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1. Introduction
The history of the studies of fractional order calculus is nearly as old as the classical integer order analysis. However, it was not used in physical sciences for many years. But, during the last few decades, applications of the fractional calculus in applied mathematics, viscoelasticity [1], control [2], electrochemistry [3] and electromagnetic [4] have become more and more prevalent. The evolution of the symbolic computation programs also helped this development. Various interdisciplinary applications could be expressed with the help of fractional derivatives and integrals.
Some fundamental descriptions and applications of fractional calculus are given in [5] and [6]. Existence and uniqueness of the solutions are also studied in [7].
Parallel to the work in physical sciences, fractional order partial differential equations (FPDEs) gave scientists the opportunity of identifying and modeling many substantial and practical physical problems.
Therefore, in recent years significant efforts have been made to create analytical methods for the solution of FPDEs. These methods include the G’/G expansion method for biological population models [8] and Nizhnik-Novikov-Veselov equation [9], the sine-Gordon expansion method for fractional Drinfeld-Sokolov-Wilson system [10] and Biswas-Milovic equation [11], the generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan, Ginzburg-Landau, Fokas-Lenells, Zakharov-Kuznetsov and Schrödinger equations [12-19], he first integral method for Burgers’ type equations [20] and Wu-Zhang system [21] and the modified Kudryashov method for Klein-Gordon equation [22] and fractional biological population models [23], the simplest equation method for time fractional Kolmogorov-Petrovskii-Piskunov equation [24], and the Laplace and Fourier transform method for time fractional diffusion and convection-diffusion equations [25].
In this paper, the NEDAM is used to obtain analytical solutions of time-fractional SRLW equation. The present method is suitable for many different classes of equations and produces reliable exact solutions for the conformable fractional differential equations.
In the literature, there are several fractional derivative definitions of an arbitrary order such as Caputo and Riemann-Liouville derivatives. Recently, a new definition, so-called “the conformable fractional derivative", has been suggested by R. Khalil et al. [26]. This definition provides simple and proper fractional derivative and integral operators which satisfy basic properties of classical derivatives.
Definition 1.1 A μ - th order conformable fractional derivative of a function defined by
For f: [0, ∞) → R and for all t > 0, μ ε (0, 1)
If f is μ-differentiable in some (0, a) for a > 0 and
The next theorem includes some properties of this new definition [26].
Theorem 1.1 Let f and g be μ-differantiable functions at point t > 0 for 0 < μ ≤ 1. Then
1.
2.
3.
4.
5. Tμ(c) = 0 for all constant functions f(t) = c
6. Moreover, if f be differentiable, then
Definition 1.3 Let f be a function of n variables x1, …, xn. Then the conformable partial derivative of f of order μ ∈ (0, 1] in xi is defined as [27]
Definition 1.4 The conformable integral of a function f for μ ≥ 0 is defined as
A large number of researchers have used this definition of fractional derivative and integral by combining them with different methods to find the approximate and exact solutions of various partial fractional differential equations. Unlike the Caputo and Riemann-Liouville definitions, the conformable derivative definition satisfies the chain rule [29], so, one can calculate reliable exact solutions for fractional differential equations using this definition.
After this introductory section, Sec. 2 is reserved for a brief review of the governing equation and in Sec. 3, the basic idea of the implemented method is illustrated. In Sec. 4, the exact solutions of the SRLW equation are presented. Graphical representations of wave solutions are given in Sec. 5. Finally the article ends with a conclusion in Sec. 6.
2. Governing equation
In this study, we examine a long wave model namely symmetric regularized-long-wave equation with the conformable time-fractional derivative as
where t > 0 and 0 < μ ≤ 1.
Long waves, also known as shallow water waves, are waves moving at a depth of less than 1/20 of their wavelength in shallow waters of the sea or ocean. Many flow types can be modeled with the help of shallow water equations. These equations are very suitable for wave propagation phenomena and describe the interaction of two long waves with different dispersion relations.
Long wave equations provide an attractive and prosperous field of research. Numerous field of applications benefits from long water theory, such as, physical oceanography, dam breaking problems, river flooding, breakwater construction and control, coastal engineering and wave propagation in tsunami estimation. Besides, this equation is highly used in flood analysis, they are very effective in determining the effects of floods affecting large areas.
In the slanted coasts and waterfront zones, these equations’ analytic solutions produce solid predictions of time-dependent variations of velocity dispersion of the shallow water waves. Long water wave equations are not only applied to the velocity and height of the currents in the channels of a water waste network but also vorticity and flow analysis in seas and oceans. Subsequently, such sort of equations gives robust solutions to the problems which arise from hydrodynamic systems.
In the literature, there are several techniques to obtain exact solutions of the SRLW equations. These techniques include the sec-csc, tan-cot, sech-csch and tanh-coth methods [30], the auxiliary equation method [31], the modified extended tanh method [32], the sub-equation method [33], the modified Kudryashov method [34], the sine-Gordon expansion method [35], the exponential rational function method [36], the (G’/G)-expansion method [37], the extended Jacobi elliptic function expansion method [38], the exp(-φ(ξ))-expansion method [39], the exp-function method [40], the improved (G’/G)-expansion method [41], and the (Dα ξ G’/G)-expansion method [42].
3. Idea of the new extended direct algebraic method
A few studies exist about the NEDAM in the literature. It has been used to obtain exact solutions for the time-fractional Phi-4 equation [43], the Schrödinger-Hirota equation [44], and the Ginzburg-Landau [45] equation.
Now we present the basic idea of the method.
Take the following nonlinear conformable fractional partial differential equation into account
Where u is the function to be determined, F is a polynomial of u which is given with its partial and conformable fractional partial derivatives.
Firstly consider the wave transformation
where k and w are arbitrary constants. Rewriting Eq. (5) as the subsequent nonlinear ODE gives
Here, primes represent the standard derivatives with respect to ξ.
Secondly, assume that Eq. (7) has solutions of the form
Where b j (0 ≤ j ≤ n) are constant coefficients to be calculated afterwards, n is a positive integer determined by balancing the nonlinear terms and the highest order derivatives
in [46] and nonlinear terms in Eq. (7) and Q(ξ) is the solution of the following ODE
Here α, β and σ are constants. Some special solutions of Eq. (7) are listed below:
1) If δ = β 2 - 4ασ < 0 and σ ≠ 0,
2) If δ = β 2 - 4ασ > 0 and σ ≠ 0,
3) If ασ > 0 and β = 0,
4) If ασ < 0 and β = 0,
5) If β = 0 and σ = α,
6) If β = 0 and σ = -α,
7) If β 2 = 4ασ
8) If β = k, α = mk (m ≠ 0) and σ = 0,
9) If β = σ = 0,
10) If β = α = 0,
11) If α = 0 and β ≠ 0,
12) If β = k, σ = mk (m ≠ 0) and α = 0,
The above mentioned generalized hyperbolic and triangular functions are defined as
where p, q > 0 are arbitrary constants called deformation parameters.
Next, subrogating Eqs.(8) and (9) into Eq.(7) and setting the b j to zero, gives a nonlinear algebraic system for j = 0,1,…, N
Finally, substituting these constants and the solutions of Eq. (9) into Eq. (8) and considering Eq.(6), yields exact solutions of Eq.(5), which will be presented in the next section.
4. Exact solutions of the time-fractional SRLW equation
Using (6), we reduce (4) to the following nonlinear ODE
Balancing the highest order derivative U’’’’ with nonlinear term UU’’ [46] we obtain n = 1. Therefore we rewrite (8) as
Combining this equation with Eq. (9) into Eq. (10), collecting the coefficients of Q j (ξ) and equating to 0 we obtain the following terms
Using these coefficients and implementing the above mentioned steps, we acquire the following solutions
Assuming that δ < 0 and σ ≠ 0,
Assuming that δ > 0 and σ ≠ 0,
where δ = β 2 - 4ασ and η = kwln 2 A.
Assuming that ασ > 0 and β = 0,
Assuming that ασ < 0 and β = 0,
Assuming that β = 0 and σ = α
Assuming that β = 0 and σ = -α,
Assuming that β 2 = 4ασ
Assuming that β = k, α = mk (m ≠ 0) and σ = 0,
Assuming that β = α = 0,
Assuming that α = 0 and β ≠ 0,
Finally, assuming that β = k, σ = mk (m ≠ 0) and α = 0,
5. Graphical representations of wave solutions
In this section, we present 3-dimensional and 2-dimensional surface plots of the chosen exact solutions obtained by the NEDAM. The plots are presented for several values of α, β, σ, A, p, q, k, w, l, m coefficients and μ, the conformable fractional derivative. While Figs. 1, 2, and 3 represent periodic wave solutions for u 1(x, t), u 2(x, t), u 4(x, t), Fig. 4 is an example for solitary wave solution u 37(x, t) for x ∈ [-5, 5] and t ∈ [0, 5]. As seen, the double periodic solutions are deteriorated to solitary wave solutions and the amplitude goes to infinity.
Figure 1 3D and 2D simulation of solution u 1(x; t) for α = 2; β = 3; σ = 1:5; A = e; p = 0:9; q = 0:9; k = 2; w = 1, and μ = 0:90:
Figure 2 3D and 2D simulation of solution u 2(x; t) for α = 3; β = 3; σ = 1; A = e; p = 0:95; q = 0:95; k = 2; w = 2, and μ = 0:90:
Figure 3 3D and 2D simulation of solution u 2(x; t) for α = 3; β = 3; σ = 1; A = e; p = 0:95; q = 0:95; k = 2; w = 2, and μ = 0:90:
6. Conclusion
In this study, the new extended direct algebraic method (NEDAM) has been successfully applied to time-fractional SRLW partial differential equation, which arises in long water flow models. The traveling wave transform is used to reduce the equation to integer order ordinary differential equation and with the aid of mathematical software Mathematica and using the conformable derivative definition, numerous new exact solutions of the equation have been obtained which do not exist in the literature. These solutions include rational, exponential, generalized trigonometric, and generalized hyperbolic function series. Thus, it is shown that the method is very effective and powerful as compared with other existing methods in the literature and could be used for different types of FPDEs arising in different areas of mathematical physics.
