Abstract and Applied Analysis

Volume 2014 (2014), Article ID 278672, 10 pages

http://dx.doi.org/10.1155/2014/278672

Research Article

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

Xiao-Jun Yang,¹ Jordan Hristov,² H. M. Srivastava,³ and Bashir Ahmad⁴

¹Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

²Department of Chemical Engineering, University of Chemical Technology and Metallurgy, Sofia, 8 Kliment Ohridsky Boulevard, 1756 Sofia, Bulgaria

³Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, V8W 3R4

⁴Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 15 May 2014; Accepted 29 May 2014; Published 12 June 2014

Academic Editor: Dumitru Baleanu

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.

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

Alternatively

Notice 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

From (32) and (37), 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

From (33) and (64), we have

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.

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