Abstract and Applied Analysis

Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation

Xiao-Jun Yang, Jordan Hristov, H. M. Srivastava, and Bashir Ahmad

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A mathematical model of fractal waves on shallow water surfaces is developed by using the concepts of local fractional calculus. The derivations of linear and nonlinear local fractional versions of the Korteweg-de Vries equation describing fractal waves on shallow water surfaces are obtained.

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Abstr. Appl. Anal., Volume 2014 (2014), Article ID 278672, 10 pages.

First available in Project Euclid: 2 October 2014

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Yang, Xiao-Jun; Hristov, Jordan; Srivastava, H. M.; Ahmad, Bashir. Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation. Abstr. Appl. Anal. 2014 (2014), Article ID 278672, 10 pages. doi:10.1155/2014/278672. https://projecteuclid.org/euclid.aaa/1412277044

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  • J. Boussinesq, Essai sur la Théorie des Eaux Courantes, vol. 23 of Mémoires Présentés par Divers Savants à l\textquotesingle Académie des Sciences de l'Institut National de France, Imprimerie Nationale, 1877.
  • D. J. Korteweg and G. de Vries, “XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 39, no. 240, pp. 422–443, 1895.
  • J. Canosa and J. Gazdag, “The Korteweg-de Vries-Burgers equation,” Journal of Computational Physics, vol. 23, no. 4, pp. 393–403, 1977.
  • J. P. Boyd, “Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves,” Physica D: Nonlinear Phenomena, vol. 21, no. 2-3, pp. 227–246, 1986.
  • M. Tsutsumi, T. Mukasa, and R. Iino, “On the generalized Korteweg-de Vries equation,” Proceedings of the Japan Academy, vol. 46, no. 9, pp. 921–925, 1970.
  • R. Dodd and A. Fordy, “The prolongation structures of quasipolynomial flows,” Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, vol. 385, no. 1789, pp. 389–429, 1983.
  • F. Calogero and A. Degasperis, Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations, North-Holland, New York, NY, USA, 1982.
  • R. S. Johnson, “On the inverse scattering transform, the cylindrical Korteweg-de Vries equation and similarity solutions,” Physics Letters A, vol. 72, no. 3, pp. 197–199, 1979.
  • G. Segal, “The geometry of the KdV equation,” International Journal of Modern Physics A, vol. 6, no. 16, pp. 2859–2869, 1991.
  • M. Wadati, “Stochastic Korteweg-de Vries equation,” Journal of the Physical Society of Japan, vol. 52, no. 8, pp. 2642–2648, 1983.
  • R. M. Miura, “Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,” Journal of Mathematical Physics, vol. 9, no. 8, pp. 1202–1204, 1968.
  • J. W. Miles, “The Korteweg-de Vries equation: a historical essay,” Journal of Fluid Mechanics, vol. 106, pp. 131–147, 1981.
  • S. Momani, Z. Odibat, and A. Alawneh, “Variational iteration method for solving the space- and time-fractional KdV equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 1, pp. 262–271, 2008.
  • S. A. El-Wakil, E. Abulwafa, M. Zahran, and A. Mahmoud, “Time-fractional KdV equation: formulation and solution using variational methods,” Nonlinear Dynamics, vol. 65, no. 1-2, pp. 55–63, 2011.
  • A. Atangana and A. Secer, “The time-fractional coupled-Korteweg-de-Vries equations,” Abstract and Applied Analysis, vol. 2013, Article ID 947986, 8 pages, 2013.
  • O. Abdulaziz, I. Hashim, and E. S. Ismail, “Approximate analytical solution to fractional modified KdV equations,” Mathematical and Computer Modelling, vol. 49, no. 1-2, pp. 136–145, 2009.
  • X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  • X.-J. Yang, D. Baleanu, and J.-H. He, “Transport equations in fractal porous media within fractional complex transform method,” Proceedings of the Romanian Academy A, vol. 14, no. 4, pp. 287–292, 2013.
  • Y.-J. Hao, H. M. Srivastava, H. Jafari, and X.-J. Yang, “Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates,” Advances in Mathematical Physics, vol. 2013, Article ID 754248, 5 pages, 2013.
  • Y. Zhao, D. Baleanu, C. Cattani, D. F. Cheng, and X.-J. Yang, “Maxwell\textquotesingle s equations on Cantor sets: a local fractional approach,” Advances in High Energy Physics, vol. 2013, Article ID 686371, 6 pages, 2013.
  • X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of Navier-Stokes equations on cantor sets,” Mathematical Problems in Engineering, vol. 2013, Article ID 769724, 8 pages, 2013.
  • A. M. Yang, C. Cattani, C. Zhang, G. N. Xie, and X. J. Yang, “Local fractional Fourier series solutions for non-homogeneous heat equations arising in fractal heat flow with local fractional derivative,” Advances in Mechanical Engineering, vol. 2014, Article ID 514639, 5 pages, 2014.
  • Y. Y. Li, Y. Zhao, G. N. Xie, D. Baleanu, X. J. Yang, and K. Zhao, “Local fractional Poisson and Laplace equations with applications to electrostatics in fractal domain,” Advances in Mathematical Physics, vol. 2014, Article ID 590574, 5 pages, 2014.
  • C. Y. Long, Y. Zhao, and H. Jafari, “Mathematical models arising in the fractal forest gap via local fractional calculus,” Abstract and Applied Analysis, vol. 2014, Article ID 782393, 6 pages, 2014.
  • S. Q. Wang, Y. J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,” Abstract and Applied Analysis, vol. 2014, Article ID 176395, 7 pages, 2014.
  • X.-J. Yang, H. M. Srivastava, J.-H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,” Physics Letters A, vol. 377, no. 28–30, pp. 1696–1700, 2013.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematical Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  • X.-J. Ma, H. M. Srivastava, D. Baleanu, and X.-J. Yang, “A new Neumann series method for solving a family of local fractional Fredholm and Volterra integral equations,” Mathematical Problems in Engineering, vol. 2013, Article ID 325121, 6 pages, 2013.
  • A.-M. Yang, Z.-S. Chen, H. M. Srivastava, and X.-J. Yang, “Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving local fractional derivative operators,” Abstract and Applied Analysis, vol. 2013, Article ID 259125, 6 pages, 2013.
  • W. Wei, H. M. Srivastava, L. Wang, P.-Y. Shen, and J. Zhang, “A local fractional integral inequality on fractal space analogous to Aderson\textquotesingle s inequality,” Abstract and Applied Analysis, vol. 2014, Article ID 797561, 7 pages, 2014. \endinput