Families of propagating soliton solutions for (3+1)-fractional Wazwaz-BenjaminBona-Mahony equation through a novel modification of modified extended direct algebraic method

The article presents a new modification to the modified Extended Direct Algebraic Method (mEDAM) namely r+mEDAM to effectively and precisely acquire propagating soliton and other travelling wave solutions to the Fractional Wazwaz-Benjamin-Bona-Mahony (FWBBM) equation. By using this updated approach, we are able to find more and new families of propagating soliton solutions for the FWBBM problem, such as soliton, kink, lump-like singular, trigonometric, hyperbolic, periodic, shock, singular & non-singular wave solutions. We also provide 3D and 2D graphs that visually illustrate the obtained solutions. By obtaining accurate propagating soliton solutions, our r+mEDAM proves to be practical while also revealing important details about the dynamics of the equation and suggesting possible applications in the fields of optics, materials research, and water waves.


Introduction
Nonlinear Partial Differential equations (NPDEs) receive applications in many areas of physics, mathematics and engineering.A wide variety of phenomena, such as the dispersion of organic compounds, thermal conductivity, wave bending, fluid mechanics, magnetism, oceanic wave propagation, quantum field theory, and hydrodynamics, are all effectively described by these equations.The domain of NLPDEs is shown by the advection equation, Burgers' equation, Boussinesq equation, linked nonlinear equations, Fisher's equation, and several more equations [1][2][3][4][5].
In the context of NPDEs, well known equations like the the modified Korteweg-de-Vries (KdV) model, extended Benjamin-Bona-Mahony model, the Boussinesq model, and the regularized long waves model have been extensively investigated [6,7].In order to characterise wave motion below the surface in a uniform channel, Benjamin, Bona, and Mahony first proposed the Korteweg-de-Vries model in 1972.This resulted in the creation of the Benjamin-Bona-Mahony (BBM) model, a shallow wave propagation-focused normalised KdV model.The analysis of Rossby and drifting waves in rotating fluid systems makes use of the BBM model [8][9][10].
The BBM model is given as [11]: & the Kdv model is presented below [12]: ( ) The BBM equation and the KdV equation are both used in the modelling of various waves, such as surface pulses in water, extended signals in nonlinear diffraction processes, acoustic energy in periodic pebbles, resonating waves in compressible fluids and magnetic hydrodynamic waves in erythrocytes.But these two equations differ significantly from one another.Only when low-frequency waves are taken into account is the BBM equation equal to the KdV equation, and the two equations provide different wave-related results.The KdV equation functions as an existence equation, whereas the BBM equation is classified as a regularity equation [14].In [15,16], the theory of the BBM model, such as its uniqueness, stability, & consistency, is examined.In recent years, the BBM model has also been the subject of extensive study.A consensus exists that solving the BBM model is considerably easier than calculating the KdV model, but both equations have stable solutions for solitary waves [17,18].The BBM equation is occasionally referred to as the long wave equation.In equation (1), when n = 2, the BBM equation becomes the modified BBM equation [19] examined the solution to several problems including initial values for the modified BBM model.In 2017, Wazwaz suggested a new equation for characterizing the three-dimensional dynamics of water waves.The Wazwaz-Benjamin-Bona-Mahony (WBBM) equation is a modification of the Benjamin-Bona-Mahony equation.In a result, Wazwaz  In fields such as the mechanical & electrical properties of real materials, fractional order models have surmounted the limitations imposed by conventional integer order models [4,5].In this study, we will focus on the 3-dimensional FWBBM model, which is written as follows: y & b D z represent the conformable fractional derivative of order β, where 0 < β 1, t 0. Numerous numerical and analytical approaches have been established in the literature to deal with FPDEs.Analysts frequently choose for analytical methods despite the availability of numerical approaches because they may provide a full understanding of the underlying physical processes and precise insights into the behaviour of the system.Finding analytical answers for FPDEs, meanwhile, is difficult.Therefore, it is still important to do research in developing analytical solutions to FPDEs.Numerous analytical methods have been developed to deal with this issue.These techniques include the Adomian decomposition method [9], Khater method [10], Variable separable method [12], (G'/G)-expansion method [13], integral transforms method [14], Exp-function method [15], Differential transform method [16], homotopy perturbation method [17], and EDAM [18].
The mEDAM is one of these analytical techniques that is specifically utilised to find families of solitons solutions for FPDEs.With the use of variable transformation, the FPDE is converted into a Nonlinear Ordinary Differential equation (NODE) using this technique.The NODE is then transformed into a collection of algebraic equations under the supposition of a series form solution.The soliton solutions for the relevant FPDEs can be obtained by solving the ensuing system of algebraic equations.The non-local and non-classical nature of these equations may be handled utilising the mEDAM, which is particularly helpful for FPDEs that are difficult to solve using traditional analytical methods.The soliton solutions produced by mEDAM can help us better understand the underlying physical processes and the behaviour of the system [20][21][22].
The aim of this study is to explore propagating soliton and other travelling wave solutions for 3-dimensional FWBBM model presented in (4) using a novel modification of mEDAM namely r+mEDAM.These soliton solutions describe the soliton phenomena in FWBBM model.A soliton is a self-sustaining wave packet which retains its shape & speed throughout propagation, due to the delicate equilibrium between dispersion & nonlinearity.Some examples of solitons include lump waves (localized solitons), kink waves (topological solitons), shock waves (nonlinear waves with discontinuities), damped waves (Low-frequency oscillation) & periodic waves (solitons with periodic behavior) [23][24][25].
The fractional derivatives presented in (4) are conformable fractional derivatives.This derivative operator of order β is described as [26]: The following properties of this derivative are used in transformation of FPDE in (4) into NODE: where ñ(χ) & υ(χ) are arbitrary differentiable functions, whereas k, k 1 & k 2 signify real constants in the current context.
2. The methodical strategy of r+mEDAM [20] We introduce the r+mEDAM approach in this section.Let's think about the FPDE in the following manner: where F has derivatives with respect to υ in (10).Occasionally, (10) can be integrating once or multiple times in order to determine the constant(s) of integration.
2. Afterwards, we suggest that equation (10) holds the subsequent solution: Here, the values of the constants d i (where i = − N,K,N) are to be driven & W(υ) satisfies the ensuing ODE: where A ≠ 0, 1 & j, k, l are constants.
3. We determine the positive integer N as stipulated in equation (11) by looking for a homogenous balance between the main nonlinear component & the highest order derivative in equation (10).

4.
In order to create a polynomial expression using (W(υ)), we first substitute equation (11) into equation (10), or the equation produced from integrating (10), & then arrange all of the terms of (W(υ)) in the same order.In order to calculate d i (where i = − N,K,N) & other related parameters, the coefficients of this derivation polynomial are then equal to zero.
5. Using the MAPLE programme, we solve this set of algebraic equations.
6.The unknown parameters are then accounted for & substituted into equation (11), together with the solution for W(υ) (obtained from equation ( 12)), to produce the analytical solutions for equation (9).We may produce the families of soliton solutions shown below by using the general solution of equation (12): Family 10: Family 11: Family 12: q, p > 0 & are known as deformation parameters when P = k 2 − 4jl is satisfied.The following is a description of the hyperbolic & trigonometric functions that are generalized:

Application of r+mEDAM
This section emphasizes on utilizing our proposed method r+mEDAM to confirm its efficacy & dependability.
As a consequence, we will receive families of the soliton solutions for FWBBM specified in (4).Applying the following transformation: (4) now becomes by inserting (13) into it: We get an expression in W(υ) when we put (16) in (15).Collecting all terms with same powers of W(υ) & equating the coefficient of each term to zero gives a system of seven algebraic equations.We use Maple to solve the system.And in a result we get the following five cases of results for the system of algebraic equations: Case. 1 Case. 2 Case. 4 Family 1: When E < 0l ≠ 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: tan sec 2 24 ) and Family 2: When E > 0 l ≠ 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: 2 29 2 30 and Family 3: When jl > 0 and k = 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: Family 4: When jl < 0 and k = 0, then (13) and (16) suggest the subsequent families of propagating soliton solutions: Family 5: When l = j and k = 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: Family 6: When l = − j and k = 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: Family 7: When E = 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: Family 22: When k = j = 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: Family 8: When j = 0, k ≠ 0 and l ≠ 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: Family 8: When k = σ, l = sσ(s ≠ 0) and j = 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: ( ) Taking case 2, we obtain the ensuing families of propagating soliton solutions for (4): Family 9: When E < 0 l ≠ 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: and Family 10: When E > 0 l ≠ 0, then (13) and (16) suggest the subsequent families of propagating soliton solutions: and tanh 1 4 coth 1 4 4 66 Assuming case.3, we get the subsequent families of propagating soliton solutions (4): Family 18: When E < 0 l ≠ 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: and Family 19: When E > 0 l ≠ 0, then (13) and (16) suggest the subsequent families of propagating soliton solutions: ( ) Assuming case.4, we get the following families of propagating soliton solutions for (4): Family 28: When E < 0Ã¢l ≠ 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: ) and Family 29: When E > 0 ≠ 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: ) and )) Family 30: When jl > 0 and k = 0, then (13) and ( 16) suggest the subsequent families of propagating soliton solutions: .hx ht

Discussion and graphs
In this section, various travelling wave structures found in the system under study are visually shown.We employ the improved EDAM, r+mEDAM, to extract and illustrate several wave structures: soliton, kink, lumplike singular, hyperbolic, trigonometric, periodic, shock, singular, and non-singular waves in 2-D, 3-D, and contour forms.Solitons and singular and nonsingular waves are examples of travelling wave solutions that are essential to comprehending the dynamic behaviour of many physical systems that are represented by partial differential equations.Solitons are self-reinforcing solitary waves that can hold their shape throughout propagation.They are distinguished by their stability and localised nature.Singular waves reveal extreme circumstances or critical spots in a system by discontinuities or singularities in their amplitude or shape.Conversely, nonsingular waves signify more regular, smoother waveforms.These travelling wave solutions are important because they may simulate a wide range of phenomena, such as fluid dynamics and nonlinear optics, providing a greater comprehension of intricate physical processes and enabling the investigation of nonlinear behaviours in mathematical and scientific contexts.Differing aspects can be more easily illustrated by introducing different parameters.Regarding the fractional WBBM equation, solitons were the main subject of a similar study conducted by Shafqat-Ur-Rehman et al [26].Specifically, ten findings were derived for hyperbolic and periodic functions using their approach, which involved the conformable fractional derivative and the basic DAM technique.After examination, we discovered that when particular values for r, k, l, , and j are taken into account, our results agree with some of their case results.Our work thus offers better generalisations of their results.Moreover, we note that the results achieved in this work are new, and as far as we know, no one has before documented in the literature the application of these mathematical tools to the 3D-FWBBM equation.This may help us comprehend the behaviour of physical occurrences.It is anticipated that these responses will significantly influence our ability to explain and understand the physics underlying time evolution processes.Furthermore, the methods have proven to be highly efficient, reliable, powerful, and suitable for nonlinear problems encountered in many different fields of the natural sciences.The soliton, kink, lump-like singular, trigonometric, hyperbolic, periodic, shock, singular, and non-singular wave solutions that are described in equations (23), (88) (106) and (162), respectively, are among the results that are graphically depicted in figures 1, 2, 3, and 4. Remark 1.A kink wave is seen in figure 1.A wave profile with an abrupt shift or discontinuity in its amplitude or form is referred to as a kink wave.Physically, it indicates a small area where the wave exhibits a clear curvature or departure from its usual behaviour.Different factors, such a change in boundary conditions or nonlinear wave interactions, might cause this shift.
Remark 2. In figure 2, a lump-like singular wave is shown.A non-trivial localised wave structure that exhibits singular behaviour in its amplitude or shape is called a singular lump-like wave.Remark 3. figure 3 shows a singular periodic profile.A non-trivial periodic waveform that exhibits pronounced singularities or discontinuities in its amplitude or shape is referred to as a singular periodic wave.

Conclusion
In conclusion, this work offers a unique r+mEDAM version of the mEDAM approach to produce exact families of soliton solutions for the (3+1)-dimensional FWBBM.These soliton solutions display a variety of wave forms, including soliton, kink, lump-like singular, trigonometric, hyperbolic, periodic, shock, singular & non-singular waves.These insights into the dynamics of the FWBBM model under varied beginning circumstances are quite helpful.It is demonstrated that the r+mEDAM approach is a flexible and effective method for solving nonlinear PDEs and FPDEs.It makes it possible to generate a variety of soliton families, advancing our knowledge of the underlying physical processes and the interactions between various wave phenomena in a nonlinear environment.The derived soliton solutions are also useful for understanding the FWBBM model's interactions, stability, and soliton propagation properties.Overall, the r+mEDAM modification greatly advances our understanding of nonlinear wave dynamics and provides a solid groundwork for future study in related fields.However, the method's flaw appears when the highest derivative term and any nonlinear term are not homogeneously balanced.

Remark 4 .
The wave portrayed in the figure4.can be characterised as an uniform wave with an abrupt change or disruption in its amplitude.This kind of wave is also known as a shock wave or a jump discontinuity.It symbolises a quick shift or transition in the wave's characteristics across a relatively small distance, such as pressure, density, or velocity.
In recent years, Fractional Partial Differential equations (FPDEs) have gained more importance due to their significance & utilization in numerous fields such as shear modulus, electrostatics, quantum dynamics & many other.In order to simulate technical & physical phenomena that cannot be adequately explained by integer order Partial Differential equations (PDEs), FPDEs have been utilized.Also, fractional derivatives are employed to model systems that require precise damping modeling.The objective of employing fractional derivatives & their utilization in mathematical applications & other sciences is to investigate the most recent findings in these fields.
The homogeneous method between the maximum derivative U″ & the maximum nonlinear term U3 suggests that N = 1.Putting N = 1 in (11) suggests the following solution to (15):