THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

SHALLOW WATER WAVES IN POROUS MEDIUM FOR COAST PROTECTION

ABSTRACT
This paper extends the Hirota-Satsuma equation in continuum mechanics to its fractional partner in fractal porous media in shallow water for absorbing wave energy and preventing tsunami. Its derivation is briefly introduced using the fractional momentum law and He's fractional derivative. The fractional complex transform is adopted to elucidate its basic solution properties, and a modification of the exp-function method is used to solve the equation. The paper concludes that the kinetic energy of the travelling wave tends to be vanished when the value of the fractional order is less than one.
KEYWORDS
PAPER SUBMITTED: 2017-03-12
PAPER REVISED: 2017-05-15
PAPER ACCEPTED: 2017-05-24
PUBLISHED ONLINE: 2017-12-02
DOI REFERENCE: https://doi.org/10.2298/TSCI17S1145W
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE Supplement 1, PAGES [S145 - S151]
REFERENCES
  1. Wang, Y., et al., A Short Review on Analytical Methods for Fractional Equations with He's Fractional Derivative, Thermal Science, 21 (2017), 4, pp. 1567-1574
  2. Wang, Y., et al., A Fractional Whitham-Broer-Kaup Equation and its Possible Application to Tsunami Prevention, Thermal Science, 21 (2017), 4, pp. 1847-1855
  3. Yang, X. J., et al., A New Fractional Operator of Variable Order: Application in the Description of Anomalous Diffusion, Physica A, 481 (2017), Sept., pp. 276-283
  4. Yang, X. J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New York, USA, 2015
  5. 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
  6. Yang, X. J., et al., Exact Travelling Wave Solutions for the Local Fractional Two-Dimensional Burgers- type Equations. Computers and Mathematics with Applications, 73 (2017), 2, pp. 203-210
  7. Yang, X. J., et al., On Exact Traveling-Wave Solutions for Local Fractional Korteweg-de Vries Equation. Chaos, 26 (2016), 8, pp. 110-118
  8. Yang, X. J., et al., On Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain, Fractals, 25 (2017), 4, 1740006
  9. Yang, X. J., et al., A New Fractional Derivative without Singular Kernel: Application to the Modeling of the Steady Heat Flow, Thermal Science, 20 (2016), 2, pp. 753-756
  10. Yang, X. J., Fractional Derivatives of Constant and Variable Orders Applied to Anomalous Relaxation Models in Heat-transfer Problems. Thermal Science, 21 (2017), 3, pp. 1161-1171
  11. Hu, Y., He, J.-H., On Fractal Space-Time and Fractional Calculus, Thermal Science, 20 (2016), 3 pp. 773-777
  12. He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
  13. 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
  14. Liu, H. Y., et al., Fractional Calculus for Nanoscale Flow and Heat Transfer, International Journal of Numerical Methods for Heat & Fluid Flow, 24 (2014), 6, pp. 1227-1250
  15. Li, Z. B., He, J. H., Application of the Fractional Complex Transform to Fractional Differential Equations, Mathematical & Computational Applications, 15 (2011), 5, pp. 97-137
  16. He, J.-H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334.
  17. 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
  18. 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
  19. Liu, F. J., et al., Fractional Model for Insulation Clothings with Cocoon-Like Porous Structure, Thermal Science, 20 (2016), 3, pp. 779-784
  20. 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
  21. 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
  22. Wang, K. L., Liu, S. Y., He's Fractional Derivative for Nonlinear Fractional Heat Transfer Equation, Thermal Science, 20 (2016), 3, pp. 793-796
  23. 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
  24. Liu, H. Y., et al., A Fractional Model for Heat Transfer in Mongolian Yurt, Thermal Science, 21 (2017), 4, pp. 1861-1866
  25. Hirota, R., Satsuma, J., N-Soliton Solutions of Model Equations for Shallow Water Waves, Journal of the Physical Society of Japan, 40 (1976), 2, pp. 611-612
  26. Huang, L., Chen, Y., Nonlocal Symmetry and Similarity Reductions for the Drinfeld-Sokolov-Satsuma- Hirota System, Applied Mathematics Letters, 64 (2017), Feb., pp. 177-184
  27. Lu, X., et al., A Note on Rational Solutions to a Hirota-Satsuma-Like Equation, Applied Mathematics Letters, 58 (2016), Aug., pp. 13-18
  28. Zhang, M. F., et al., Efficient Homotopy Perturbation Method for Fractional Non-Linear Equations Using Sumudu Transform, Thermal Science, 19 (2015), 4, pp. 1167-1171
  29. Yang, X. J., et al., Local Fractional Homotopy Perturbation Method for Solving Fractal Partial Differential Equations Arising in Mathematical Physics, Romanian Reports in Physics, 67 (2014), 3 pp. 752-761
  30. He, J. H., A Short Remark on Fractional Variational Iteration Method, Physics Letters A, 375 (2011), 38, pp. 3362-3364
  31. He, J. H., Generalized Equilibrium Equations for Shell Derived from a Generalized Variational Principle, Applied Mathematics Letters, 64 (2016), Feb., pp. 94-100
  32. He, J. H., et al., Variational Iteration Method for Bratu-like Equation Arising in Electrospinning, Carbohydrate Polymers, 105 (2014), 1, pp. 229-230
  33. EI-Sayed, A. M. A., et al., Adomian's Decomposition Method for Solving an Intermediate Fractional Advection-Dispersion Equation, Pergamon Press, Inc., 59 (2010) 5, pp. 1759-1765
  34. Bekir, A., Aksoy, E., A Generalized Fractional Sub-Equation Method for Nonlinear Fractional Differential Equations, International Conference on Analysis & Applied Mathematics, 1611 (2014), 1, pp. 78-83
  35. Khaleghizadeh, S., On Fractional Sub-Equation Method, Nonlinear Science Letters A, 8 (2017), 1, pp. 66-76
  36. Arshad, M., et al., Fractional Sub-Equation Method for a Generalized Space-Time Fractional Fisher Equation with Variable Coefficients, Nonlinear Science Letters A, 8 (2017), 2, pp. 162-170
  37. Rajeev, Singh, A. K., Homotopy Analysis Method for a Fractional Stefan Problem, Nonlinear Science Letters A, 8 (2017), 1, pp. 50-59
  38. Sohail, A., et al., Travelling Wave Solutions for Fractional Order KdV-Like Equations Using G′/G-Expansion Method, Nonlinear Science Letters A, 8 (2017), 2, pp. 228-235
  39. He, J. H., Asymptotic Methods for Solitary Solutions and Compactons, Abstract & Applied Analysis, 2012 (2012), ID 916793
  40. He, J. H., Exp-Function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences & Numerical Simulation, 14 (2013), 6, pp. 363-366
  41. Zhou, X. W., Exp-Function Method for Solving Huxley Equation, Mathematical Problems in Engineering, 2008 (2008), ID 538489
  42. Guner, O., Bekir, A., Exp-Function Method for Nonlinear Fractional Differential Equations, Nonlinear Science Letters A, 8 (2017), 1, pp. 41-49
  43. He, J. H., Wu, X. H., Exp-Function Method for Nonlinear Wave Equations, Chaos, Solitons & Fractals, 30 (2006), 3, pp. 700-708
  44. 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
  45. He, J.-H., A New Fractal Derivation, Thermal Science, 15 (2011), Suppl. 1, pp. S145-S147
  46. Wu, X. E., Liang, Y. S., Relationship between Fractal Dimensions and Fractional Calculus, Nonlinear Science Letters A, 8 (2017), 1, pp. 77-89
  47. Wang, Y., Control of Solitary Waves by Artificial Boundaries, Nonlinear Science Letters A, 8 (2017), 3, pp. 337-338

© 2017 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence