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The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables

The nonlinear fractional evolution equations (NLFEEs) emerge frequently in diverse research field of science and applications of engineering. The 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

The 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 [

The residual segments of the article is schematized as follows: in Section

Suppose

Consider

In addition, if

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 [

Let

The Caputo derivative is another important fractional derivative concept developed by Michele Caputo [

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

By means of wave transformation (

Step 1: In this subsection, we apply the two variables

along with the following relations

In this manner, it gives

The solutions to equation (

Case 1: when

In view of that, we obtain

where

Case 2: if

Therefore, we obtain

where

Case 3: when

Therefore, we find

where

Step 2: in agreement with two variables

where

Step 3: after balancing the maximum order of derivatives and nonlinear terms, which appear in equation (

Step 4: setting (

Step 5: in a similar manner, we can examine the values of

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

Step 2: the balancing principle between the highest-order linear and nonlinear terms presented in (

Step 3: introducing (

Step 4: substituting the values that showed up in step 3 into (

In this section, the suggested extended tanh function method has been interpreted to obtain ample exact solutions to NLFEEs which was summarized by Wazwaz [

Step 1: we consider the wave solution as follows:

wherein

where

Step 2: taking uniform balance between the maximum order nonlinear term and the derivative of the maximum order appearing in equation (

Step 3: substitute solution (

Step 4: inserting the values that appeared in step 3 into equation (

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 time-fractional Burgers’ equation as follows:

Integrating equation (

Considering the homogeneous balance of the highest-order nonlinear term and highest-order derivative showing up in equation (

Case 1: for

Inserting the top values into solution (

where

Since

where

Case 2: in a comparative way, when

The substitution of these results into solution (

where

If the unknown parameters are assigned as

where

Case 3: in the parallel algorithm when

Making use of these values into solution (

It is substantial to observe that the traveling wave solutions

Considering the homogeneous balance, the solution of equation (

Substituting equation (

From the point of view of the above results, we achieve the following generalized solitary wave solutions:

In particular, if

The choice of

It is significant to refer that the traveling wave solutions

The homogeneous symmetry allows solution equation (

Substituting (

Using the values of the parameters assembled above into solution (

The solutions established above by the extended tanh approach are advanced and progressive. These might be convenient to describe the relativistic electron and the physical processes of unidirectional propagation of weakly nonlinear acoustic waves via a gas-filled tube.

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. This segment explains the incidents by portraying 3D plots of some of the solutions that are found. The portraits are precedents of the solutions shown in Figures

3D plot of the kink type soliton of (

3D plot of the single soliton solution of (

3D plot of the periodic wave solution of (

3D plot of the singular kink type soliton of (

3D plot of the kink type soliton of (

3D plot of the singular kink type soliton of

The results of the time-fractional Burgers’ equation include the kink soliton, singular soliton, periodic soliton, and some general solitons which are displayed in Figures

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. The 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. The 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. The dynamics of solitary waves have been graphically depicted in terms of space and time coordinates which reveal the consistency of the techniques used. The 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. This 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. Therefore, 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.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.