Sim Rshid ,Rehn Ashrf ,Zki Hmmouch

a Department of Mathematics, Government College University, Faisalabad, Pakistan

b Department of Mathematis, Lahore College for Women University, Lahore, Pakistan

c Division of Applied Mathematics, Thu Dau Mot University Binh Duong Province, Vietnam

d Department of Medical Research, China Medical University Hospital, Taichung, Taiwan

e Department of Sciences, Ecole Normal Superieure, Moulay Ismail University of Meknes, 50 0 0 0 Morocco

Keywords: Fuzzy set theory Hukuhara differentiability KdV Equation Generalized integral transform Caputo fractional derivative AB-Fractional operator Homotopy perturbation method

ABSTRACT This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries (KdV) equation has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acoustic waves in plasma.Indeed,the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method (HPTM) to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness.This approach is connected with the fuzzy generalized integral transform and HPTM.Besides that,two novel results for fuzzy generalized integral transformation concerning fuzzy partial gH-derivatives are presented.Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method.Furthermore,2D and 3D simulations depict the comparison analysis between two fractional derivative operators (Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense) under generalized gH-differentiability.The projected method (GHPTM) demonstrates a diverse spectrum of applications for dealing with nonlinear wave equations in scientific domains.The current work,as a novel use of GHPTM,demonstrates some key differences from existing similar methods.

1.Introduction

Water waves are a prominent natural phenomenon.The analysis of water ripples and their various transformations is crucial to fluid dynamics in general and ocean phenomena in particular[1–3].Water waves inspired the conception of nonlinear dispersive waves and solitons from a physics perspective.Numerous researchers,including Korteweg and de Vries,have addressed the KdV equation as a mathematical model for shallow water waves in streams and oceans,laying the foundations for the idea of solitary waves.Since then,the KdV-type equations have gained significance,with applications ranging from including internal gravity waves in lakes with various cross-sections,ion-acoustic waves in plasmas,and interfacial waves in a two-layer liquid with varying depths being identified [4–6].As a result,numerous explored and developed approaches in nonlinear partial differential equations,such as symmetry groups,B?cklund transformation,Painléve analysis,and the trail equation approach,are based on the KdV-type problem [7–10].Numerous researchers have devoted their interest to developing analytical and numerical approaches for obtaining various sorts of nonlinear PDEs;see [11–15].

The enormous variety and quality of evidence that must be incorporated into the modelling process in order for it to be as realistic as possible is a major challenge.The procedure of developing mathematical methods is subject to constraints,such as appropriate comprehension,uncertainty in precision and unpredictability,and mis-classification that contribute to parametric variations.Fuzzy simulation is a valuable tool for researchers to communicate technical difficulties.Several well-known areas have potential application of fuzzy set theory,such as control systems,knowledge-based systems,image processing,power engineering,industrial automation,robotics,consumer electronics,artificial intelligence/expert systems,management,and operations research.

Fractional calculus has been acknowledged as a highly valuable framework for addressing sustainability and complex phenomena for the past thirty years,due to its advantageous qualities such as nonlocality,heritability,high reliability,and analyticity [16–18].The modified fractional notion was developed in order to address the challenges involved with processes,including inhomogeneities.Various innovators established the underlying framework,as well as their perspectives on expanding calculus,including Liouville,Hadamard,Caputo,Grunwald,Letnikov,Abel,Riez,Caputo-Fabrizio,Atangana-Baleanu (AB),researched the use of the fractional derivative and fractional differential equations (FDEs),see [19,20].Numerous essential interactions in electromagnetics,acoustics,viscoelasticity,electrochemistry,and material science are well explained by FDEs [21,22].

Some researchers have broadened the definition of derivatives in the fuzzy setting in order to use fuzzy differential equations as a modelling framework for complex systems.This enables the formulation of differential equations in a fuzzy framework,as investigated by such authors [23].

When faced with actual occurrences,partial differential equations aren’t always the appropriate choice.We require to accumulate information from millions of domains in order to simulate complex behavior.These aspects of data sets are frequently ambiguous.Modeling complex systems with uncertain data has propelled fuzzy partial differential equations to the forefront of current mathematical modeling,capturing the attention of a slew of authors [24].Fuzzy set theory has been widely applied in several domains,i.e.,fixed-point theory,topology,fractional calculus,integral inequalities,image processing,bifurcation,control theory,consumer electronics,artificial intelligence,and operations research,[25,26].

Chang and Zadeh [27] were the first to suggest the fuzzy derivative notion,which was quickly adopted by numerous other researchers [28].Hukuhara’s publication [29] is the main focus of the concept of set valued DEs and fuzzy DEs.The Hukuhara derivative served as the foundation for the investigation of set DEs and,thereafter,fuzzy fractional DEs.Agarwal et al.in [30] reported the fuzzy Riemann-Liouville fractional differential equations leveraging the notion of Hukuhara differentiability,which was the basic foundation for the theme of fuzzy fractional derivatives.To handle ambiguous fractional differential equations,they adopted the Riemann-Liouville differentiability notion,relying on Hukuhara differentiability.The stability analysis of the solution for Riemann-Liouville fuzzy fractional DEs has been expounded in [31,32].Allahviranloo et al.[33] addressed explicit solutions to unpredictable fractional DEs under Riemann-LiouvilleH-differentiability incorporating Mittag-Leffier mechanisms in [34],and formed fuzzy fractional DEs under Riemann-LiouvilleH-differentiability incorporating fuzzy Laplace transforms.They demonstrated two novel existence theorems for fuzzy fractional differential equations using Riemann-Liouville generalizedH-differentiability and fuzzy Nagumo and Krasnoselskii-Krein criteria [35].Arqub et al.[36] employed the reproducing kernel algorithm for the solution of twopoint fuzzy boundary value problems.The fuzzy Fredholm-Volterra integrodifferential equations have been solved by the adaptation of the reproducing kernel algorithm by [37].Ahmad et al.[38] studied the third order fuzzy dispersive PDEs in the Caputo,Caputo-Fabrizio,and Atangana-Baleanu fractional operator frameworks.Shah et al.[28] presented the evolution of one dimensional fuzzy fractional PDEs.To the best of our knowledge,the authors[39,40] developed the novel extended approach and new numerical solutions for fuzzy conformable fractional differential equations and constructed the numerical solutions for fuzzy differential equations using the reproducing kernel Hilbert space method.Bushneq et al.[41] explored the ‘findings of fuzzy singular integral equations with an Abel’s type kernel using a novel hybrid method.In [42],Zia et al.adopted a semi-analytical technique for obtaining the solutions of fuzzy nonlinear integral equations.Salahshour et al.[43] expounded theH-differentiability with Laplace transform to solve the FDEs.

When it comes to discovering solutions to significant challenges,researchers prefer integral transformations.The Laplace transformation was used on biological population,prey-predator and disease models in [44,45].Many researchers employed multiple integral transforms (Elzaki,Swai,Mohand,modified Laplace,Shehu,Aboodh) to develop exact solutions to important difficulties in dynamics,inorganic chemistry,and biological sciences,see [46–48].

Following this propensity,Jafari [49] contemplated the generalized integral transform (GIT) is a significant transformation in its entirety.A variant of the GIT is the,Laplace,ρ-Laplace,Elzaki,Aboodh,Natural,Swai,Sumudu,Kamal,Mohand,Pourreza,G-transform and Shehu transform,see [50–61].

The focus of this research is to suggest a sophisticated homotopy perturbation transform method [62,63] that can handle nonlinear partial fuzzy differential equations employing the fuzzy generalized integral transformation.Formulae are converted to algebraic expressions using the fuzzy general integral transform.The nonlinear components of the problem are then handled using the He’s polynomial [64,65] approach to achieve the solution.The fuzzy generalized homotopy perturbation method is the name given to the novel perturbation method.

In this research,we propose a semi-analytical approach to the generalized homotopy perturbation transform method via fuzzy set theory.The aforesaid approach is used to construct the parametric form of the fuzzy mappings and is considered to be a valuable tool for solving the fuzzy dispersive and fifth order KdV models under fuzzy initial conditions.The generic model for the investigation of magneto-acoustic waves in plasma and shallow water waves with surface tension is the fifth order KdV equations.The integral transform applied here,in general,is the refinement of several existing transforms.Moreover,numeruous applications of the proposed algorithm are presented via the different fractional order and uncertainty parameter?∈[0,1].Also,their 2D and 3D simulations show the applicability of the method over the other methods.As a consequence,each finding generates a pair of solutions that are closely in agreement with the exact one.However,we have the choice to attain the appropriate one.Finally,as a part of our concluding remarks,we discussed the accumulated facts of our findings.

2.Fundamental ideas of fractional and fuzzy calculus

This section clearly exhibits some major features connected to the stream of fuzzy set theory and FC,as well as certain key findings about the generalized integral transform.For more details,we refer [16,66].

Throughout this investigation,we use the notationΨis (1)-differentiable and (2) -differentiable,respectively,if it is differentiable under the assumption (i) and (ii) defined in the above definition.

where B(α)denotes the normalize function that equals to 1 whenαassumed to be 0 and 1.Also,we suppose that type (i)-gHexists.So here is no need to consider (ii)-gHdifferentiability.

Recently,Jafari [49] defined the generalized integral transform.We extend this concept to fuzzy set theory.

Remark 2.1.Definition 2.11 leads to the following conclusions:

Also,considering the classical generalized integral transform proposed by Jafari [49],we get

Then,the aforesaid expressions can be written as

Proof.Employing Definition 2.11,we conclude

Again,taking into consideration of Definition 2.11,yields

Consequently,using the fact ofTheorem 1in Salahshour et al.[33] and for any arbitrary fixed 0≤?≤1, we have

This gives the desired result.□

Meddahi et al.[71] proposed the ABC fractional derivative operator in the generalized integral transform sense.Furthermore,we leverage the notion of fuzzy ABC fractional derivative in a fuzzy generalized integral transform sense as follows:

Remark 2.2.Definition 2.12 leads to the following conclusions:

3.Methodology of the fuzzy homotopy perturbation transform method via the generalized integral transform

In this unit,we exhibit the fundamental technique of the fuzzy generalized integral transform to establish the general solution for the one-dimensional fuzzy fractional fifth order KdV model.

Here,we employ the following generic form of time-fractional fuzzy PDE to implement this technique:

Also,the transformed function in the fuzzy ABC derivative sense

It follows that

Furthermore,we have

By employing the perturbation method,we acquire the solution of the first case of (3.3) as

The nonlinear term in (3.3) can be calculated from

Substituting (3.8) and (3.9) into (3.6),we attain the iterative terms which yields the solution for the fuzzy fractional CFD operator:

and again,plugging (3.8) and (3.10) into (3.7),we attain the iterative terms which yields the solution for fuzzy AB fractional derivative operator in the Caputo sense:

Then,by equating powers ofηin (3.11),we compute the following CFD homotopies:

Moreover,by equating powers ofηin (3.12),we compute the following ABC operator homotopies:

Repeating the same procedure for the upper case of (3.3).Therefore,we mention the solution in parameterized version as follows:

4.Physical interpretation by means of generalized homotopy perturbation transform method

Here,we elaborate the approximate-analytical solution of fuzzy fractional fifth order KdV models via the generalized homotopy perturbation transform method involving the CFD and ABC fractional derivative operators,respectively.Throughout this investigation,the MATLAB/MAPLE 2021 software package has been considered for graphical representation and complex computation processes.

Problem 4.1.Consider the nonlinear fuzzy fractional fifth order KdV model:

The parameterized formulation of (4.1) is presented as

Case I.Firstly,taking into consideration the CFD coupled with the generalized homotopy perturbation transform method on the first case of (4.3).

In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

Equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows:

Case 2.Now,we employ the fuzzy ABC derivative operator on the first case of (4.3) as follows: In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

By the virtue of (4.5),we can calculate the following iterative terms by equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows:

From Fig.4.1 represents the analysis of Problem 4.1 (a) the lower and upper accuracy for the approximate solutions undergHdifferentiabilty of CFD when?=1 (b) the lower and upper accuracy for the approximate solutions undergH-differentiabilty of CFD when?=1, 0.9and0.8.It is worth mentioning that as the accuracies increase,the order of approximations increases.

Fig.4.2 displays the analysis of Problem 4.1 undergH-differentiabilty of CFD (a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameters?=. 7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Fig.4.2. (a) Solution profiles by Caputo fractional derivative of Problem 4.1 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=. 7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.3 illustrates the analysis of Problem 4.1 undergH-differentiabilty of ABC fractional derivative (a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameter?=. 7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Fig.4.3. Solution profiles by ABC fractional derivative of Problem 4.1 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=. 7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.4 defines the comparison analysis of Problem 4.1 undergH-differentiabilty of Caputo and ABC fractional derivatives (a) 3D simulation (b) 2D simulation.At the third term approximation,the values of the numerical results of multiple grid points generated by the GHPTM are comparable to the values of the exact solution with high accuracy,indicating a high level of conformity with the two findings of different fractional operators with excellent convergence between them.Furthermore,by implementing inferential statistical testing,the proposed method will aid scientists working on magneto-acoustic waves occurring in plasma and fullid mechanics in assessing competency.

Fig.4.4. Comparisons solution profiles by Caputo and ABC fractional derivative of Problem 4.1 demonstrates (a) three dimensional lower and upper case solutions,(b) two dimensional lower and upper case solutions,when ?=. 7, ?=1 and λ=0.5.

The parameterized formulation of (4.6) is presented as

Case I.Firstly,taking into consideration the CFD coupled with the generalized homotopy perturbation transform method on the first case of (4.3).In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

Equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows

Case 2.Now,we employ the fuzzy ABC derivative operator on the first first case of (4.3) as follows: In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

By the virtue of (4.10),we can calculate the following iterative terms by equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows:

In this analysis,we compared the lower and upper accuracies ofΨ(w1,λ;?)and the HPM solutions [72] in Table 1.We assumed the fractional order to be?=1 when?=. 01 with varying values ofw1andλ.Where,values written for HPM solution have been achieved by reference [72].In Table 2,we showed the absolute error between different values ofw1andλ1when?=1 and?∈[0,1] employing JHPTM for the non-homogeneous fifth order KdV equation,showing how the approximate solution is compared with the exact solution.

From Fig.4.5 represents the analysis of Problem 4.2 (a) the lower and upper accuracy for the approximate solutions undergHdifferentiabilty of CFD when?=1 (b) the lower and upper accuracy for the approximate solutions undergH-differentiabilty of CFD when?=1, 0.9and0.8.It is worth mentioning that as the accuracies increase,the order of approximations increases.

Fig.4.5. Solution profiles of Problem 4.2 demonstrate (a) Lower and upper case solutions for ?=1 (b) Lower and upper case solutions for ?=1, 0.9, and 0.8.

Fig.4.6 displays the analysis of Problem 4.2 undergH-differentiabilty of CFD (a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameters?=. 7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Table 4.2 The absolute error analysis among HPM [72],the absolute error of the lower and upper accuracies of JHPT MCFD and the lower and upper accuracies of JHPT MABC of Example 4.2 for approximated results of Ψ(w1, λ;?) at ?=1 with varying values of w1 and λ..

Fig.4.6. Solution profiles by Caputo fractional derivative of Problem 4.2 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=.7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.7 illustrates the analysis of Problem 4.2 undergH-differentiabilty of ABC fractional derivative (a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameter?=. 7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Fig.4.7. Solution profiles by ABC fractional derivative of Problem 4.2 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=.7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.8 defines the comparison analysis of Problem 4.2 undergH-differentiabilty of Caputo and ABC fractional derivatives (a) 3D simulation (b) 2D simulation.

Fig.4.8. Comparisons solution profiles by Caputo and ABC fractional derivative of Problem 4.1 demonstrates (a) three dimensional lower and upper case solutions,(b) two dimensional lower and upper case solutions,when ?=.7, ?=1 and λ=0.5.

At the third term approximation,the values of the numerical results of multiple grid points generated by the GHPTM are comparable to the values of the exact solution with high accuracy,indicating a high level of conformity with the two findings of different fractional operators with excellent convergence between them.Furthermore,by implementing inferential statistical testing,the proposed method will aid scientists working on magneto-acoustic waves occurring in plasma and fullid mechanics in assessing competency.

Remark 4.2.Letting, then the approximate solution of Problem 4.2 reduces toΨ(w1,λ)=exp (w1-λ).

Problem 4.3.Consider the fuzzy fractional dispersive KdV model:

subject to fuzzy ICs

The parameterized formulation of (4.11) is presented as

Case I.Firstly,taking into consideration the CFD coupled with the generalized homotopy perturbation transform method on the first case of (4.3).

In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

Equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows

implies that

Finally,we have

Case 2.Now,we employ the fuzzy ABC derivative operator on the first first case of (4.3) as follows:

In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

Equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows:

Table 4.3 The exact,the lower and upper accuracies via JHPT MCFD and the lower and upper accuracies via JHPT MABC solutions of Example 4.3 with varying values of w1 and λ in comparison with the solution profile of [73] obtained by Laplace Adomian decomposition method.

Table 4.4 The absolute error analysis among Laplace Adomina decomposition method [73],the absolute error of the lower and upper accuracies of JHPT MCFD and the lower and upper accuracies of JHPT MABC of Example 4.3 for approximated results of Ψ(w1, λ;?) at ?=1 with varying values of w1 and λ..

In this analysis,we compared the lower and upper accuracies ofΨ(w1,λ;?) and the LADM solutions [73] in Table 3.We assumed the fractional order to be?=1 when?=. 01 with varying values ofw1andλ.Where,values written for LADM solution have been achieved by reference [73].

In Table 4,we showed the absolute error between different values ofw1andλ1when?=1 and?∈[0,1] employing JHPTM for the non-homogeneous fifth order KdV equation,showing how the approximate solution is compared with the exact solution.

From Fig 4.9 represents the analysis of Problem 4.3 (a) the lower and upper accuracy for the approximate solutions undergH-differentiabilty of CFD when?=1 (b) the lower and upper accuracy for the approximate solutions undergH-differentiabilty of CFD when?=1, 0.9and0.8.It is worth mentioning that as the accuracies increase,the order of approximations increases.

Fig.4.10 displays the analysis of Problem 4.3 undergH-differentiabilty of CFD (a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameters?=. 7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Fig.4.10. (a) Solution profiles by Caputo fractional derivative of Problem 4.3 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=. 7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.11 illustrates the analysis of Problem 4.3 undergH-differentiabilty of ABC fractional derivative (a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameter?=. 7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Fig.4.11. Solution profiles by ABC fractional derivative of Problem 4.3 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=.7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.12 defines the comparison analysis of Problem 4.3 undergH-differentiabilty of Caputo and ABC fractional derivatives (a) 3D simulation (b) 2D simulation.

Fig.4.12. Comparisons solution profiles by Caputo and ABC fractional derivative of Problem 4.1 demonstrates (a) three dimensional lower and upper case solutions,(b) two dimensional lower and upper case solutions,when ?=.7, ?=1 and λ=0.5.

At the third term approximation,the values of the numerical results of multiple grid points generated by the GHPTM are comparable to the values of the exact solution with high accuracy,indicating a high level of conformity with the two findings of different fractional operators with excellent convergence between them.Furthermore,by implementing inferential statistical testing,the proposed method will aid scientists working on magneto-acoustic waves occurring in plasma and fullid mechanics in assessing competency.

Remark 4.3.Letting, then the approximate solution of Problem 4.3 reduces toΨ(w1,λ)=sin(w1-λ).

Problem 4.4.Consider the fuzzy fractional dispersive KdV model:

The parameterized formulation of (4.14) is presented as

Case I.Firstly,taking into consideration the CFD coupled with the generalized homotopy perturbation transform method on the first case of (4.3).

In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

Equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows

Case 2.Now,we employ the fuzzy ABC derivative operator on the first first case of (4.3) as follows:

In view of the process stated in Section 3,we have

In view of fuzzy IC and making use of the inverse generalized integral transform implies

Now implementing the HPM,we have

Equating the coefficients of the same powers ofη, we have

The series form solution is presented as follows:

From Fig.4.13 represents the analysis of Problem 4.4 (a) the lower and upper accuracy for the approximate solutions undergHdifferentiabilty of CFD when?=1 (b) the lower and upper accuracy for the approximate solutions undergH-differentiabilty of CFD when?=1, 0.9and0.8.It is worth mentioning that as the accuracies increase,the order of approximations increases.Fig.4.14 displays the analysis of Problem 4.4 undergH-differentiabilty of CFD(a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameters?=.7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Fig.4.13. Solution profiles of Problem 4.1 demonstrate (a) Lower and upper case solutions for ?=1 (b) Lower and upper case solutions for ?=1, 0.9, and 0.8.

Fig.4.14. Solution profiles by Caputo fractional derivative of Problem 4.4 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=.7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.15 illustrates the analysis of Problem 4.4 undergHdifferentiabilty of ABC fractional derivative (a) the two dimensional approximate solution with various fractional orders but fixed uncertain parameter?=.7 (b) the two dimensional approximate solution with various uncertain parameters but fixed fractional order?=0.7.

Fig.4.15. Solution profiles by ABC fractional derivative of Problem 4.4 demonstrate (a) Lower and upper case solutions for different fractional orders with uncertainty parameter ?=.7 and λ=0.5.(b) Lower and upper case solutions for various uncertainties with ?=0.7 and λ=0.5.

Fig.4.16 defines the comparison analysis of Problem 4.4 undergH-differentiabilty of Caputo and ABC fractional derivatives (a) 3D simulation (b) 2D simulation.

At the third term approximation,the values of the numerical results of multiple grid points generated by the GHPTM are comparable to the values of the exact solution with high accuracy,indicating a high level of conformity with the two findings of different fractional operators with excellent convergence between them.Furthermore,by implementing inferential statistical testing,the proposed method will aid scientists working on magneto-acoustic waves occurring in plasma and fullid mechanics in assessing competency.

Remark 4.4.Letting, then the approximate solution of Problem 4.4 reduces toΨ(w1,λ)=cos (w1+w2+2λ).

5.Conclusion

We have successfully combined the generalized integral transform with a HPTM in a limited setting for the approximation of nonlinear dispersive and fifth order KdV models with the CFD and ABC fractional derivatives undergHdifferentiability in this investigation.The generalized integral transform transform simplified the problem to a time independent problem,alleviating the necessity for the traditional time stepping approach.For various fractional orders and uncertainty?∈[0,1],numerical tests were conducted for KdV equations with a CFD and ABC fractional derivative.Furthermore,2D and 3D simulations have been carried out and comparison graphs are presented that demonstrate the effectiveness and supremacy of the proposed method.From the findings,we can deduce that GHPTM is reliable in approximating the phenomenon of atmospheric internal waves.The proposed findings provide a thorough and pragmatic look into the behaviour of these waves,which can be leveraged to refine weather and climate models,as the waves have a significant influence on climate change as well as beachfront ocean and ocean engineering,as has been observed in recent years.

Declarations

Availability of supporting data Not applicable.Competing interests The authors declare that they have no competing interests.FundingNot ApplicableAuthor’s contributionsAll authors contributed equally to the writing of this paper.All authors read and approved the nal manuscript.

Declaration of Competing Interest

The authors declare no conflict of interest.