

International Scientific Journal

### A FRACTIONAL WHITHAM-BROER-KAUP EQUATION AND ITS POSSIBLE APPLICATION TO TSUNAMI PREVENTION

**ABSTRACT**

A fractional Whitham-Broer-Kaup equation is suggested using He's fractional
derivative to model solitary waves in shallow water in porous medium near a
dam. A modification of the exp-function method, the generalized exponential rational function method, is adopted to elucidate the basic solution properties of the equation, revealing that the value of the fractional order can be used effectively to control the wave velocity, the wave height, and the wave morphology. This theoretical result can be used for possible tsunami prevention.

**KEYWORDS**

PAPER SUBMITTED: 2016-05-10

PAPER REVISED: 2016-06-28

PAPER ACCEPTED: 2016-08-29

PUBLISHED ONLINE: 2017-09-09

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**Issue 4**, PAGES [1847 - 1855]

- He, J.-H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
- He, J.-H., Liu, F. J., Local Fractional Variational Iteration Method for Fractal Heat Transfer in Silk Cocoon Hierarchy, Nonlinear Science Letters A, 4 (2013), 1, pp. 15-20
- Liu, F. J., et al., He's Fractional Derivative for Heat Conduction in a Fractal Medium Arising in Silkworm Cocoon Hierarchy, Thermal Science, 19 (2015), 4, pp. 1155-1159
- Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628
- He, J.-H., et al., A New Fractional Derivative and its Application to Explanation of Polar Bear Hairs, Journal of King Saud University Science, 28 (2016), 2, pp. 190-192
- Wang, K. L., Liu, S. Y., A New Solution Procedure for Nonlinear Fractional Porous Media Equation Based on a New Fractional Derivative, Nonlinear Science Letters A, 7 (2016), 4, pp. 135-140
- Wang, K. L., Liu, S. Y., He's Fractional Derivative for Nonlinear Fractional Heat Transfer Equation, Thermal Science, 20 (2016), 3, pp. 793-796
- Sayevand, K., Pichaghchi, K., Analysis of Nonlinear Fractional KdV Equation Based on He's Fractional Derivative, Nonlinear Science Letters A, 7 (2016), 3, pp. 77-85
- Liu, F. J., et al., A Fractional Model for Insulation Clothings with Cocoon-Like Porous Structure, Thermal Science, 20 (2016), 3, pp. 779-784
- Zhu, W. H., et al., An Analysis of Heat Conduction in Polar Bear Hairs Using One-Dimensional Fractional Model, Thermal Science, 20 (2016), 3, pp. 785-788
- Hu, Y., et al., On Fractal Space-Time and Fractional Calculus, Thermal Science, 20 (2016), 3, pp. 773-777
- Sayevand, K., Pichaghchi, K., Analysis of Nonlinear Fractional KdV Equation Based on He's Fractional Derivative, Nonlinear Science Letters A, 7 (2016), 1, pp. 77-85
- He, J.-H., et al., Geometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus, Physics Letters A, 376 (2012), 4, pp. 257-259
- Wang, L., Chen, X., Approximate Analytical Solutions of Time Fractional Whitham-Broer-Kaup Equations by a Residual Power Series Method, Entropy, 17 (2015), 9, pp. 6519-6533
- Ma, H. C., et al., Exact Solutions of Nonlinear Fractional Partial Differential Equations by Fractional Sub-Equation Method, Thermal Science, 19 (2015), 4, pp. 1239-1244
- Li, Z. B., He, J.-H., Fractional Complex Transform for Fractional Differential Equations, Mathematical & Computational Applications, 15 (2010), 5, pp. 970-973
- He, J.-H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
- Li, Z. B., et al., Exact Solutions of Time-Fractional Heat Conduction Equation by the Fractional Complex Transform, Thermal Science, 16 (2012), 2, pp. 335-338
- He, J.-H., Maximal Thermo-Geometric Parameter in a Nonlinear Heat Conduction Equation, Bulletin of the Malaysian Mathematical Sciences Society, 39 (2016), 2, pp. 605-608
- He, J.-H., Exp-Function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation, 14 (2013), 6, pp. 363-366
- Guner, O., Bekir, A., Exp-Function Method for Nonlinear Fractional Differential Equations, Nonlinear Science Letters A, 8 (2017), 1, pp. 41-49
- Aksoy, E., et al., Exponential Rational Function Method for Space-Time Fractional Differential Equations, Waves in Random and Complex Media, 26 (2016), 2, pp. 142-151