Abstract
This research addresses the modified F-expansion, the newly established extended modified F-expansion, and the unified method for computing the exact solution of the time-fractional regularized long-wave (Tf-RLW) model with Conformable fractional derivative. The Tf-RLW model is widely applicable to solitary wave propagation in shallow water waves, plasma waves, and ion-acoustic waves. The three analytical techniques are employed to simplify this nonlinear model and justify the obtained waves in relation to the existing waves of this model. Fractional derivatives are non-integer-order operators widely used to model complex phenomena in science and engineering, with notable applications in ocean and coastal engineering for tsunami-wave mitigation. This research presented the bright-dark bell waves, periodic rogue waves, periodic waves, singular kink waves, and kinky-periodic bell waves as solutions to the governing model. 3-D, density, and 2-D sketches are given to analyze the intense behavior of the attained waves. Stability investigation of the obtained solutions is also included. All the solutions obtained in this research are substantiated using Maple 18.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-026-37284-6.
Keywords: Tf-RLW equation, Conformable derivative, Soliton, Periodic wave
Subject terms: Engineering, Mathematics and computing, Physics
Introduction
The world’s maximum phenomena can be modeled by nonlinear partial differential equations (NLPDEs). For instance, NLPDEs are widely used in modeling to describe intricate physical phenomena in numerous fields of engineering and science, particularly in plasma physics, such as ion-acoustic waves, nonlinear long-wave mechanics, and water wave mechanics, which are especially relevant in coastal engineering. That is why the solutions of such types of NLPDE are so important. However, deriving an exact solution for NLPDEs is complicated for researchers because there is no unique technique to solve any NLPDEs. Many operative methods are exposed for solving NLPDEs by the mathematician, such as the generalized Kudryashov’s method1, the Hirota bilinear method2–4, the improved modified simplest equation method5, the auxiliary equation method6, the enhanced modified simple equation method7, the direct algebraic scheme8, the unified method9, the modified F-expansion10,11, the exp 
-expansion method12, the bifurcation technique13,14, the modified extended tanh-function method15, the unified solver method16, the Sine–Gordon expansion method17, symbolic computation with neural network method18, the novel PH method19, the Maclaurin series method20, the generalized projective Riccati equation method21,22, the variational principal method23, the 
-expansion method24, and so on25–27. Side by side, some well-established computational software, such as Mathematica, Maple, MATLAB, and COMSOL, help us solve problems, verify the obtained solutions, and create 3D and 2D plots of the solutions. Fractional derivatives are a crucial technique for describing the behavior and internal mechanisms of waves. Many researchers implement different types of fractional derivatives in their research work. Some commonly used fractional derivatives are conformable derivatives28,29, truncated M-fractional derivative30, beta-derivative31,32, Caputo-Fabrizio derivative33,34, fractal generalized-variational structure35, local-fractional derivative36–39, Atangana-conformable time-derivative40, He’s fractional derivative41, Riemann–Liouville42 fractional derivative, etc.
The conformable derivatives physically provide the fractional order dynamics while maintaining the structure of classical calculus43. It describes how it modifies the rate of change of a physical quantity by introducing fractional space or time scaling. Conformable derivatives are a trend of fractional calculus that overcomes some limitations of other fractional derivatives, such as Caputo and Riemann–Liouville type derivatives.
This research aims to derive an exact travelling wave soliton of the integrable fractional NLPDE, namely the Tf-RLW model applicable in shallow water-wave mechanics44,45 with time-fractional conformable derivatives effects. We are rendering with the time-fractional RLW (Tf-RLW) model as12:
 |
1 |
The main objective of this article is to obtain soliton solutions of the Tf-RLW model by the unified method9, the modified F-expansion method10,11, and the extended modified F-expansion method (newly established) with time-fractional derivatives of conformable sense12.
This research is formulated as follows: Section 2 contains “Preliminaries and methodology”. “Solution of the Tf-RLW model” is presented in Section 3. “Results and discussion” are provided in Section 4. Section 5 describes the “Stability investigation” of the obtained solutions. “Novelty and comparison” are included in Section 6. Finally, “Conclusions” are provided in Section 7.
To the best of our knowledge, this is the first study to employ these three analytical approaches in investigating exact solutions of the Tf-RLW model.
Preliminaries and methodology
Conformable derivative
T. Abdeljawad46 and Khalil et al.47 introduce time-fractional derivatives of conformable sense as:
For 
, the fractional 
-order conformable derivatives defined by:
 |
Established theorems about time-fractional derivatives of conformable sense as:
Theorem 1:
For 
and 
conformable differentiable48 with 
:
-
(i)
-
(ii)
-
(iii)
-
(iv)
-
(v)
For differentiable
Theorem 2:
If 
be a real-valued function as 
be 
-conformable differentiable49 and 
be differentiable then:
 |
Modified F-expansion scheme
Consider the f-NLEE as:
 |
2 |
where 
is the function of 
, and 
is a polynomial of 
with its derivatives.
Step 1: The combined transformation 
where 
and 
are constants, transfer Eq. (2) as:
 |
3 |
where, 
is a function of 
.
Step 2: Presume the trial solution of Eq. (3) as:
 |
4 |
where 
and 
are arbitrary constants. 
is the integer attained by balancing the highest order nonlinear and derivative terms of Eq. (3).
Step 3: The auxiliary equation of Eq. (4) be well-thought-out as:
 |
5 |
Step 4:
and 
are related to 
as (see Table 1):
Table 1.
Relations Among 
and 
.
| Sl. No | Values of  and 
|
The function 
|
|---|---|---|
| 1 |
 and 
|
 |
| 2 |
 and 
|
 |
| 3 |
 and 
|
|
| 4 |
 and 
|
 Or 
|
| 5 |
 and 
|
|
| 6 |
 and 
|
|
| 7 |
 and 
|
 |
| 8 |
 and 
|
 |
| 9 |
 and 
|
 |
| 10 |
 and 
|
 |
| 11 |
 and 
|
 |
Step 5: Substituting 
in Eq. (3), a polynomial of 
will be obtained. Likening the coefficients of 
equal zero and solving, the values of 
are attained for the deserved solutions.
Extended modified F-expansion scheme
Presume the trial solution of Eq. (3) as:
 |
6 |
where 
and 
are arbitrary constants. 
are integers. 
is attained by balancing the highest order nonlinear and derivative terms of Eq. (3).
The auxiliary equation of Eq. (6) be well-thought-out as:
 |
7 |
where 
and 
are related to 
as Table 1 and the values of 
are attained for the deserved solutions, step 5 as above.
Remark:
The modified F-expansion scheme familiarized result in a series for 
, i.e., provides new types of waves in the literature, while the extended modified F-expansion method is used 
. In both cases, we use the same auxiliary equation 
The unified scheme
Presume the trial solution of Eq. (3) as:
 |
8 |
where 
and 
are arbitrary constants. 
is the integer attained by balancing the highest order nonlinear and derivative terms of Eq. (3). The auxiliary equation of Eq. (8) be well-thought-out as:
 |
9 |
Here, 
is related to 
as:
If 
then trigonometric function solutions occur as:
 |
10 |
If 
then hyperbolic function solutions occur as:
 |
11 |
If 
then rational function solutions occur as:
 |
12 |
where 
are constants. Substituting 
with its derivatives in Eq. (3), a polynomial of 
will obtained. Likening the coefficients of 
equal zero and solving, the values of 
are attained for the deserved solutions.
Solution of the Tf-RLW model
This sub-section integrates the following Tf-RLW model50 for a conformable derivative with time fraction using three intrigation techniques:
 |
13 |
Using wave transformation, 
where 
and 
are constants, reduces Eq. (13) as:
 |
14 |
Solution of the Tf-RLW model by the modified F-expansion scheme
By the modified F-expansion scheme, the trial solution with 
of Eq. (14) is,
 |
15 |
Substitute Eq. (15) in Eq. (14), and equating the coefficient of 
, a system of equations is obtained (See APPENDIX I). Solving them, the obtained two sets of constants are:
Set-I: 
 |
Set -II: 
 |
Here we seek solutions of the Tf-RLW model Eq. (13) according to Table 1 for the above two sets.
Solution for Set-I
 |
16 |
 |
17 |
 |
18 |
 |
19 |
 |
20 |
 |
21 |
 |
22 |
 |
23 |
 |
24 |
 |
25 |
where 
and 
with 
are constant.
Solution for Set-II
 |
26 |
 |
27 |
 |
28 |
 |
29 |
 |
30 |
 |
31 |
 |
32 |
 |
33 |
 |
34 |
where 
and 
with 
are constant.
Solution of the Tf-RLW model by the extended modified F-expansion scheme
By the extended modified F-expansion scheme, the trial solution with 
and 
of Eq. (14) is,
 |
35 |
Substitute Eq. (35) in Eq. (14) and equating the coefficient of 
, a system of equations is obtained (See APPENDIX II). Solving them, the obtained set of constants is:
 |
 |
Here we seek solutions of the f-RLW model Eq. (13) according to Table 1 for the above sets.
 |
36 |
 |
37 |
 |
38 |
 |
39 |
 |
40 |
 |
41 |
 |
42 |
 |
43 |
 |
44 |
 |
45 |
 |
46 |
where 
and 
with 
are constant.
Solution of the Tf-RLW model by the unified scheme
By the modified F-expansion scheme, the trial solution with 
of Eq. (14) is,
 |
47 |
Substitute Eq. (47) in Eq. (14) and equating the coefficient of 
, a system of equations is obtained (See APPENDIX III). Solving them, the obtained two sets of constants are:
Set-I:
Set -II:
Here we seek solutions of the Tf-RLW model Eq. (13) according to the pronounced scheme for the above two sets.
Solution for Set-I
 |
48 |
 |
49 |
 |
50 |
 |
51 |
 |
52 |
 |
53 |
 |
54 |
where 
and 
with 
are constant.
Solution for Set-II
 |
55 |
 |
56 |
 |
57 |
 |
58 |
 |
59 |
 |
60 |
 |
61 |
where 
and 
with 
are constant.
Results and discussion
This section represents the dynamical behavior of the obtained solutions with their 3-D and density plots with different fractional effects.
In Fig. 1, we observe a kink wave without a fractional effect as 
. A different type of local breather wave arises in the soliton due to different values of the conformable fractional parameter 
Corresponding density plots under each graph present the banding effect for more fractionality. The 2-D sketch expresses the position of the wave at 
for a different fractional parameter 
for the free parameter 
Fig. 1.
The 3-D, density, and 2-D sketch for the kink and local breather wave Eq. (16).
Figure 2 signifies a bright-bell wave for the free parameter 
. The pattern of bright-bell waves is turning to band for more fractionality as 
The 2-D sketch expresses wave shapes at 
for different fractionality and being wider parameter with increases.
Fig. 2.
The 3-D, density, and 2-D sketch for the bright-bell wave solution of Eq. (18).
In Fig. 3, we observe a dark-bell wave without a fractional effect as 
The dark bell wave makes a rogue wave with an increase of fractionality for the free parameter 
. The 2-D sketch expresses the position of the wave at 
and 
for a different fractional parameter 
Fig. 3.
The 3-D, density, and 2-D sketch dark-bell wave solution of Eq. (27) at real slice.
In Fig. 4, we observe a singular-kink wave without a fractional effect as 
for the free parameter 
. Different types of rogue waves in different positions are created for the different values of the conformable fractional parameter 
The 2-D sketch expresses the position of the wave at 
and 
for a different fractional parameter 
Fig. 4.
The 3-D, density, and 2-D sketch of the singular-kink wave solution Eq. (27) at an imaginary slice.
Figure 5 represents periodic rogue waves for the free parameter 
at Eq. (46). The 2-D sketch expresses the position of the wave at 
and 
which expresses that the amplitude decreases with the increase of fractionality.
Fig. 5.
The 3-D, density, and 2-D sketch of the periodic rogue waves solution for Eq. (46) of a real slice.
Figure 6 represents kinky-periodic rogue waves for the free parameter 
at Eq. (46). The 2-D sketch expresses that the amplitude varies with different fractionality at 
and 
.
Fig. 6.
The 3-D, density, and 2-D sketch of the kinky-periodic rogue waves solution for Eq. (46) of an imaginary slice.
In Fig. 7, we observe a different type of periodic bright-bell wave for the free parameter 
at Eq. (49) of real slices with different fractional parameters 
The 2-D sketch expresses that the width increases with the increase of fractionality at 
and 
.
Fig. 7.
The 3-D, density, and 2-D sketch of the periodic bright-bell wave solution Eq. (49) of real slice.
In Fig. 8, a kinky-periodic wave is obtained for 
at an imaginary slice of Eq. (49) with a different fractional parameter 
The 2-D sketch expresses the change of width at 
and 
for different fractionality.
Fig. 8.
The 3-D, density, and 2-D sketch of the kinky-periodic wave solution Eq. (49) of an imaginary slice.
Figure 9 represents the bright-bell wave for Eq. (60) with 
. The shape of the bell is bending with the increase of fractionality. The 2-D sketch expresses the width are change at 
and 
for a different fractional parameter 
Fig. 9.
The 3-D, density, and 2-D sketch of the bright-bell wave for Eq. (60).
Stability investigation
This section corresponds to the stability investigation of the governing equation by considering the next Hamiltonian factor51,52.
 |
62 |
which satisfies the following condition.
 |
63 |
with momentum 
and velocity 
.
Now, we apply the solution Eq. (20) and Eq. (62) with constraints 
, then the stability condition can be expressed as.
 |
Accordingly, 
is the stable solution. In a similar way, one can acquire the stability conditions of our other outcomes.
Novelty and comparison
In the prior research, Ejaz et al.53 smear subdivision collocation scheme on the RLW model and derive the analytical and numerical solutions with 2-D, 3-D plots as periodic bright-bell type waves. Hossain et al.54 obtained kink, anti-kink, bright and dark bell, double periodic, and periodic types of waves of the RLW model by the EMSE technique. By smearing predictor–corrector and quartic-B-spline collocation method, DAG et al.55 derive a numerical algorithm for solving the RLW model and obtained single solitary wave solutions in bright-bell and periodic bright-bell types of solutions. They also discuss the properties of the bell wave for different values of parameters. Shah et al.56 smear integral-transform and ET method on the fractional RLW model and derive bright-bell and kink waves with the effects of fractional derivatives. Yavuz et al.57 smear the MLDM method with the sense of Caputo on the fractional RLW model and acquire approximate-analytical solutions in the form of bright-bell waves. Hassan et al.58 derive a single bright-bell wave for the RLW model by inserting the RPA method. Aminikhah et al.59 derive hyperbolic, trigonometric, and rational function solutions of the fractional RLW model with the effects of conformable fractional-derivative. Bright-dark bell wave was obtained with the VIM method for the RLW model by Hosseini et al.60. Besides this, we implemented three integration methods, namely, the unified method9, the modified F-expansion10,11, and the extended modified F-expansion (newly established) with time-fractional derivatives of conformable sense 12. Our implemented procedure provides bright, dark bell, periodic bright-dark bell wave, kinky-periodic wave, and double-periodic waves with the effects of conformable sense, which are novel and exceptional to prior works.
Conclusions
In this research, we magnificently apply the modified F-expansion scheme, the extended modified F-expansion scheme, and the unified scheme to integrate the time-fractional nonlinear Tf-RLW equation with conformable fractional derivative. As an upshot, we attained periodic rogue wave, dark-bell wave, bright-bell wave, kinky-singular wave, and double periodic wave. Some 3-D and density profiles of the attained wave are sketched with various values of the parametric constant to illustrate the internal mechanism of the attained waves. Side by side, some 2-D profiles are sketched for clarity of fractional effects on gained solutions. The applied three schemes may fail to solve the higher-order nonlinear models. Furthermore, the applied three schemes are first, easier, and can be handled by a computer easily. The computational software MAPLE 18 was applied to solve the problem and verify our obtained solutions.
Supplementary Information
Below is the link to the electronic supplementary material.
Author contributions
Mohammad Mobarak Hossain: conceptualization, visualization, software, methodology, data-curation, writing – original draft. Harun-Or-Roshid: formal analysis, validation, methodology, writing – review and editing. Mohammad Safi Ullah: formal analysis, review – original draft. Md. Abu Naim Sheikh: resources, investigation, validation.
Funding
No funding was received for this work.
Data availability
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Mohammad Mobarak Hossain, Email: mobarak4074@gmail.com.
Mohammad Safi Ullah, Email: safi@cou.ac.bd.
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This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.