Abstract

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

1. Introduction

The differential equation of fractional order is the new form of classical integer order differential equations. Different types of differential equations of both ordinary differential equations (ODEs) and partial differential equations (PDEs) in various fields of science like fluid mechanics and biological systems are expressed in fractional forms [1]. There is no any particular method for accessing the exact type of solutions of fractional PDEs, but some approximate solutions are determined by using the Adomian decomposition approach, the homotopy perturbation approach, and the homotopy analysis approach [2–4]. The analytical method and its various forms are well known for determining soliton wave-type solutions of the nonlinear PDEs. With chronology, some investigators have utilized the new analytical approach on fractional-type nonlinear PDEs for obtaining the solitary solutions. This work is related to the fractional order generalized Korteweg-de Vries (gKdV) equation [5]; for this, the fractional form of the gKdV equation [5] is taken as

Equation (1) is a fractional form of the classical generalized Korteweg-de Vries equation which exists by changing the 1st-order time derivatives by fractional derivatives.

In the past two decades, the fractional calculus theory gained great attention and popularity in various fields of science and engineering due to its demonstrated applications. These contributions to the fields of sciences and engineering are based on mathematical analysis. They cover widely known classical fields such as Abel’s integral equation and viscoelasticity. They also include the analysis of feedback amplifiers, the fractional-order Chua-Hartley systems, electrode-electrolyte interface models, fractional-order models of neurons, electric conductance of biological systems, generalized voltage dividers, fitting of experimental data, capacitor theory, and the fields of special functions [6–9].

Several robust methods have been used to solve the FDEs, the fractional differential equations and dynamic systems containing fractional derivatives. Some of the most important methods are Adomian’s decomposition method [10–12], the exp-function method [13], He’s variational iteration method [14, 15], the fractional subequation method [16], the first integral method [17], the homotopy analysis method [18], the (/)-expansion method [19], the homotopy perturbation method [20, 21], the spectral methods [22], and the transform methods [23]. In [24], the authors presented two methods, which are the exp()-expansion method and the Kudryashov method. In [25], the authors demonstrated three methods, which are the csch function method, the tanh-coth method, and the modified simple equation method. In [26–29], the authors introduced the semi-inverse variational principle method, the extended Kudryashov method, the modified simple equation method, and the expanded trail equation method, respectively. Moreover, the fractional differential equations have been studied by powerful authors, and their applications were introduced in sciences and engineering branches [30–32].

One of the well-known equations is the ZK equation, first obtained as a description of weakly nonlinear ion-acoustic modes in a strongly magnetized plasma; it is of particular interest as it is the simplest equation that admits cylindrical and spherical solitary wave solutions in addition to the planar KdV soliton solutions [33]. Another powerful analytical method is called the exp-function method (EFM), which was first presented by He [34]. EFM has successfully been applied to many situations. For example, He and Wu [34] solved the nonlinear wave equations via EFM. Abdou [35] solved generalized solitonary and periodic solutions for nonlinear partial differential equations by EFM. For further information, refer to vigorous references in ([35–40]). In the following years, this proposed method was improved by many researchers. Yang developed general fractional derivatives, along with theory, methods, and applications, to some nonlinear fractional differential equations [41]. Recently, there has been a new fractal nonlinear Burgers’ equation arising from the acoustic signal propagation studied by Yang [42]. Also, Yang et al. investigated fundamental solutions of anomalous equations implemented with the decay exponential kernel [43]. A new integral transform operator for solving the heat-diffusion problem has been utilized by Yang [44]. In [45], Liu et al. probed the group analysis to the time fractional nonlinear wave equation and found many exact solutions. Moreover, Liu et al. worked on the time-fractional nonlinear diffusion equation [46]. Same authors proposed the fractional symmetry group method for the time fractional nonlinear heat conduction equation which usually appears in mathematics, physics, integrable systems, fluid mechanics, and nonlinear areas [47]. Furthermore, in the last decade, many real world problems have been explained using fractional partial differential equations which can be linked to the following valuable researches: utilizing the -homotopy analysis transform method and the fractional natural decomposition method for the fractional Benney-Lin equation [48]; applying the quasiaffine biframelets to the fractional differential equations [49]; analyzing the moderate epidemiological model to describe computer viruses with an arbitrary order derivative [50]; solving the fractional Kaup-Kupershmidt equation with the Atangana-Beleanu derivative and the Caputo-Fabrizio derivative [51]; employing the Riemann-Liouville integral, the Atangana-Baleanu integral operator, and the nonlinear Telegraph equation [52]; solving singular fractional integrodifferential equations with applications to hematopoietic stem cell modeling [53]; applying the truncated-fractional derivative to fractional differential equations [54]; applying the-expansion method to the study of the ()-dimensional hyperbolic nonlinear Schrödinger equation [55], the study of unreported cases of 2019-nCOV epidemic outbreaks [56–58], and the study of the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation [59]; and employing a reliable hybrid numerical method for a time-dependent vibration model [60]. To make this paper more self-contained, a general trigonometric, hyperbolic, and exponential function of the gKdV equation is constructed with the help of an expansion approach, which is crucial in obtaining the lump solution of equation (1).

The pattern of this article is summarized as follows: Section 2 gives the details of the initial definitions; Section 3 gives an introduction of the direct truncation method which is utilized for getting the exact solutions of the gKdV equation in Section 4; Section 5 gives the numerical simulation and graph details; and Section 6 gives some conclusions at the end.

2. Initial Definitions

Definition 1. Definition of -derivative: suppose , then the derivative of of order is defined as

The following properties and new theorems will be used.

Theorem 2. Suppose ; let be -differentiable at point . Therefore, we have

Theorem 3 (see [61–63]). Suppose is a function such that is differentiable and also -differentiable. Assume is a differentiable function defined in the range of . Therefore, we havewhere a prime denotes the classical derivatives with respect to .

3. Methodology

In this section, we give a description of the direct truncation method and introduce it to a partial differential equation.

For a given partial differential equation

Using a transformation aswhere and are constants to be determined later. We can rewrite equation (5) in the following nonlinear ODE:where the prime denotes the derivative with respect to . If possible, we integrate equation (7) term by term one or more times. This yields constants of integration. For simplicity, the integration constants can be set to zero. Suppose has the following truncation form:in which and are introduced in the following forms:where , , , , , , and are constants to be determined, is given, and eitherandare functions determined by an ordinary differential system or and are functions given by direct ansatz such that their derivations are combinations of and , and is determined by an ordinary differential equation:in which the function is also given by a direct ansatz according to the context, and the exponent is determined by utilizing the homogeneous balance method in equation (5). The value is determined by equalizing the maximum order nonlinear term and the maximum order partial derivative term appearing in (7). Ifis rational, then the appropriate transformations can be applied to conquer these hurdles. Substituting (9) and (10) into (8) leads to a polynomial in and ; then, we set the coefficients of and the constant term to zero to get a system of algebraic equations on the unknown parameters in together with the unknown numbers , , , , , , and for . By solving the system, one can get , , , , , , and and the unknown parameters in ; then, by solving equation (10) to get, the solutions of equation (5) can be obtained.

4. Traveling Wave Solution Fractional Order Form of Generalized KdV Equation

The given section deals with the application of a new powerful expansion technique by determining the traveling wave form solutions of the fractional order generalized KdV equation:by applying the aforementioned method. By using the fractional beta complex transform , equation (11) is reduced to the following ODE:

Integrating equation (12) once and setting the constant of integration equal to zero results in

Balancing and by employing the homogenous principle, we get

To get a closed form solution, we use the transformation

Substituting (15) into equation (13), we get

Balancing and , we get

Case I. Then, the exact solution will be asInserting (18) into equation (16), we obtain the below relationwhere are polynomial statements in terms of , , , , , , , and . Hence, solving the resulting system simultaneously, we acquire the following sets of parameters of solutions.

Set I. We, therefore, gained the following generalized solitary solution:in which

Set II. We, therefore, gained the following generalized solitary solution:in which

Set III. We, therefore, gained the following generalized solitary solution:in which

Set IV. We, therefore, gained the following generalized solitary solution:in which

Set V. We, therefore, gained the following generalized solitary solution:in which

Set VI. We, therefore, gained the following generalized solitary solution:in which