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

and Applied Analysis 3 Alternatively ∂ α ψ (x, y, z, t) ∂t + ∂ α φ 1 (x, y, z, t) ∂x + ∂ α φ 2 (x, y, z, t) ∂y + ∂ α φ 3 (x, y, z, t) ∂z = H (x, y, z, t) , (17a) φ (x, y, z, t) = φ 1 (x, y, z, t) i α + φ 2 (x, y, z, t) j α + φ 3 (x, y, z, t) k α . (17b) Notice that the quantity φ(x, y, z, t) can represent mass, energy, or momentum in fractal media. If ρ denotes the fractal mass density, then the function φ = ρυ is themass fractal flux and H(x, y, z, t) = 0. In this case, the local fractional conservation ofmass in fractal media reads as ∂ α ρ ∂t + ∇ α ⋅ (ρυ) = 0. (18) In passing e 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 E in fractal media is ∂ α E ∂t + ∇ α ⋅ (Eυ) = H (x, y, z, t) . (19) The function φ = Eυ in (19) is the fractal flux vector of the energy in fractal media. Further, if the function E = ρC α T denotes the amount of heat energy per unit fractal volume in fractal media, then the transport flux is φ = ρCTυ. (20) Thus, the conservation of thermal energy in fractal media can be expressed as ∂ α ρC α T ∂t + ∇ α ⋅ (ρC α Tυ) = ∇ α ⋅ k (∇ α T) + H (x, y, z, t) . (21) As a consequence of (21), the local fractional Fourier law (with fractal thermal conductivity k) reads as F = −k∇ α T. (22) For constant C α and k, (21) can be rewritten as [17, 22] ∂ α T ∂t + υ ⋅ ∇ α T = a∇ 2α T + H (x, y, z, t) ρC α , (23) where the fractal thermal diffusivity a is a = k ρC α . (24) The local fractional conservation of momentum in fractal media is ∂ α M ∂t + ∇ α ⋅ (Mυ) = H (x, y, z, t) , (25) where the quantity M represents the momentum in fractal media while the function φ = Mυ is the fractal momentum flux vector fractal media. If the momentum per unit fractal volume is M = ρυ, then the sources due to fractal stresses and fractal body forces (gravity generated) are ∇ ⋅ θ and H(x, y, z, t) = ρg, respectively. With this terminology, we have ∂ α ρυ ∂t + ∇ α ⋅ (ρυυ) = ∇ α ⋅ θ + ρg. (26) In view of the local fractional conservation of mass (18), (26) takes the form ∂ α υ ∂t + (υ ⋅ ∇ α ) υ = 1 ρ ∇ α ⋅ θ + g. (27) In (26) (υ ⋅ ∇)υ is the nondifferentiable advection of momentum in fractal media. For compressible fluids, the general form of the NavierStokes equation on Cantor sets is [21] ρ ∂ α υ ∂t = − ∇ α p + 1 3 μ∇ α ((∇ α ⋅ υ)) + μ∇ 2α υ + ρb − ρυ (∇ α ⋅ υ) , (28) where υ is the fractal fluid velocity, μ is the dynamic viscosity, p is the thermodynamic pressure, and b denotes the specific fractal body force. If the term (1/3)μ∇((∇ ⋅ υ)) + μ∇υ is zero, then (28) reduces to ρ ∂ α υ ∂t = −∇ α p + ρb − ρυ (∇α ⋅ υ) , (29) which is known as Cauchy’s equation of motion of flows on Cantor sets [21]. For 1 ρ ∇ α ⋅ θ = − 1 ρ ∇ α p, b = −g, (30) the Navier-Stokes equation on Cantor sets for a compressible fluid becomes ∂ α υ ∂t + υ (∇ α ⋅ υ) = − 1 ρ ∇ α p − g. (31) 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): ∂ α ρ ∂t + ∇ α ⋅ (ρυ) = 0, (32) ∂ α υ ∂t + υ (∇ α ⋅ υ) = − 1 ρ ∇ α p − g. (33) 4 Abstract and Applied Analysis If the fractal fluid is incompressible and locally fractional irrotational, then we have

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 timefractional derivatives [15].Abdulaziz et al. [16] discussed the modified KdV equations with different space-and timefractional 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][18][19][20][21][22][23][24][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][27][28][29][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 Kortewegde Vries equation is derived from local fractional calculus in Section 3. The conclusions are outlined in Section 4.

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 quantity (, ) which varies within the fractal volume  () .Observe that the variations in (, ) with respect to the fractal time corresponds to the variation in the flux through the fractal boundary S () or by a source inside the volume  () .The integral form of local fractional conservation of the quantity (, ) is given by [17,19,21] (, )  () , (9) 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][20][21][22] ∬  (  ) S where is the local fractal surface and  denote elements of the surface with unit normal local fractional vector n  .When Δ ()  → 0 as  → ∞, the local fractional volume integral of the function u takes the form [17,[19][20][21][22][23] The local fractional derivative of a function () of order  is 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 function  emerging from ( 14) is [17] ∇   = lim In the Cantorian coordinates, the local fractional conservation equation (14)  If  denotes the fractal mass density, then the function  =  is the mass fractal flux and (, , , ) = 0.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 function  =  in ( 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 constant   and , (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 quantity  represents 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) are ∇  ⋅  and (, , , ) = , 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] where  is the fractal fluid velocity,  is the dynamic viscosity,  is the thermodynamic pressure, and b denotes the specific fractal body force.
If the term (1/3)∇  ((∇  ⋅ )) + ∇ 2  is 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 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  0 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  =  0 , () = 0, 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 have  = −ℎ 0 (, ) which leads to or Therefore, at the free surface, we have where (, , ) = .
For  = 0, 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 where  is a constant.From (43), we get Then From ( 33) and (64), we have Consequently If the conditions ( 2 /    ) = 0 and ( 2 /     ) = 0 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

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 Abstract and Applied Analysis 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  0 = √ℎ is the linear wave velocity in shallow water.The two small parameters are  = /ℎ  and  = ℎ/ with depth of the water ℎ, while  and  are 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 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 ( 2 ,  4 ) are omitted, then from (101), we obtain (106) From ( 93) and (95), we have which can alternatively be written as Substituting ( 108) in ( 97) and (106), we get where  = ℎ/ ≪ 1 and  = /ℎ  ≪ 1.
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 where  =  and  → 1.
We notice that (126) is the local fractional Kortewegde Vries equation.When there are coefficient relations, namely,  = 1 and  = 1, we obtain a new local fractional Korteweg-de Vries equation.When neglecting the nonlinear term of (127), we obtain the linear local fractional Kortewegde Vries equation as follows: where (, ) is a nondifferentiable function.

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.