New Explicit Solutions to the Fractional-Order Burgers’ Equation

The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables (( G ′ / G ) , ( 1/ G )) -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional diﬀerential equations (NLFDEs) into ordinary diﬀerential equations. In this article, we ascertain the solutions in terms of tanh, sech, sinh, rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting deﬁnite values of the parameters in the general solutions. Mathematical analysis of the solutions conﬁrms the existence of diﬀerent soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory eﬀect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.


Introduction
e nonlinear fractional evolution equations (NLFEEs) emerge frequently in diverse research field of science and applications of engineering. e fractional derivative has been happening in numerous physical problems, for example, recurrence subordinate damping conduct of materials, motion of an enormous meager plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, and PI λ D μ controller for the control of the dynamical system. Fractional-order differential equations describe the phenomena. e fractional-order differential equations are broadly used as generalizations of conventional differential equations with the integral order to explain different intricate phenomena in numerous fields including the diffusion of biological populations, electric circuit, fluid flow, chemical kinematics, control theory, signal processing, optical fiber, plasma physics, solid-state physics, and other areas [1][2][3][4][5]. e concepts of dissipation, dispersion, diffusion, convection, and reaction are closely related to the abovestated phenomena, and nonlinear fractional partial differential equations (NLFPDEs) can be used to evaluate them exactly. Wave shape has an effect on sediment transport and beach morpho dynamics, while wave skewness has an impact on radar altimetry signals, and asymmetry has an impact on ship responses to wave impacts. Traveling wave solutions are a special class of analytical solutions for NLFEEs. Solitary waves are transmitted traveling waves with constant speeds and shapes that achieve asymptotically zero at distant locations. e appearance of solitary waves in nature is rather frequent in plasmas, fluids dynamics, solidstate physics, condensed matter physics, chemical kinematics, optical fibers, electrical circuits, bio-genetics, elastic media, etc. Consequently, it is important to search for the exact traveling wave solutions of NLFPDEs to understand the facts. erefore, many researchers have been motivated on finding the exact solutions to nonlinear fractional-order differential equations, and significant progress has been made in analyzing the exact solutions of these types of equations. e major challenges, however, are that there is no unified numerical or analytical approach that can investigate all sorts of nonlinear fractional-order differential equations. us, several numerical and theoretical methods for finding solutions for NLFDEs have been established, for example, the differential transformation method [6,7], the variational iteration method [8][9][10], the fractional subequation method [11], the Kudryashov [12] method, the homotopy perturbation method [13,14], the homotopy analysis method [15], the exp-function method [16,17], the (G ′ /G)-expansion method and its various modification [18][19][20][21][22], the Chelyshkov polynomial method [23,24], the multiple exp-function method [25], the finite difference method [26], the finite element method [27], the first integral method [28,29], the modified simple equation method [30], the reproducing kernel method [31], the two variables ((G ′ /G), (1/G))-expansion method [32,33], and the Picard technique [34]. e time-fractional Burgers' equation is crucial for modeling shallow water waves, weakly nonlinear acoustic waves propagating unidirectionally in gas-filled tubes, and bubbly liquids. Inc [9] studied the approximate and exact solutions to the time-fractional Burgers' equation by the variational iteration method. Bekir and Guner [35] established the exact solution to the mentioned equation by using the (G′/G)-expansion method. Bulut et al. [36] examined the analytical approximate solution to the suggested equation through the modified trial equation method. Recently, Saad and Al-Sharif [37] studied the exact and analytical solutions to this equation. As far as is known, the stated equation has not been investigated through the two variables (((G ′ /G), (1/G)))-expansion technique, exp-function strategy, and expanded tanh function method. erefore, the aim of this study is to establish further general and some fresh solutions of the abovementioned equation using the suggested methods. e residual segments of the article is schematized as follows: in Section 2, definition and preliminaries have been introduced; in Section 3, the two variables ((G ′ /G), (1/G))-expansion method, the exp-function method, and the extended tanh function method have been described. In Section 4, the exact solutions to the suggested equation have established. In Section 5, physical interpretation and explanation of the extracted solutions are provided. In the lattermost part, the conclusions are given.

Definition and Preliminaries
Supposef: [0, ∞) ⟶ R be a function. e α-order conformable derivative of f is interpreted as [38] T α (f)(t) � lim for every t > 0 and α ∈ (0, 1). If f is α-differentiable in some (0, a), a > 0, and exists; then, e following theorems point out few axioms that are satisfied conformable derivatives. Theorem 1. Consider α ∈ (0, 1] and let us suppose f and g be α-differentiable at a point t > 0. erefore, Some more properties including the chain rule, Gronwall's inequality, some integration techniques, Laplace transform, Tailor series expansion, and exponential function with respect to the conformable fractional derivative are explained in [38].
Theorem 2. Let f be an α-differentiable function in conformable differentiable, and suppose that g is also differentiable and defined in the range of f. en, (2) e Caputo derivative is another important fractional derivative concept developed by Michele Caputo [39]. is definition is particularly useful for finding numerical solutions. e definition of Riesz [40,41] in relation to the fractional derivative, on the contrary, is also important for extracting numerical solutions. e two concepts are not discussed in depth here since the aim of this article is to establish exact solutions.

Outline of the Methods
In this part, we summarize the principal parts of the suggested methods to analyze exact traveling wave solutions to the NLFEEs. Assume the general NLFEE is of the form where u represents an unknown function, consisting the spatial derivative x and temporal derivative t, and P represents a polynomial of u(x, t) and its derivatives where the highest order of derivatives and nonlinear terms of the highest order are associated. Take into account the wave transformation where c and k are nonzero arbitrary constants. By means of wave transformation (4), equation (3) can be rewritten as 2 Mathematical Problems in Engineering where the superscripts specify the ordinary derivative of u relating to ξ.
Step 1: In this subsection, we apply the two variables ((G ′ /G), (1/G))-expansion method to acquire the wave solutions of the NLFEEs. Take into account the second order ODEs along with the following relations In this manner, it gives e solutions to equation (6) depend on λ as λ < 0, λ > 0, and λ � 0. Case 1: when λ < 0, the general solution to equation (6) is In view of that, we obtain where σ � A 2 1 − A 2 2 . Case 2: if λ > 0, the solution to (6) is given as follows: erefore, we obtain where σ � A 2 1 + A 2 2 . Case 3: when λ � 0, the solution of equation (6) is erefore, we find where A 1 and A 2 are arbitrary constants.
Step 2: in agreement with two variables ((G ′ /G), (1/G))-expansion scheme, the solution of (5) is presented as a polynomial of ϕ and ψ of the form where a i and b i are arbitrary constants to be determined later.
Step 3: after balancing the maximum order of derivatives and nonlinear terms, which appear in equation (5), it can be fixed the positive integer N.
Step 4: setting (15) into (5) along with (8) and (10), this modifies to a polynomial in ϕ and ψ having the degree of ψ as one or less than one. If we compare the polynomial of similar terms to zero, then it will give a set of mathematical equations which can be unraveled by computational software and finally yield the values of a i , b i , μ, A 1 , A 2 , and λ, where λ < 0; this condition provides solutions of the hyperbolic function.
Step 5: in a similar manner, we can examine the values of a i , b i , μ, A 1 , A 2 , and λ, and trigonometric and rational solutions can be established separately for the case of λ > 0 and λ � 0.

e Exp-Function Method.
Within this section, the key components of the exp-function method are described for searching the traveling wave solution to the NLFDEs.
Step 1: the arrangement is to be communicated in the shape as indicated by the exp-function method: where c, d, p, and q are unknown positive integers, which can be evaluated later, and p n and q m are unidentified constants.
Step 2: the balancing principle between the highestorder linear and nonlinear terms presented in (5) and substituting (16) into (5) yield c and p, and the balance of lowest-order linear and nonlinear terms yields the values of d and q.
Step 3: introducing (16) into (5) and setting the coefficient of exp(nξ) to zero provides an arrangement of set of mathematical equations for p n , q m , c, and k. en, unraveling the set with the aid of computer software, such as Maple, we attain the constants.
Step 4: substituting the values that showed up in step 3 into (16), we ascertain exact solutions to the NLFEEs in (3).

e Extended Tanh Function Method.
In this section, the suggested extended tanh function method has been interpreted to obtain ample exact solutions to NLFEEs which was Mathematical Problems in Engineering summarized by Wazwaz [42].
e basic concept of this method is to present the solution as a polynomial of hyperbolic functions, and then, solving the coefficient of tanh(μξ) implies solving a system of algebraic equations. e core steps of the extended tanh function method for finding exact analytic solutions of nonlinear PDEs of the fractional order are as follows: Step 1: we consider the wave solution as follows: wherein where μ is any arbitrary constant.
Step 2: taking uniform balance between the maximum order nonlinear term and the derivative of the maximum order appearing in equation (5) to determine the positive constant N.

Analysis of the Solutions
Here, we search further comprehensive exact analytic wave solutions for the stated time-fractional Burgers' equation by means of the suggested methods. Let us consider the timefractional Burgers' equation as follows: where p and v are arbitrary constants. e physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe are described by the timefractional Burgers' equation. e fractional derivative results are obtained from the memory effect of the wall friction through the boundary layer. e similar formation can be found in several systems, namely, waves in bubbly liquids and shallow water waves. For equation (19), we recommend the subsequent wave transformation: where c be the velocity of the traveling wave. For wave transformation (20), time-fractional Burgers' equation (19) reduced to the ensuing integral order differential equation: Integrating equation (21) with zero constant, we obtain

Solutions through Two Variables ((G ′ /G), (1/G))-Expansion Method.
Considering the homogeneous balance of the highest-order nonlinear term and highest-order derivative showing up in equation (22), the arrangements of equation (15) accept the shape where a 0 , a 1 , and b 1 are constants to be determined.
Case 1: for λ < 0, embedding solution (23) into (22) along with equations (8) and (10) yields a set of algebraic equations, and by explaining these equations by computer algebra such as Maple, we achieve the following results: Inserting the top values into solution (23), we find the solution to equation (19) in the form )/(v(λ 2 σ + μ 2 ))) (t α /α). Since A 1 and A 2 are basic constants, one might have picked self-assertively their values. If we take μ � 0 and where ))(t α /α). Case 2: in a comparative way, when λ > 0, substituting (23) into (22) together with (8) and (12) yields an arrangement of algebraic equations for a 0 , a 1 , b 1 , and ω, and we acquire the following results by working out these equations: e substitution of these results into solution (23) possesses the following expression for the general solution of equation (19): where If the unknown parameters are assigned as μ � 0 and A 1 � 0 and A 2 ≠ 0 or A 1 ≠ 0 and A 2 � 0 in solution (29), it provides the next solitary wave solution: ))(t α /α). Case 3: in the parallel algorithm when λ � 0, using equations (22) and (21) along with (8) and (14), we achieve a set of mathematical equations whose solutions are a 0 � 0, Making use of these values into solution (23) produces the solution to equation (19) as where ξ � ± (pb 1 /v)

Mathematical Problems in Engineering
It is substantial to observe that the traveling wave solutions u 1 1 -u 1 7 of the studied equation are inclusive and standard. e attained solutions have not been noted in the earlier study. ese solutions are convenient to designate the physical processes of unidirectional propagation of weakly nonlinear acoustic waves via a gas-filled tube, shallow-water waves, and waves in bubbly liquids.

Solution by the Exp-Function Method.
Considering the homogeneous balance, the solution of equation (16) takes the form Substituting equation (34) into (22) leads a equation in exp(nξ); here, n represents any whole number. Inserting each coefficient of this equation to zero yields a cluster of mathematical equations (for straightforwardness, here, we have discarded) for p i ′ s, q i ′ s, and ω. ese mathematical equations are solved by computer algebra, namely, Maple, which gives the following outcomes: From the point of view of the above results, we achieve the following generalized solitary wave solutions:

Physical Interpretation and Explanation
In this section, we mainly discuss about the physical interpretation of the determined solitary wave solutions, including kink, singular solitons, singular kink, and periodic wave of the NLFEEs. A graph is an effective approach for explaining mathematical concepts. It is capable of describing any circumstances in a straightforward and understandable manner. is segment explains the incidents by portraying 3D plots of some of the solutions that are found. e portraits are precedents of the solutions shown in Figures 1-6 using the computational software, namely, Mathematica.

Conclusion
In this article, using three reliable approaches referring conformable the fractional derivative, we have established scores of advanced, further general, and wide-ranging solitary wave solutions to the time-fractional Burgers' equation. e ascertained closed-form solutions of the considered equation include kink, single solitons, periodic solitons, singular kink, and some other kinds of solutions, including some free parameters. e obtained solutions are capable to analyze the phenomena of weakly nonlinear acoustic waves propagating unidirectionally in gas-filled tubes, shallow water waves, and bubbly liquids. e dynamics of solitary waves have been graphically depicted in terms of space and time coordinates which reveal the consistency of the techniques used. e accuracy of the results obtained in this study has been verified using the computational software Maple by placing them back into NLFPDEs and found correct. is study shows that all the methods implemented are reliable, effective, functional, and capable of uncovering nonlinear fractional differential equations arising in the field of nonlinear science and engineering. erefore, we can firmly claim that the implemented methods can be used to    compute exact wave solutions of other nonlinear fractional equations associated with real-world problems, and this is our next contrivance.

Data Availability
No data were used to support this study.