Pure Traveling Wave Solutions for Three Nonlinear Fractional Models
Three nonlinear fractional models, videlicet, the space-time fractional Boussinesq equation, -dimensional breaking soliton equations, and SRLW equation, are the important mathematical approaches to elucidate the gravitational water wave mechanics, the fractional quantum mechanics, the theoretical Huygens’ principle, the movement of turbulent flows, the ion osculate waves in plasma physics, the wave of leading fluid flow, etc. This paper is devoted to studying the dynamics of the traveling wave with fractional conformable nonlinear evaluation equations (NLEEs) arising in nonlinear wave mechanics. By utilizing the oncoming -expansion technique, a series of novel exact solutions in terms of rational, periodic, and hyperbolic functions for the fractional cases are derived. These types of long-wave propagation phenomena played a dynamic role to interpret the water waves as well as mathematical physics. Here, the form of the accomplished solutions containing the hyperbolic, rational, and trigonometric functions is obtained. It is demonstrated that our proposed method is further efficient, general, succinct, powerful, and straightforward and can be asserted to install the new exact solutions of different kinds of fractional equations in engineering and nonlinear dynamics.
The solutions of fractional partial differential equations (FPDEs) are often of interest and are applied in practical life. One of the main purposes of mathematical physics is to determine the exact solutions. Some scholars had perused the plenty of physical wave equations. Therefore, several analytical methods were systematically developed and applied to achieve exact and approximate solutions of fractional ordinary and partial differential equations with applications in various fields of sciences like fluid flow, mechanics, biology, nonlinear optics, substance energy, system identification, and geooptical filaments which are expressed in fractional forms [1–11].
In the past few decades, a lot of studies have been executed to find the new and further exact traveling wave solution of space-time fractional PDEs by many research. With the collaboration of potential symbolic computer programming software, they have been appointed for researching appropriate solution to the nonlinear space-time fractional PDEs by executing powerful techniques, for example, the tan- expansion method, the sin-Gordon expansion method, the expansion method, and the advanced exponential expansion method [12–25].
This work mainly investigates three nonlinear fractional models by utilizing the oncoming -expansion method [17, 25]. Recently, Bashar and Roshid  and Rahhman et al.  have studied this technique to some fractional and nonfractional NLEEs. They found that this introduced method provides some simple general form of traveling wave solutions. Rahhman et al.  did not give any fruitful discussion about fractional NLEEs with this proposed method. The important idea of this method is too explicit: the exact solutions of NLEEs satisfy the nonlinear ODE, , where and are real parameters.
The space-time fractional Boussinesq equation with the -derivative [26, 27] is presented as follows: where is the vertical deflection. In Ref. , authors constructed an analytical solution for both linear and nonlinear time-fractional Boussinesq equations by an iterative method. Also, Hemeda  studied the fractional Boussinesq-like equation via a new iterative method. Authors of  investigated nonlinear two-point boundary value problem to the fractional Boussinesq-like equation. Furthermore, the space-time fractional breaking soliton equations [27, 31] are taken in the following form: Meng and Feng  used an auxiliary equation method to the space-time fractional -dimensional breaking soliton equation. Authors of [27, 33, 34] discussed the space-time fractional SRLW equation (STFSRLWE) in the following case: The functional variable method, exp-function method, and -expansion method to the fractional SRLW equation in the sense of the modified Riemann-Liouville derivative were utilized in Ref. . The interested readers can see more works in Refs. ([36–50]). Inc and coworkers presented the new soliton structures to some time-fractional nonlinear differential equations with a conformable derivative via the Ricatti–Bernoulli sub-ODE method . Salahshour and colleagues worked on the truncated -fractional derivative as a novel and effective derivative under interval uncertainty and investigated the existence and uniqueness conditions of the solution . Authors of  studied a coupled nonlinear Maccari’s system which describes the motion of isolated waves localized in a small part of space. Authors of  employed the exponential function method for the combined KdV-mKdV equation.
One technique, videlicet, the oncoming -expansion technique, was employed by many researchers for solving the number of nonlinear PDEs or fractional PDEs. For both methods, the interested readers can refer for the first technique to Refs. ([55–59]).
The pattern of this article is summarized as follows. In Sections 2 and 3, the properties and the detail of technique are given, which are to be utilized for getting the exact solutions of the fractional Boussinesq, breaking soliton, and SRLW equations along with numerical simulation and details of graph in Sections 4–7. Finally, some conclusions are given in the end.
2. Analysis of -Derivative
Definition 1 (see ). Definition of -derivative: let ; then, the -derivative of of order is defined as
The features and novel theorems will be utilized as follows:The proofs of the above -derivative properties are obviously given in .
Theorem 2. Let be -differentiable at point ; therefore, we get (1)(2)(3)(4)(5)We have the following features as follows:
Theorem 3. Let be a function such that is differentiable and also -differentiable. Also, let be a differentiable function defined in the range of . Then, we get where prime denotes the classical derivatives with respect to .
3. The Oncoming -Expansion Technique
This method was summarized and improved for achieving the analytic solutions of NLPDEs.
Step 1. Assume that a nonlinear partial differential equation is given in the general form as follows: After simple algebraic operations, this equation is transformed into an ordinary differential equation (ODE) with the below transformation: as well as into nonlinear ODE:
Step 2. Then, assume that the searched wave solutions of equation (9) have the following representation:
where and are constants to be determined, such that , and is the solution of the following first-order differential equation:
If we try to find the solution of (11), then we obtain special solutions that vary according to the state of the coefficients:
Solution 1 (hyperbolic function solution). If and , then we achieve where is the integral constant.
Solution 2 (trigonometric function solution). If and , afterward we achieve Solution 3. If , , and , afterward we achieve Solution 4. If , , and , afterward we achieve Solution 5. If , , and , afterward we achieve where , , , and are also the constants to be explored later. As usual, for determining , the highest-order derivative should be balanced with the highest-order nonlinear terms in equation (10). However, the positive integer can be determined in this way.
Step 3. Following these operations, according to the value obtained above, let (11) be substituted into equation (10). Therefore, we obtain a set of algebraic equations that contains . Then, setting each coefficient of to zero, we can get a set of overdetermined equations for , , and . Since the obtained algebraic equation system will be difficult to solve manually, symbolic computation such as Maple can be used at this stage. Assume that the estimation of the constants can be gotten by fathoming the mathematical conditions in step 2. Substituting the estimations of the constants together with the arrangements of equation (11), we will acquire new and far reaching precise traveling wave arrangements of the nonlinear development equation (7).
The has the following features as follows: where . Balancing with yields
4. The First Equation
By utilizing the following transformation: then, equation (1) transformed to where , and by integrating equation (20) twice with respect to , it can be seen as The balance number will be obtained by using the balance principle. Then, the exact solution is given as Firstly, we substitute the expressions of in (22) into (21) and collect all terms with the same order of . Then, by equating the coefficient of each polynomial to zero, we obtain a set of algebraic equations as follows:
Second: According to Family I, (22) can be written as where According to Family II, (22) can be written as where According to Family III, (22) can be written as where According to Family IV, (22) becomes where
Third: According to Family I, (22) becomes where According to Family II, (22) can be written as where According to Family III, (22) can be written as where According to Family IV, (22) can be written as where
5. The Second Equation
By utilizing the following transformation: then, equation (1) transformed to where and . By putting into equation (66), it can be seen as The balance number will be obtained by using the balance principle. Then, the exact solution is given as Firstly, we substitute the expressions of in (69) into (68) and collect all terms with the same order of . Then, by equating the coefficient of each polynomial to zero, we obtain a set of algebraic equations as follows: