Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations

1 Introduction

Nonlinear differential equations in fractional forms (FNDEs) represent a variety of models that play a major role for many real world problems in different fields of science including engineering sciences, fluid dynamics and mechanics, biology, mathematics, and physics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Extracting exact solutions and lump solutions of such models is an interesting and a reach area of research. Different approaches are used in literature for calculating the exact solutions which include the method of exp-function [16, 17], the unfied method and its generalized form [18, 19, 20, 21, 22, 23], the expansion of G^′/G approach [24, 25], the expansion of Riccati equation [26, 27], the analysis of Lie symmetry method [28, 29], and the generalized form of the Kudryashov’s technique [30, 31], the method of first integral [32, 33] and sub equation approach [34, 35].

In the present work, we use the technique of ERF method [36, 37] to set new exact solutions to four different types of conformable fractional Boussinesq-like equations [38, 39] given in the following forms

and

where utα is the fractional derivative in conformable sense of order 0 < α ≤ 1 in the t > 0 is defined by the following [40]

Eq. (1) contains the fourth spatial derivative u_xxxx and the dissipative term u_xx. In Eq. (2), u_xxxx is replaced by a mixed spatio-temporal uxxtt2α. While in Eq. (3), u_xx is replaced by uxtα and u_xxxx is replaced by another mixed spatio-temporal uxxtα.. Finally Eq. (4) does not contain the dissipative termu_xx.

Eqs. (1)-(4) are non-integrable equations that are used as model in ocean and coastal sciences when α = 1. Some applications of these equations when they are represented by tsunami wave modeling and mathematical modeling of tidal oscillations. Furthermore, they can be used in studying the dynamics of the thin in viscid layers with free surface, the wave propagation in elastic rods, and in the continuum limit of lattice dynamics or some particular forms of electrical circuits [41, 42, 43]. It worth mentioning that, the aforementioned equations were studied first by the Hirota’s method for constructing single and singular soliton solutions [43].

Based on these ideas, present paper is designed as follows. A brief description of the method of ERF is given in the next section. Sections 3-6 discusses the application of the method of ERF to the conformable fractional Boussinesq-Like equations represent by (1)-(4) respectively. The physical explanation for the obtained solution is given in section 7. Finally, section 8 outlines the main conclusions.

2 The method of ERF

The method of ERF is a recent technique to construct the solutions to nonlinear forms of fractional PDEs. The method has been used to determine various kinds of nonlinear PDEs. Mohyud-Din et al. set solutions in various forms to the space-time fractional Boussinesq, symmetric form of the RLW and breaking soliton equations defined in two space dimension. The Focas, Zakharov-Kuznetsov, BBM equations and coupled Burger’s system in both space and time fractional forms were also solved by implementation of this method [37]. Bekir et al., [44] proposed the method of ERF to the Kuramoto-Sivashinsky and Oiao equations. In [45], Kaplan et al. considered the same approach to construct exact solutions to the Tzitzeica-Dodd-Bullough, the Tzitzeica and the Dodd-Bullough-Mikhailov equations. Tala-Tebue et al., [46] applied the method of ERF to obtain the new explicit solutions of Zhiber-Shabat and related equations.

The fundamentals of the ERF are given below.

We consider the following nonlinear conformable fractional PDE in two independent, one time and one space, and one dependent variable u

where F is a polynomial-like structure in u and its partial derivatives and includes nonlinear term(s).

With the use of the compatible wave transformation

where c ≠ 0 is constant to be found in the next steps. This transformation enables the following changes

Substituting Eq.(7) in Eq.(6) yields a nonlinear ODE as following

where U′ demonstrates the derivation with respect to ξ.

The most foremost steps of ERF method are as follows:

1. Let the solution of Eq. (8) be of the form

   U(ξ)=∑i=0nai(1+eξ)i.(8)

   where a_i,(0 ≤ i ≤ n) are constants to be determined with the condition a_n ≠ 0.

2. The integer n>0 can be determined by the homogeneous balance between the derivative term of highest order and one of the nonlinear terms in Eq. (8). Let the degree of U(ξ) be D[U(ξ)] = n, then, the degree of other expressions in derivative and nonlinear terms defined by the following

   DdrUdξr=n+r,DUwdrUdξrl=nw+l(n+r).

   Therefore, we obtain the value of n in Eq. (11).

3. By substituting the predicted solution (9) into Eq. (8), collecting the terms appropriately, and forcing the coefficients of each power of e^iξ to zero, an algebraic system of equations is constructed.

4. The solution of the resultant system determines the explicit relations among the parameters and coefficients that are compulsory to define the solutions to Eq. (6).

3 The first conformable fractional Boussinesq-like equation

We next study the first conformable fractional Boussinesq-like equation (1). The compatible wave transformation u(x, t) = U(ξ), ξ = kx – ctαα, reduces Eq. (1) to the following ODE:

Integrating Eq. (10) twice and eliminating the integration constants by assuming both of them zero, we get

We assume that Eq. (11) has a solution of the form

Where all coefficients a_i(0 ≤ i ≤ n) are constants satisfying a_n ≠ 0.

The balance between U″ and U³ in equation (11) forces n = 1. In so doing, Eq. (11) takes the following form

Substituting the predicted solution (13) into Eq. (11) and rearranging the coefficients of e^iξ(0 ≤ i ≤ 3) and by equating each coefficient to zero, we find

The following results are obtained upon solving the above algebraic equations using Maple

Using (14), substituting these results into Eq. (13), we can determine the following explicit exact solutions of Eq. (1) as follows

where ξ=kx∓12α4−2k2tα.

4 The second conformable fractional Boussinesq-like equation

We choose the second conformable fractional Boussinesq-like equation (2) to study in this section. The compatible wave transformation u(x, t) = U(ξ), ξ = kx – ctαα, reduces Eq. (2) to the following ODE

Integrating Eq. (15) twice and eliminating both integration constants by equating them to zero, we get

We suppose that Eq. (16) is represented in the form given below

where a_i(0 ≤ i ≤ n) are constants with a_n ≠ 0.

The balance between U″ and U³ in equation (16) gives n = 1. In so doing, Eq. (16) takes the following form

Substituting the predicted solution (18) into Eq. (16) and collecting the coefficients of e^iξ(0 ≤ i ≤ 3) and by equating each coefficient to zero, we find

The following results are obtained upon solving the above algebraic equations using Maple

Using (19), substituting these results into Eq. (18), we can determine the following exact solutions of Eq. (2) as follows

where ξ=kx∓2α2k2+4tα.

5 The third conformable fractional Boussinesq-like equation

We next study the third conformable fractional Boussinesq-like equation (3). By making the compatible wave transformation

Eq. (3) becomes

Integrating Eq. (21) twice and setting the integration constants to zero yield

We suppose that Eq. (22) has a solution in the form given below

where a_i(0 ≤ i ≤ n) are constants with a_n ≠ 0.

Balancing U″ and U³ in equation (22), we obtain n = 1. In so doing, Eq. (22) takes the following form

Substituting Eq. (24) into Eq. (22) and collecting the coefficients of e^iξ(i = 0, 1, 2, 3) and by equating each coefficient to zero, we find

The following results are obtained upon solving the above algebraic equations using Maple

Using (25), substituting these results into Eq. (24), we can find the following exact solutions of Eq. (3) as follows

where ξ = k(x – 12α(k² – 2)t^α).

6 The fourth conformable fractional Boussinesq-like equation

We first study the for the conformable fractional Boussinesq-like equation (4). By making the transformation

Eq. (26) becomes

Integrating Eq. (27) twice and setting the integration constants to zero yield

We suppose that Eq. (28) has a solution in the form given below

where a_i(i = 0, 1, 2, …, n) are constants.

Balancing U″ and U³ in equation (28), we obtain n = 1. In so doing, Eq. (28) takes the following form

Substituting Eq. (30) into Eq. (28) and collecting the coefficients of e^iξ(i = 0, 1, 2, 3) and by equating each coefficient to zero, we find

The following results are obtained upon solving the above algebraic equations using Maple

Using (31), substituting these results into Eq. (30), we can find the following exact solutions of Eq. (4) as follows

where ξ=kx∓i22αktα.

7 Comparison and physical explanation

In this study, we derived several types of exact solutions for Eqs. (1)-(4) by using the ERF method.

Figs. 1-4 represent hyperbolic rational solutions for Eqs. (1)-(4) respectively. In Fig. 1, there are two separated rogue waves with the same height and amplitude propagating regularly indifferent directions parallel to the t-axis. In Fig. 2, we have periodic rogue waves with different heights and the same amplitude propagating regularly around breather waves and moving parallel to the t-axis. In Figs. 3-4, we have dark solitary waves. Comparing our results with other results in [28, 29, 32], it can be seen that our results are different and new when α = 1. Moreover, the other results are special case of our solutions by choosing the arbitrary parameters in the solutions with suitable values.

Fig. 1

The solution of Eq. (1) (|u|): α = 0.95; k = –2; –10 ≤ x ≤ 10; 0 ≤ t ≤ 5.

Fig. 2

The obtained soliton solution of Eq. (2) (|u|): α = 1; k = –1; –10 ≤ x ≤ 10;0 ≤ t ≤ 5.

Fig. 3

The obtained soliton solution of Eq. (3) (|u|): α = 0.95; k = 2; –10 ≤ x ≤ 10;0 ≤ t ≤ 5.

Fig. 4

The obtained soliton solution of Eq. (4) (|u|): α = 0.9; k = 1; –10 ≤ x ≤ 10;0 ≤ t ≤ 5.

We mention that the ERF method provides us with the priceless information about the conformable fractional Boussinesq-Like equations given by Eqs. (1)-(4). The obtained solutions given by this method are important to determine the structure and the dynamical behavior of the long ve problem, the range of them and their velocities in different aspects. The method suggested here is simple, direct, reliable and effective that can be extended to study and solve many of the NLEEs in different branches of science.

8 Conclusion

Investigation has been made on conformable fractional Boussinesq-like equations in four different forms. Using the ERF approach and symbolic computations, we constructed distinct hyperbolic solutions which are classified into rogue wave, periodic rogue wave and dark wave solutions. We verified that the ERF procedure is simple and direct to use with the help of symbolic computation software, so it will undoubtedly be useful for studying other FNDEs. The results show that the arbitrary functions have significant effect on the wave behavior and can be used to give a deeper insight into many complex phenomena that occur in different scientific areas.