Research Article | Open Access

Xiao-Jun Yang, Jordan Hristov, H. M. Srivastava, Bashir Ahmad, "Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation", *Abstract and Applied Analysis*, vol. 2014, Article ID 278672, 10 pages, 2014. https://doi.org/10.1155/2014/278672

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

**Academic Editor:**Dumitru Baleanu

#### Abstract

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.

#### 1. Introduction

The mathematical model of shallow water waves, conceived by Boussinesq [1], was rediscovered by Korteweg and de Vries [2]. It is commonly known as Korteweg-de Vries equation (KdV) [2, 3] and is given by

Several versions of the KdV equations found in the literature are listed below.(i)Generalized KdV equation (GKdV) [4] (ii)Generalized-generalized KdV equation (GGKdV) [5] (iii)Deformed KdV equation (DKdV) [6] (iv)Modified-modified KdV equation (MM KdV) [6] whereare constants. (i)Modified KdV equation (M KdV) [7] (ii)Spherical KdV (SKdV) was [7] (iii)Cylindrical KdV equation (CKdV) was [8]

For Further versions of KdV equation, we refer the reader to [9–12] and the references cited therein.

Recently, the fractional KdV equations have been discussed by several authors. Momani et al. [13] studied the KdV equation with both space- and time-fractional derivatives, while the time-fractional derivative case has been considered by El-Wakil et al. [14]. Atangana and Secer [15] developed solutions for coupled Korteweg-de Vries equations with time-fractional derivatives [15]. Abdulaziz et al. [16] discussed the modified KdV equations with different space- and time-fractional derivatives.

It is imperative to note that the above mentioned works are based on the fractional calculus of differentiable functions. However, there are certain nondifferentiable physical quantities describing the physical parameters locally, where the concept of differentiable functions is not applicable. In such cases the local fractional calculus (LFC) concept allows to obtain solutions adequate to such nondifferentiable problems [17–25] such as local fractional Helmholtz and diffusion equations [19], local fractional Navier-Stokes equations in fractal domain [21], local fractional Poisson and Laplace equations arising in the electrostatics in fractal domain [23], fractional models in forest gap [24], inhomogeneous local fractional wave equations [25], local fractional heat conduction equation [26], and other results [26–30].

In the present work, we focus on the derivation of the linear and the nonlinear local fractional versions of the Korteweg-de Vries equation describing fractal waves on shallow water surfaces.

The paper is organized as follows. In Section 2, we recall the local fractional conservation laws for the quantities in mathematical physics while the local fractional Korteweg-de Vries equation is derived from local fractional calculus in Section 3. The conclusions are outlined in Section 4.

#### 2. Theoretical Background

##### 2.1. Local Fractional Conservation Laws Arising in Mathematical Physics

First of all, we discuss the local fractional conservation laws of mass, energy, and momentum in fractal media.

Let us consider the quantitywhich varies within the fractal volume. Observe that the variations inwith respect to the fractal time corresponds to the variation in the flux through the fractal boundaryor by a source inside the volume. The integral form of local fractional conservation of the quantityis given by [17, 19, 21] where is the fractal flux vector and is the source (sink) for a nondifferentiable quantity.

The local fractional surface integral is defined by [17, 19–22] where is the local fractal surface and denote elements of the surface with unit normal local fractional vector. Whenas, the local fractional volume integral of the functiontakes the form [17, 19–23]

The local fractional derivative of a function of orderis defined by [17, 24, 25] with

Using (9), the local fractional differential form of the local fractional conservation balance of the quantity can be expressed as

The local fractional gradient of the scale functionemerging from (14) is [17]

In the Cantorian coordinates, the local fractional conservation equation (14) with respect tocan be written as AlternativelyNotice that the quantitycan represent mass, energy, or momentum in fractal media.

If denotes the fractal mass density, then the functionis the mass fractal flux and . In this case, the local fractional conservation of mass in fractal media reads as

In passing remark that (18) is used to describe fractal physical problems [17, 19, 20].

In the context of the present analysis, the local fractional conservation of energy in fractal media is

The functionin (19) is the fractal flux vector of the energy in fractal media. Further, if the function denotes the amount of heat energy per unit fractal volume in fractal media, then the transport flux is

Thus, the conservation of thermal energy in fractal media can be expressed as

As a consequence of (21), the local fractional Fourier law (with fractal thermal conductivity) reads as

For constantand, (21) can be rewritten as [17, 22] where the fractal thermal diffusivity is

The local fractional conservation of momentum in fractal media is where the quantityrepresents the momentum in fractal media while the function is the fractal momentum flux vector fractal media.

If the momentum per unit fractal volume is, then the sources due to fractal stresses and fractal body forces (gravity generated) areand, respectively. With this terminology, we have

In view of the local fractional conservation of mass (18), (26) takes the form

In (26) is the nondifferentiable advection of momentum in fractal media.

For compressible fluids, the general form of the Navier-Stokes equation on Cantor sets is [21] whereis the fractal fluid velocity,is the dynamic viscosity,is the thermodynamic pressure, and denotes the specific fractal body force.

If the termis zero, then (28) reduces to

which is known as Cauchy’s equation of motion of flows on Cantor sets [21].

For the Navier-Stokes equation on Cantor sets for a compressible fluid becomes

##### 2.2. Fractal Water Waves

###### 2.2.1. Linear Theory for Fractal Water Waves

Let us consider the following local fractional conservation equations of fluid motion in fractal media (Cauchy’s equation of motion of flows on Cantor sets):

If the fractal fluid is incompressible and locally fractional irrotational, then we have

The local fractional Laplace operator is

We notice that (38) is the local fractional Laplace equation (see [21, 23]).

If the following relationship is valid [21]

Then, we have

Hence, from (33) and (41), we get which leads to

Equation (43) can be rewritten in terms of local fractional gradient as or

From (45), we have where is the initial pressure.

Let us suggest that the velocity of the fractal flow normal to the fractal interface can be described as and is equal to the velocity of the fractal interface normal to itself. With these suggestions, we obtain

which is the fractal kinematic equation on the fractal boundary with

When the fractal boundary condition at the free surface is specified, then it follows from (46) and (48) that where , and .

If the bottom section of the flow is considered, then

Further, if the normal velocity of the flow is zero at the fixed solid boundary, (50) gives

For a horizontal bottom, we havewhich leads to or

Therefore, at the free surface, we have where.

For , we find from (56) and (57) that

Therefore, we define the line problem for a water wave as follows:

From (57), we may present the fractal surface as

###### 2.2.2. Nonlinear Theory of Fractal Water Waves

The linear wave equation given in [21] is whereis a constant.

From (43), we get or

Then

Consequently

If the conditions and are satisfied, then from (66), we get Further, from (32) and (35), we obtain where

Using (51), (68), and (69), we obtain the local fractional conservation equations for one-dimensional waves on the bottom given by which lead to

Furthermore, from (38), (50), (51), and (59), we get with

#### 3. Local Fractional Korteweg-de Vries Equation

Using (74) and (76), it is possible to expand the fractal velocity potential into a nondifferentiable series with respect to in the following form:

Then, it follows from (74) and (78) that

Hence,

Thus, from (77), we get

Equations (70) and (73) lead to

Hence, we get where

Similarly, from (84) and (85), we can get

In order to obtain a dimensionless form of (74)–(77), we make the following scale transformations: so that

In this context, the equation for the free water surface is

Here is the linear wave velocity in shallow water. The two small parameters are and with depth of the water, whileandare the typical height and length of the solitary wave, respectively.

Equations (84), (88), and (91) allow developing a nondifferentiable series with respect to in the form

In view of (84), (89), and (90), we have so that which leads to

Hence, from (97) and (98), we obtain

To this end, let us consider the following relations: Then, from (99) and (103), we have

From (101), we get which yield where

If the terms of are omitted, then from (101), we obtain

From (93) and (95), we have which can alternatively be written as

Substituting (108) in (97) and (106), we get where and.

In view of (109), we obtain which result into the form

We notice that (111) is the linear local fractional wave equation for water waves when.

We may also transform (110) into the following forms: which yield whereand.

For developing the problem further, by (113), (105), and (106) can be expressed as
which lead to
whereand*.*

In view of (108), (115), and (116), we obtain

Now, from (113), we may look for a solution of the form whereandare functions ofand its local fractional derivatives.

Utilizing (115), (116), and (118), we obtain which lead to

Moreover, from (121), we have such that (119) and (120) are valid withandas constants.

Therefore, from (115), (119), and (122), we have such that where

From (124), we arrive at the following formula: which leads to or (neglecting the low term) where is a nondifferentiable function, and .

We notice that (126) is the local fractional Korteweg-de Vries equation. When there are coefficient relations, namely,and, we obtain a new local fractional Korteweg-de Vries equation. When neglecting the nonlinear term of (127), we obtain the linear local fractional Korteweg-de Vries equation as follows: where is a nondifferentiable function.

#### 4. Conclusions

In this work, we have derived the local fractional Korteweg-de Vries equation related to fractal waves on shallow water surfaces from the local fractional calculus view point. The linear and nonlinear theories for fractal water wave are presented and the linear and nonlinear local fractional Korteweg-de Vries equations are also obtained.

#### Conflict of Interests

The authors declare that they have no conflict of interests regarding this paper.

#### References

- J. Boussinesq,
*Essai sur la Théorie des Eaux Courantes*, vol. 23 of*Mémoires Présentés par Divers Savants à l'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. View at: Publisher Site | Google Scholar - J. Canosa and J. Gazdag, “The Korteweg-de Vries-Burgers equation,”
*Journal of Computational Physics*, vol. 23, no. 4, pp. 393–403, 1977. View at: Google Scholar - 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. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Google Scholar - G. Segal, “The geometry of the KdV equation,”
*International Journal of Modern Physics A*, vol. 6, no. 16, pp. 2859–2869, 1991. View at: Publisher Site | Google Scholar - M. Wadati, “Stochastic Korteweg-de Vries equation,”
*Journal of the Physical Society of Japan*, vol. 52, no. 8, pp. 2642–2648, 1983. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar - J. W. Miles, “The Korteweg-de Vries equation: a historical essay,”
*Journal of Fluid Mechanics*, vol. 106, pp. 131–147, 1981. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar - Y. Zhao, D. Baleanu, C. Cattani, D. F. Cheng, and X.-J. Yang, “Maxwell's equations on Cantor sets: a local fractional approach,”
*Advances in High Energy Physics*, vol. 2013, Article ID 686371, 6 pages, 2013. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - 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. View at: Publisher Site | Google Scholar - W. Wei, H. M. Srivastava, L. Wang, P.-Y. Shen, and J. Zhang, “A local fractional integral inequality on fractal space analogous to Aderson's inequality,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 797561, 7 pages, 2014. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2014 Xiao-Jun Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.