# An Efficient Technique for Time-Fractional Water Dynamics Arising in Physical Systems Pertaining to Generalized Fractional Derivative Operators

Authors: Saima Rashid; Bushra Kanwal; Muhammad Attique; Ebenezer Bonyah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study proposed theq˜-homotopy analysis transform method (q˜-HATM) as a revolutionary mathematical method for addressing nonlinear time-fractional Boussinesq and approximate long wave dynamics models with the Caputo and Atangana–Baleanu fractional derivatives in the Caputo sense. Through a specific velocity distribution, these models play an essential role in explaining the physics of wave propagation. The q˜-HATM is a new improvement to the Elzaki transform (ET) that simplifies the computations. The presented scheme addresses computational complexity by avoiding the use of Adomian and He’s polynomials, which is a distinguishing feature of this innovative methodology over decomposition and the homotopy perturbation transform method. The convergence analysis and error analyses are carried out in the current investigation for the upcoming strategy. We provide illustrations to exemplify the suggested system’s strength and efficacy, and the error estimates are described to ensure reliability. The analytical and graphic illustrations show that the projected methodology is numerically very precise and pragmatic to analyze the solution of fractional associated dynamics that arise in physics and engineering.


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## Body

## 1. Introduction

In 1965, L’Hopital was the first one to propose fractional calculus (FC). FC has gained a significant amount of dedication and acclaim over the last few centuries. It is attributed to its potential to include an accurate representation of chaotic, dynamic events. In the development of applications and analysis of physical phenomena, fractional-order nonlinear evolution frameworks have recently acquired interest. Fc is a refinement of classical calculus that has dynamical and reproductive implications in physical strength.Numerous studies have recognized the essential characteristics of fractional calculus [1–5]. FC is used to analyze and solve issues in a variety of disciplines, including hydrodynamics, continuum biomechanics, quantum physics, bioengineering, electrochemistry, and several related domains. The result is an indeterminate order difference representing the foregoing occurrences. This is crucial in identifying the performance of complicated issues that arise in essence [6–15].Whitham [16], Broer [17], and Kaup [18] introduced the Whitham–Broer–Kaup (WBK) model in the 1920s, which states the transmission of superficial water wave forms has varying dispersal characteristics. Taking into account the time-fractional coupled WBK models [19],(1)Dt1δΘ+ΘΘx+Υx+cΘxx=0,0<δ≤1,Dt1δΥ+ΘΥx+ΥΘx+cΘxxx−dΥxx=0,where Υ=Υx,t1 is the altitude that differs significantly from the convective steady state, and Θ=Θx,t1 is the transverse speed. Furthermore, c and d are constants that represent different diffusion strengths; thus, (1) generates a refined Boussinesq equation if c=1 and c=1. Accordingly, the scheme indicates the traditional long wave equation for c=0 and d=1. These formulas are employed in marine and coastline science to demonstrate the transmission of shocks in viscous dissipation and dynamic substrates, and they are intended for contexts that necessitate fluid infiltration in permeable subterranean strata. Furthermore, (1) provides the basis for a number of formulations that depict the unrestrained subterranean, such as discharge and underground water incompressible flows.The last several decades have witnessed the invention of numerous innovative strategies for solving complex fractional DEs, as well as revolutionary processing techniques using representational scripting. In this regard, Liao presented the homotopy analysis approach (HAM), which is premised on generating a homotopy that constantly reshapes a random initial estimate into the effective response of the specified subject [20]. It does not need to be discretized, linearized, or perturbed. To handle difficult and complicated challenges in related scientific and technological fields, meanwhile, massive computing and additional computing capacity are required. To alleviate these constraints, an agglomeration of transformation strategies is required.Theq˜-HATM was used to discover an approximation-analytical expression for coupled MB and ALW models of fractional order in this study. Numerous researchers have identified and analyzed these formulas using procedures such as the Adomian decomposition method (ADM) [21], the variational iteration method (VIM) [22], the coupled fractional reduced differential transform method (CFRDTM) [19], the Laplace Adomian decomposition method (LADM) [23], and other tools [24–26]. The researchers of such reference lists used a complicated solution approach to locate the problem, whereas the solution process of the prospective method is quite simple and straightforward. Furthermore, to handle coupled DEs, these approaches need a large amount of processing and a lengthy treatment process. The described procedure is an improved methodology that combines the q˜-HAM and the Elzaki transform [27] in an impressive way. As a result, it does not necessitate convolutions, interpolation, or modification; it also reduces large mathematical processes, requires less computing capacity, and does not involve getting complicated integrals, multipliers, or physical dimensions. The prospective approach includes a number of strong characteristics, notably a simple treatment process, a broad convergence zone, and the absence of any assumptions, deconvolution, or modification. With only a few repetitions, the current approach produces extremely precise parameter estimates. This methodology has the privilege of being able to combine two effective methodologies for achieving exact-analytical solutions for complicated DEs. It is interesting to note that using semianalytical approaches to analyze solutions for complex and complicated structures and phenomena found in engineering and innovation takes less C.P.U. time. The ℏ and n supplementary variables in the q˜-HATM method allow us to modify and regulate the solution’s convergence, which rapidly approaches the exact model within appropriate limits. It is noteworthy to reiterate that, when contrasted to other conventional methodologies, the suggested trajectory can reduce computing tasks and effort while maintaining better precision.Numerous scholars have extensively employedq˜-HATM to investigate different sorts of complex problems because of its reliability and effectiveness. For instance, in [28], the research evaluated and discovered the estimated approximate expression for a fractional model of the vibrational problem, demonstrating the precision of q˜-HATM. Al-Qurashi et al. [29] used the q˜-HATM method to solve time-fractional PDEs in shallow water wave propagation based on power and Mittag-Leffler memory. Researchers in [30] examined the framework of Lienard’s equation. Several individuals have investigated and solved many difficult challenges resulting in connected scientific fields [31–34]. In addition to the conventional time derivative, the time-fractional derivative exhibits advantageous features, which can be investigated using the described methodology. Furthermore, the aforementioned research review demonstrates that the prospective methodology is relatively rigorous and efficient and that it can be used to explore complex nonlinear structures of any level, with the implementation of fractional-order differential yielding very intriguing and appropriate information. As a result of our experiments, we developed the estimated analytical expression for paired revolutionary DEs modelling shallow groundwater disturbances via separate velocity distributions. Furthermore, we attempted to recreate the performance of the paired interface for the system concentrations, and we investigated the behaviour of the q˜-HATM solution for different fractional-order levels.


## 2. Preliminaries

In this segment, we present basic definitions and notions which will be used in the present framework.Definition 1 (see [1]).
Suppose there be a Caputo fractional derivative operator, defined as(2)Dζδ0cvζ=1Γr−δ∫0ζvrw1ζ−w1δ+1−rdw1,r−1<δ<r,drdζrvζ,δ=r.Definition 2 (see [3]).
Suppose there be an Atangana–Baleanu fractional derivative operator, described as(3)Dζδη1ABCvζ=Mδ1−δ∫η1ζv′ζEδ−δζ−w1δ1−δdw1,where v∈Δ1a1,a2Sobolev space,a1<a2,δ∈0,1 and Mδ indicate normalization function as Mδ=M0=M1=1.Definition 3 (see [3]).
Suppose there be a Atangana-Baleanu fractional integral operator, defined as(4)I¯ζδη1ABCvζ=1−δMδvζ+δΓδMδ∫η1ζvw1ζ−w1δ−1dw1.Definition 4 (see [27]).
Suppose there be a setP having an exponential function such that z is a finite number and j1,j2 may be finite or infinite, then(5)P=vζ:∃z,j1,j2>0,vζ<zexpζjι,ifζ∈−1j×0,∞.Definition 5 (see [27]).
Suppose there be an ET for a functionvζ¯, defined as follows:(6)Evζ¯ϱ=Vϱ=ϱ∫0∞expζ¯ϱvζ¯dζ¯,ζ¯≥0,ϱ∈j1,j2.Definition 6 (see [35]).
Suppose there be an ET of a Caputo fractional derivative, defined as follows:(7)EDζδ0cvζ¯ϱ=ϱ−δVϱ−∑j=0n−1vj0ϱ2−δ+j,n−1<δ<n.Definition 7 (see [36]).
Suppose there be an ET of the Atangana–Baleanu fractional derivative operator, defined as follows:(8)EDζ¯δ0ABCvζ¯ϱ=ϱNδϱδδ+1−δVϱϱ−ϱv0,where Evζ¯=Vϱ.


## 3. Configuration ofq˜-Homotopy Analysis Transform Method

The suggested approach’s technique is illustrated by utilizing the operator defined in (7) and (8), respectively, for the fractional-order framework (1)). We propose a fractional-order system dynamics of the following form.


### 3.1. Type I (Caputo Fractional Derivative Operator)

(9)Dζ¯δa1cΘx,ζ¯+RΘx,ζ¯+NΘx,ζ¯=ℱx,ζ¯,n−1<δ≤n,underlying initial conditions (ICs)(10)Θx,0=Θ0x,where Dζδ0c presents the Caputo derivative of Θx,ζ¯ while RΘx,ζ¯ is the linear and NΘx,ζ¯ denotes the generic nonlinear operator. Also, ℱx,ζ¯ is continuous mapping.On implementing the differentiation formula of the ET on (9), we find(11)ϱ−δEΘx,ζ¯−∑j=0n−1Θj0ϱ2−δ+j+ERΘx,ζ¯+NΘx,ζ¯=Eℱx,ζ¯.Following the generalization, we have(12)EΘx,ζ¯−ϱδ∑j=0n−1Θj0ϱ2−δ+j+ERΘx,ζ¯+NΘx,ζ¯=ϱδEℱx,ζ¯.The nonlinear operator is defined as follows:(13)Nψx,ζ¯;q˜=Εψx,ζ¯;q˜−ϱδ∑j=0n−1Θj0ϱ2−δ+j+ϱδΕRψx,ζ¯;q˜+ϱδΕNψx,ζ¯;q˜−ϱδΕℱx,ζ¯,where q˜∈0,1/n. Then, using the proven methodology, we illustrate the nonzero auxiliary and embedding component ℏ, and q˜, respectively.(14)1−nq˜Eψx,ζ¯;q˜−Θ0x,ζ¯=ℏq˜ℌx,ζ¯Nψx,ζ¯;q˜,where E indicates ET, ℌx,ζ¯ signifies the nonzero auxiliary function, and ψx,ζ¯;q˜ is an unidentified function. Moreover, Θ0x,ζ¯ is the IC of Θx,ζ¯.Also, forq˜=0 and q˜=1/n, then, the following hypotheses are valid:(15)ψx,ζ¯;q˜=Θ0x,ζ¯,ψx,ζ¯;1n=Θx,ζ¯.As a result, increasingq˜ from 0 to 1/n, the solution ψx,ζ¯;q˜ varies from Θ0x,ζ¯ to the solution Θx,ζ¯. Expand the function ψx,ζ¯;q˜ with the assistance of Taylor’s series, one can obtain near to q˜ as(16)ψx,ζ¯;q˜=Θ0x,ζ¯+∑r=1∞Θrx,ζ¯q˜r,where(17)Θ0x,ζ¯=1r!∂rψx,ζ¯;q˜∂q˜r|q˜=0.Taking into account the auxiliary linear operator,Θ0x,ζ¯,n,ℏ, then the (17) tends at q˜=1/n. Therefore, it gives a solution of (11), of the form(18)Θx,ζ¯=Θ0x,ζ¯+∑r=0∞Θrx,ζ¯1nr.Now, differentiatingr-times (15), surmising q˜ and dividing by r!, plugging subsequently q˜=0, we have(19)ΕΘrx,ζ¯−KrΘr−1x,ζ¯=ℏℛrΘ⟶r−1,here, we state the vector as(20)Θ⟶r=Θ0x,ζ¯+Θ1x,ζ¯+⋯+Θrx,ζ¯.In view of the inverse ET on (19), produces(21)Θrx,ζ¯=KrΘr−1x,ζ¯+ℏΕ−1ℛrΘ⟶r−1,where(22)ℛrΘ⟶r−1=ΕΘr−1x,ζ¯−1−Krn∑j=0n−1Θj0ϱ2−δ+j+ϱδΕℱx,ζ¯+ϱδΕℛΘr−1+ℌr−1,and(23)js=0,s≤1,1,s>1.In (22), ℌs is the homotopy polynomial, which is defined as(24)ℌs=Θ0x,ζ¯=1s!∂sψx,ζ¯;q˜∂q˜s|q˜=0,and(25)ψx,ζ¯;q˜=ψ0+q˜ψ1+q2ψ2+⋯.Amalgamating (22) and (24), ones obtain(26)Θsx,ζ¯=js+ℏΘs−1x,ζ¯−1−jsnΕ−1∑j=0n−1Θj0ϱ2−δ+j+ϱδΕℱx,ζ¯+ℏΕ−1ϱδΕℛΘs−1+ℌs−1.The series solution byq˜-HATM can be written as(27)Θx,ζ¯=∑s=0∞Θsx,ζ¯.


### 3.2. Case II (ABC Fractional Derivative Operator)

(28)Dζ¯δa1ABCΘx,ζ¯+RΘx,ζ¯+NΘx,ζ¯=ℱx,ζ¯,n−1<δ≤n,underlying ICs(29)Θx,0=Θ0x,where Dζ¯δa1ABC denotes the Atangana–Baleanu fractional derivative in the Caputo context of Θx,ζ¯.On implementing the differentiation formula of the ET on (27), we find(30)Mδϱδδ+1−δΕΘx,ζ¯−ϱ2Mδϱδδ+1−δΘ0+ΕRΘx,ζ¯+NΘx,ζ¯=Εℱx,ζ¯.Following the generalization, gives(31)EΘx,ζ¯−ϱ2Θ0+ϱδδ+1−δMδERΘx,ζ¯+NΘx,ζ¯=ϱδδ+1−δMδEℱx,ζ¯.The nonlinear operator is defined as(32)Nψx,ζ¯;q˜=Εψx,ζ¯;q˜−ϱ2Θ0+ϱδδ+1−δMδΕRψx,ζ¯;q˜+ϱδδ+1−δMδΕNψx,ζ¯;q˜−ϱδδ+1−δMδΕℱx,ζ¯,where q˜∈0,1/n. Then, using the proven methodology, we illustrate the nonzero auxiliary and embedding component ℏ, and q˜, respectively.(33)1−nq˜Εψx,ζ¯;q˜−Θ0x,ζ¯=ℏq˜ℌx,ζ¯Nψx,ζ¯;q˜,where E indicates the ET, ℌx,ζ¯ signifies the nonzero auxiliary function, and ψx,ζ¯;q˜ is an unidentified function. Moreover, Θ0x,ζ¯ is the IC of Θx,ζ¯.Also, forq˜=0 and q˜=1/n, then the following hypothesis is valid:(34)ψx,ζ¯;0=Θ0x,ζ¯,ψx,ζ¯;1n=Θx,ζ¯.As a result, increasingq˜ from 0 to 1/n, the solution ψx,ζ¯;q˜ varies from Θ0x,ζ¯ to the result Θx,ζ¯. Expand the function ψx,ζ¯;q˜ with the assistance of the Taylor theorem, one can obtain near to q˜ as(35)ψx,ζ¯;q˜=Θ0x,ζ¯+∑s=1∞Θsx,ζ¯q˜s,where(36)Θ0x,ζ¯=1s!∂sψx,ζ¯;q˜∂q˜s|q˜=0.Taking into account the auxiliary linear operator,Θ0x,ζ¯,n,ℏ, then the (36) tends at q˜=1/n; therefore, it gives a solution of (32), of the form(37)Θx,ζ¯=Θ0x,ζ¯+∑s=0∞Θsx,ζ¯1ns.Now, differentiatings-times (34), surmising q˜ and dividing by s!, plugging subsequently q˜=0, we have(38)ΕΘsx,ζ¯−KsΘs−1x,ζ¯=ℏℛsΘ⟶s−1.Now, we state the vector as(39)Θ⟶s=Θ0x,ζ¯+Θ1x,ζ¯+⋯+Θsx,ζ¯.In view of the inverse ET on (37), produces(40)Θsx,ζ¯=KsΘs−1x,ζ¯+ℏΕ−1ℛsΘ⟶s−1,where(41)ℛsΘ⟶s−1=ΕΘs−1x,ζ¯−1−Ksnϱ2Θ0+ϱδδ+1−δMδΕℱx,ζ¯+ϱδδ+1−δMδΕℛΘs−1+ℌs−1,and(42)js=0,s≤1,1,s>1.In (22), ℌs is the homotopy polynomial, defined as(43)ℌs=Θ0x,ζ¯=1s!∂sψx,ζ¯;q˜∂q˜s|q˜=0,and(44)ψx,ζ¯;q˜=ψ0+q˜ψ1+q2ψ2+⋯.Amalgamating (41) and (43), we have(45)Θsx,ζ¯=js+ℏΘs−1x,ζ¯−1−jsnΕ−1ϱ2Θ0+ϱδδ+1−δMδΕℱx,ζ¯+ℏΕ−1ϱδδ+1−δMδΕℛΘs−1+ℌs−1.The series solution byq˜-HATM is written as(46)Θx,ζ¯=∑s=0∞Θsx,ζ¯.


## 3.1. Type I (Caputo Fractional Derivative Operator)

(9)Dζ¯δa1cΘx,ζ¯+RΘx,ζ¯+NΘx,ζ¯=ℱx,ζ¯,n−1<δ≤n,underlying initial conditions (ICs)(10)Θx,0=Θ0x,where Dζδ0c presents the Caputo derivative of Θx,ζ¯ while RΘx,ζ¯ is the linear and NΘx,ζ¯ denotes the generic nonlinear operator. Also, ℱx,ζ¯ is continuous mapping.On implementing the differentiation formula of the ET on (9), we find(11)ϱ−δEΘx,ζ¯−∑j=0n−1Θj0ϱ2−δ+j+ERΘx,ζ¯+NΘx,ζ¯=Eℱx,ζ¯.Following the generalization, we have(12)EΘx,ζ¯−ϱδ∑j=0n−1Θj0ϱ2−δ+j+ERΘx,ζ¯+NΘx,ζ¯=ϱδEℱx,ζ¯.The nonlinear operator is defined as follows:(13)Nψx,ζ¯;q˜=Εψx,ζ¯;q˜−ϱδ∑j=0n−1Θj0ϱ2−δ+j+ϱδΕRψx,ζ¯;q˜+ϱδΕNψx,ζ¯;q˜−ϱδΕℱx,ζ¯,where q˜∈0,1/n. Then, using the proven methodology, we illustrate the nonzero auxiliary and embedding component ℏ, and q˜, respectively.(14)1−nq˜Eψx,ζ¯;q˜−Θ0x,ζ¯=ℏq˜ℌx,ζ¯Nψx,ζ¯;q˜,where E indicates ET, ℌx,ζ¯ signifies the nonzero auxiliary function, and ψx,ζ¯;q˜ is an unidentified function. Moreover, Θ0x,ζ¯ is the IC of Θx,ζ¯.Also, forq˜=0 and q˜=1/n, then, the following hypotheses are valid:(15)ψx,ζ¯;q˜=Θ0x,ζ¯,ψx,ζ¯;1n=Θx,ζ¯.As a result, increasingq˜ from 0 to 1/n, the solution ψx,ζ¯;q˜ varies from Θ0x,ζ¯ to the solution Θx,ζ¯. Expand the function ψx,ζ¯;q˜ with the assistance of Taylor’s series, one can obtain near to q˜ as(16)ψx,ζ¯;q˜=Θ0x,ζ¯+∑r=1∞Θrx,ζ¯q˜r,where(17)Θ0x,ζ¯=1r!∂rψx,ζ¯;q˜∂q˜r|q˜=0.Taking into account the auxiliary linear operator,Θ0x,ζ¯,n,ℏ, then the (17) tends at q˜=1/n. Therefore, it gives a solution of (11), of the form(18)Θx,ζ¯=Θ0x,ζ¯+∑r=0∞Θrx,ζ¯1nr.Now, differentiatingr-times (15), surmising q˜ and dividing by r!, plugging subsequently q˜=0, we have(19)ΕΘrx,ζ¯−KrΘr−1x,ζ¯=ℏℛrΘ⟶r−1,here, we state the vector as(20)Θ⟶r=Θ0x,ζ¯+Θ1x,ζ¯+⋯+Θrx,ζ¯.In view of the inverse ET on (19), produces(21)Θrx,ζ¯=KrΘr−1x,ζ¯+ℏΕ−1ℛrΘ⟶r−1,where(22)ℛrΘ⟶r−1=ΕΘr−1x,ζ¯−1−Krn∑j=0n−1Θj0ϱ2−δ+j+ϱδΕℱx,ζ¯+ϱδΕℛΘr−1+ℌr−1,and(23)js=0,s≤1,1,s>1.In (22), ℌs is the homotopy polynomial, which is defined as(24)ℌs=Θ0x,ζ¯=1s!∂sψx,ζ¯;q˜∂q˜s|q˜=0,and(25)ψx,ζ¯;q˜=ψ0+q˜ψ1+q2ψ2+⋯.Amalgamating (22) and (24), ones obtain(26)Θsx,ζ¯=js+ℏΘs−1x,ζ¯−1−jsnΕ−1∑j=0n−1Θj0ϱ2−δ+j+ϱδΕℱx,ζ¯+ℏΕ−1ϱδΕℛΘs−1+ℌs−1.The series solution byq˜-HATM can be written as(27)Θx,ζ¯=∑s=0∞Θsx,ζ¯.


## 3.2. Case II (ABC Fractional Derivative Operator)

(28)Dζ¯δa1ABCΘx,ζ¯+RΘx,ζ¯+NΘx,ζ¯=ℱx,ζ¯,n−1<δ≤n,underlying ICs(29)Θx,0=Θ0x,where Dζ¯δa1ABC denotes the Atangana–Baleanu fractional derivative in the Caputo context of Θx,ζ¯.On implementing the differentiation formula of the ET on (27), we find(30)Mδϱδδ+1−δΕΘx,ζ¯−ϱ2Mδϱδδ+1−δΘ0+ΕRΘx,ζ¯+NΘx,ζ¯=Εℱx,ζ¯.Following the generalization, gives(31)EΘx,ζ¯−ϱ2Θ0+ϱδδ+1−δMδERΘx,ζ¯+NΘx,ζ¯=ϱδδ+1−δMδEℱx,ζ¯.The nonlinear operator is defined as(32)Nψx,ζ¯;q˜=Εψx,ζ¯;q˜−ϱ2Θ0+ϱδδ+1−δMδΕRψx,ζ¯;q˜+ϱδδ+1−δMδΕNψx,ζ¯;q˜−ϱδδ+1−δMδΕℱx,ζ¯,where q˜∈0,1/n. Then, using the proven methodology, we illustrate the nonzero auxiliary and embedding component ℏ, and q˜, respectively.(33)1−nq˜Εψx,ζ¯;q˜−Θ0x,ζ¯=ℏq˜ℌx,ζ¯Nψx,ζ¯;q˜,where E indicates the ET, ℌx,ζ¯ signifies the nonzero auxiliary function, and ψx,ζ¯;q˜ is an unidentified function. Moreover, Θ0x,ζ¯ is the IC of Θx,ζ¯.Also, forq˜=0 and q˜=1/n, then the following hypothesis is valid:(34)ψx,ζ¯;0=Θ0x,ζ¯,ψx,ζ¯;1n=Θx,ζ¯.As a result, increasingq˜ from 0 to 1/n, the solution ψx,ζ¯;q˜ varies from Θ0x,ζ¯ to the result Θx,ζ¯. Expand the function ψx,ζ¯;q˜ with the assistance of the Taylor theorem, one can obtain near to q˜ as(35)ψx,ζ¯;q˜=Θ0x,ζ¯+∑s=1∞Θsx,ζ¯q˜s,where(36)Θ0x,ζ¯=1s!∂sψx,ζ¯;q˜∂q˜s|q˜=0.Taking into account the auxiliary linear operator,Θ0x,ζ¯,n,ℏ, then the (36) tends at q˜=1/n; therefore, it gives a solution of (32), of the form(37)Θx,ζ¯=Θ0x,ζ¯+∑s=0∞Θsx,ζ¯1ns.Now, differentiatings-times (34), surmising q˜ and dividing by s!, plugging subsequently q˜=0, we have(38)ΕΘsx,ζ¯−KsΘs−1x,ζ¯=ℏℛsΘ⟶s−1.Now, we state the vector as(39)Θ⟶s=Θ0x,ζ¯+Θ1x,ζ¯+⋯+Θsx,ζ¯.In view of the inverse ET on (37), produces(40)Θsx,ζ¯=KsΘs−1x,ζ¯+ℏΕ−1ℛsΘ⟶s−1,where(41)ℛsΘ⟶s−1=ΕΘs−1x,ζ¯−1−Ksnϱ2Θ0+ϱδδ+1−δMδΕℱx,ζ¯+ϱδδ+1−δMδΕℛΘs−1+ℌs−1,and(42)js=0,s≤1,1,s>1.In (22), ℌs is the homotopy polynomial, defined as(43)ℌs=Θ0x,ζ¯=1s!∂sψx,ζ¯;q˜∂q˜s|q˜=0,and(44)ψx,ζ¯;q˜=ψ0+q˜ψ1+q2ψ2+⋯.Amalgamating (41) and (43), we have(45)Θsx,ζ¯=js+ℏΘs−1x,ζ¯−1−jsnΕ−1ϱ2Θ0+ϱδδ+1−δMδΕℱx,ζ¯+ℏΕ−1ϱδδ+1−δMδΕℛΘs−1+ℌs−1.The series solution byq˜-HATM is written as(46)Θx,ζ¯=∑s=0∞Θsx,ζ¯.


## 4. Convergence Outcomes ofq˜-HATM Solution

In what follows, we illustrate the uniqueness theorem for the general PDE defined in (9).Theorem 1.
Suppose there is a unique solution of nonlinear PDE (9) derived by q˜-HATM for every α∈0,1, where α=n+ℏ+ℏϵ+μT.Proof.
By means of (9), the solution is described as(47)Θx,ζ¯=∑s=0∞Θsx,ζ¯,where(48)Θsx,ζ¯=js+ℏΘs−1x,ζ¯−1−jsnΕ−1ϱ2Θ0+ϱδδ+1−δMδΕℱx,ζ¯+ℏΕ−1ϱδδ+1−δMδΕℛΘs−1+ℌs−1.
Surmise thatΘ and Θ∗ are two solutions of (9), then it is necessary to demonstrate that Θ=Θ∗. To illustrate the hypothesis, by means of (45), we find(49)Θ−Θ∗=n+ℏΘ−Θ∗+ℏE−1ϱδδ+1−δMδENΘ−Θ∗+RΘ−Θ∗.
On implementing the convolution theorem for ET, we have(50)Θ−Θ∗≤n+ℏΘ−Θ∗+ℏ∫0t1NΘ−Θ∗+RΘ−Θ∗δt1−ξδΓδ+1+1−δdξ≤n+ℏΘ−Θ∗+ℏ∫0t1∫∈Θ−Θ∗+μΘ−Θ∗δt1−ξδΓδ+1+1−δdξ.
Utilizing the fact of integral mean value theorem, the aforesaid expression can be written as(51)Θ−Θ∗≤n+ℏΘ−Θ∗+ℏ∫Θ−Θ∗+μΘ−Θ∗T.
Lettingα=n+ℏ+ℏ∈+μT, therefore, we have(52)Θ−Θ∗≤αΘ−Θ∗⇒1−αΘ−Θ∗≤0.
Since0<α<1, then Θ−Θ∗=0⇒Θ=Θ∗.
Thus, (9) has a unique solution.Our next result is the convergence theorem for (9).Theorem 2.
Suppose there be a Banach spaceY and a nonlinear functional f1:Y↦Y. Surmise that(53)f1w−f1v≤αw−v,∀c,d∈Y.
Then,f1 has a fixed point (see [37]). Also, for arbitrarily chosen c0,d0∈Y, the q˜-HATM sequence ultimately converges on a fixed point of f1 and(54)wm−wn≤αn1−αw1−w0,∀c,d∈Y.Proof.
Let us assume a Banach spaceℂℐ,‖.‖ on ℐ having the norm Gη=maxη∈ℐGη for all continuous functions. First, we illustrate that wm in Y is a Cauchy sequence. Considering(55)wm−wn=maxη∈ℐwm−wn=maxη∈ℐn+ℏwm−1−wn−1+ℏΕ−1ϱδδ+1−δMδΕNwm−1−wn−1+Rwm−1−wn−1≤maxη∈ℐn+ℏwm−1−wn−1+ℏΕ−1ϱδδ+1−δMδΕNwm−1−wn−1+Rwm−1−wn−1.
On implementing the convolution theorem for ET, we have(56)wm−wn≤maxη∈ℐn+ℏwm−1−wn−1+ℏ∫0t1Nwm−1−wn−1+Rwm−1−wn−1δt1−ξδΓδ+1+1−δdξ.
Utilizing the fact of the integral mean value theorem, the aforesaid expression can be written as(57)wm−wn≤n+ℏwm−1−wn−1+ℏ∈wm−1−wn−1+μwm−1−wn−1T≤αwm−1−wn−1.
Takingm=n+1, one has expression can be written as(58)wn+1−wn≤αwn−wn−1≤α2wn−1−wn−2≤α3wn−2−wn−3≤⋯≤αnw1−w0.
By employing the triangular inequality, we find(59)wm−wn≤wn+1−wn+wn+2−wn+2+⋯+wm−wm+1≤αn+αn+1+⋯+αm+1w1−w0≤αn1−αm−n−11−αw1−w0.
Note that1−αm−n−1<1As0<α<1. As a result, the aforementioned inequality yields(60)wm−wn≤αn1−αw1−w0.
However,w1−w0<∞, ultimately, as m↦∞, then wm−wn↦0, which shows that wn is a Cauchy sequence in ℂℐ, and every Cauchy sequence is convergent. So, wn is a convergent sequence.


## 5. Experimental Examples

Example 1.
([19, 21, 26]). Assuming the fractional-order modified Boussinesq model is given as(61)Dζ¯δΘx,ζ¯+Θx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x=0,0<δ≤1,Dζ¯δΥx,ζ¯+Θx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3=0,subject to ICs(62)Θx,0=ϖ−2σcothσx+λ,Υx,0=−2σ2csch2σx+λ.Proof.
First and foremost, we find the solution of (61) in two different ways.Case I: Here, we commence in view of the Caputo fractional derivative operator, which we comprise alongside the ET andq¯-HATM.By implementing the ET on (60), one can obtain(63)ΕDζ¯δΘx,ζ¯=−ΕΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x,ΕDζ¯δΥx,ζ¯=−ΕΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3.Consequently, we have(64)ϱ−δΕΘx,ζ¯−∑j=0n−1ϱ2−δ+jΘj0=−ΕΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x,ϱ−δΕΥx,ζ¯−∑j=0n−1ϱ2−δ+jΥj0=−ΕΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3.As a result, we have(65)ΕΘx,ζ¯=ϱ2Θx,0−ϱδΕΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x,ΕΥx,ζ¯=ϱ2Υx,0−ϱδΕΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3.The nonlinearity can be stated using (64) as follows:(66)N1ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Εψ1x,ζ¯;q˜−ϱ2ψ1x,0+ϱδΕΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x,N2ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Εψ2x,ζ¯;q˜−ϱ2ψ2x,0+ϱδΕΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3.Using the technique outlined in Section3, the s th order deformation equation is summarized as(67)ΕΘsx,ζ¯−KsΘs−1x,ζ¯=ℏℛ1,sΘ⟶s−1,Υ⟶s−1,where(68)ℛ1,sΘ⟶s−1,Υ⟶s−1=ΕΘs−1x,ζ¯−1−jsnϱ2ϖ−2σcothσx+λ+ϱδ∑j=0s−1Θjx,ζ¯∂Θs−1−j∂x+∂Υs−1−j∂x,ℛ2,sΘ⟶s−1,Υ⟶s−1=ΕΘs−1x,ζ¯−1−jsnϱ2−2σ2csch2σx+λ+ϱδ∑j=0s−1Θjx,ζ¯∂Υs−1−j∂x+∑j=0s−1Υjx,ζ¯∂Θs−j−1∂x+∂3Θs−1−j∂x3.Making the use of inverse ET, we acquire(69)Θsx,ζ¯=KsΘs−1x,ζ¯+ℏΕ−1ℛ1,sΘ⟶s−1,Υ⟶s−1,Υsx,ζ¯=KsΥs−1x,ζ¯+ℏΕ−1ℛ2,sΘ⟶s−1,Υ⟶s−1.Utilizing the ICS, then we have(70)Θ0x,ζ¯=ϖ−2σcothσx+λ,Υ0x,ζ¯=−2σ2csch2σx+λ.Inserting theΘ0x,ζ¯ and Υ0x,ζ¯, letting s=1 in (67), then one can find(71)Θ1x,ζ¯=K1Θ0x,ζ¯+ℏΕ−1ℛ1,1Θ⟶0,Υ⟶0,Υ1x,ζ¯=K1Υ0x,ζ¯+ℏΕ−1ℛ2,1Θ⟶0,Υ⟶0.Whens=1, then (68) reduces to(72)ℛ1,1Θ⟶0,Υ⟶0=ΕΘ0x,ζ¯−1−j1nϱ2ϖ−2σcothσx+λ+ϱδΕΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x,ℛ2,1Θ⟶0,Υ⟶0=ΕΥ0x,ζ¯−1−j1nϱ2−2σ2csch2σx+λ+ϱδΕΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x+∂3Θ0x,ζ¯∂x3.Inserting (22) and (71) in (70), then one can obtain(73)Θ1x,ζ¯=ℏΕ−1ϱ2ϖ−2σcothσx+λ−1−0nϱ2ϖ−2σcothσx+λ+ϱδΕΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x,Υ1x,ζ¯=ℏΕ−1ϱ2−2σ2csch2σx+λ−1−0nϱ2−2σ2csch2σx+λ+ϱδΕΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x+∂3Θ0x,ζ¯∂x3.Therefore, we have(74)Θ1x,ζ¯=2ℏσ2ϖcsch2σx+λζ¯δΓδ+1,Υ1x,ζ¯4ℏσ3ϖcothσx+λcsch2σx+λζ¯δΓδ+1.Similarly, fors=2, then (70) and (71) produces(75)Θ2x,ζ¯=nΘ1x,ζ¯+ℏΕ−1ΕΘ1x,ζ¯−1−nnϱ2ϖ−2σcothσx+λ+ϱδΕΘ0x,ζ¯∂Θ1x,ζ¯∂x+Θ1x,ζ¯∂Θ0x,ζ¯∂x+∂Υ1x,ζ¯∂x,Υ2x,ζ¯=nΥ1x,ζ¯+ℏΕ−1ΕΥ1x,ζ¯−1−nnϱ2−2σ2csch2σx+λ+ϱδΕΘ0x,ζ¯∂Υ1x,ζ¯∂x+Θ1x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ1x,ζ¯∂x+Υ1x,ζ¯∂Θ0x,ζ¯∂x+∂3Θ1x,ζ¯∂x3.The subsequent process is simplified after modification, as previously stated(76)Θ2x,ζ¯=2n+ℏℏσ2ϖcsch2σx+λζ¯δΓδ+1−4ℏ2σ3ϖ2cothσx+λcsch2σx+λζ¯2δΓ2δ+1,Υ2x,ζ¯=4n+ℏℏσ3ϖcothσx+λcsch2σx+λζ¯δΓδ+1−4ℏ2σ4ϖ22+coshσx+λcsch4σx+λζ¯2δΓ2δ+1,Θ3x,ζ¯=2n+ℏ2ℏσ2ϖcsch2σx+λζ¯δΓδ+1−8ℏ+nℏ2σ3ϖ2cothσx+λcsch2σx+λζ¯2δΓ2δ+1+2ℏ3σ4ϖ2csch5σx+λζ¯3δΓα+12Γ3δ+1−4σcoshσx+λΓ2δ+1+Γα+128σcoshσx+λ+ϖ3sinhσx+λ+sinh3σx+λ,Υ3x,ζ¯=4n+ℏ2ℏσ3ϖcsch2σx+λcothσx+λζ¯δΓδ+1−8ℏ+nℏ2σ5ϖ2+cosh2σx+λcsch4σx+λζ¯2δΓ2δ+1+2ℏ3σ5ϖ2csch6σx+λζ¯3δΓα+12Γ3δ+1−4σ3+2cosh2σx+λΓ2δ+1+Γα+1224σ+16σcosh2σx+λ+10ϖsinh2σx+λ+ϖsinh4σx+λ.Analogously, in view of the iterative scheme (26), we derive the various iterates as(77)Θx,ζ¯=Θ0x,ζ¯+∑s=1∞Θsx,ζ¯1ns,Υx,ζ¯=Υ0x,ζ¯+∑s=1∞Υsx,ζ¯1ns.Case II: now, we construct the solution scheme via the Atangana–Baleanu fractional derivative operator in the Caputo context coupled alongside the ET andq¯-HATM.
By implementing the ET on (60), one can obtain(78)Mδδϱδ+1−δΕΘx,ζ¯−ϱ2Mδδϱδ+1−δΘj0=−ΕΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x,Mδδϱδ+1−δΕΥx,ζ¯−ϱ2Mδδϱδ+1−δΥj0=−ΕΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3.
As a result, we have(79)EΘx,ζ¯=ϱ2Θx,0−δϱδ+1−δMδEΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x,EΥx,ζ¯=ϱ2Υx,0−δϱδ+1−δMδEΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3.
The nonlinearity can be stated using (78) as follows:(80)N1ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Εψ1x,ζ¯;q˜−ϱ2ψ1x,0+δϱδ+1−δMδΕΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x,N2ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Εψ2x,ζ¯;q˜−ϱ2ψ2x,0+δϱδ+1−δMδΕΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x+∂3Θx,ζ¯∂x3.
Using the technique outlined in Section3, the s th order deformation equation is summarized as(81)ΕΘsx,ζ¯−KsΘs−1x,ζ¯=ℏℛ1,sΘ⟶s−1,Υ⟶s−1,where(82)ℛ1,sΘ⟶s−1,Υ⟶s−1=ΕΘs−1x,ζ¯−1−jsnϱ2ϖ−2σcothσx+λ+δϱδ+1−δMδ∑j=0s−1Θjx,ζ¯∂Θs−1−j∂x++∂Υs−1−j∂x,ℛ2,sΘ⟶s−1,Υ⟶s−1=ΕΘs−1x,ζ¯−1−jsnϱ2−2σ2csch2σx+λ+δϱδ+1−δMδ∑j=0s−1Θjx,ζ¯∂Υs−1−j∂x+∑j=0s−1Υjx,ζ¯∂Θs−j−1∂x+∂3Θs−1−j∂x3.
Making the use of inverse ET, we acquire(83)Θsx,ζ¯=KsΘs−1x,ζ¯+ℏΕ−1ℛ1,sΘ⟶s−1,Υ⟶s−1,Υsx,ζ¯=KsΥs−1x,ζ¯+ℏΕ−1ℛ2,sΘ⟶s−1,Υ⟶s−1.
Utilizing the ICS, then we have(84)Θ0x,ζ¯=ϖ−2σcothσx+λ,Υ0x,ζ¯=−2σ2csch2σx+λ.
Inserting theΘ0x,ζ¯ and Υ0x,ζ¯, letting s=1 in (81), then one can find(85)Θ1x,ζ¯=K1Θ0x,ζ¯+ℏΕ−1ℛ1,1Θ⟶0,Υ⟶0,Υ1x,ζ¯=K1Υ0x,ζ¯+ℏΕ−1ℛ2,1Θ⟶0,Υ⟶0.
Whens=1, then (82) reduces to(86)ℛ1,1Θ⟶0,Υ⟶0=ΕΘ0x,ζ¯−1−j1nϱ2ϖ−2σcothσx+λ+δϱδ+1−δMδΕΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x,ℛ2,1Θ⟶0,Υ⟶0=ΕΥ0x,ζ¯−1−j1nϱ2−2σ2csch2σx+λ+δϱδ+1−δMδΕΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x+∂3Θ0x,ζ¯∂x3.
Inserting (22) and (86) in (85), then one can obtain(87)Θ1x,ζ¯=ℏΕ−1ϱ2ϖ−2σcothσx+λ−1−0nϱ2ϖ−2σcothσx+λ+δϱδ+1−δMδΕΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x,Υ1x,ζ¯=ℏΕ−1ϱ2−2σ2csch2σx+λ−1−0nϱ2−2σ2csch2σx+λ+δϱδ+1−δMδΕΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x+∂3Θ0x,ζ¯∂x3.
Therefore, we have(88)Θ1x,ζ¯=2ℏσ2ϖcsch2σx+λδζ¯δΓδ+1+1−δ,Υ1x,ζ¯=4ℏσ3ϖcothσx+λcsch2σx+λδζ¯δΓδ+1+1−δ.
Similarly, fors=2, then (85) and (86) produce(89)Θ2x,ζ¯=nΘ1x,ζ¯+ℏE−1EΘ1x,ζ¯−1−nnϱ2ϖ−2σcothσx+λ+δϱδ+1−δMδEΘ0x,ζ¯∂Θ1x,ζ¯∂x+Θ1x,ζ¯∂Θ0x,ζ¯∂x+∂ϒ1x,ζ¯∂x,ϒ2x,ζ¯=nϒ1x,ζ¯+ℏE−1Eϒ1x,ζ¯−1−nnϱ2−2σ2csch2σx+λ+δϱδ+1−δMδEΘ0x,ζ¯∂ϒ1x,ζ¯∂x+Θ1x,ζ¯∂ϒ0x,ζ¯∂x+ϒ0x,ζ¯∂Θ1x,ζ¯∂x+ϒ1x,ζ¯∂Θ0x,ζ¯∂x+∂3Θ1x,ζ¯∂x3.
The subsequent process is simplified after modification, as previously stated:(90)Θ2x,ζ¯=2n+ℏℏσ2ϖcsch2σx+λδζ¯δΓδ+1+1−δ−4ℏ2σ3ϖ2cothσx+λcsch2σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2,ϒ2x,ζ¯=4n+ℏℏσ3ϖcothσx+λcsch2σx+λδζ¯δΓδ+1+1−δ−4ℏ2σ4ϖ22+coshσx+λcsch4σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2,Θ3x,ζ¯=2n+ℏ2ℏσ2ϖcsch2σx+λδζ¯δΓδ+1+1−δ−8ℏ+nℏ2σ3ϖ2cothσx+λcsch2σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2+2ℏ3σ4ϖ2csch5σx+λΓα+12−4σcoshσx+λΓ2δ+1+Γα+128σcoshσx+λ+ϖ3sinhσx+λ+sinh3σx+λ×δ3ζ¯3δΓ3δ+1+3δ21−δζ¯2δΓ2δ+1+3δ1−δ2ζ¯δΓδ+1+1−δ3,ϒ3x,ζ¯=4n+ℏ2ℏσ3ϖcsch2σx+λcothσx+λδζ¯δΓδ+1+1−δ−8ℏ+nℏ2σ5ϖ2+cosh2σx+λcsch4σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2+2ℏ3σ5ϖ2csch6σx+λΓα+12−4σ3+2cosh2σx+λΓ2δ+1+Γα+1224σ+16σcosh2σx+λ+10ϖsinh2σx+λ+ϖsinh4σx+λ×δ3ζ¯3δΓ3δ+1+3δ21−δζ¯2δΓ2δ+1+3δ1−δ2ζ¯δΓδ+1+1−δ3.
Analogously, in view of the iterative scheme and the system of (60), we have the q˜-HATM solutions as(91)Θx,ζ¯=Θ0x,ζ¯+∑s=1∞Θsx,ζ¯1ns,Υx,ζ¯=Υ0x,ζ¯+∑s=1∞Υsx,ζ¯1ns.
Whenδ=1, then the closed form solution modified Boussinesq model (61) is(92)Θx,ζ¯=ϖ−2σcothσx+λ−ϖt1,Υx,ζ¯=−2σ2csch2σx+λ−ϖt1.The numerical modelling was carried out in an attempt to demonstrate that the forthcoming methodology improves reliability via the Caputo and Atangana–Baleanu fractional derivative operators. The acquired findings show that the prospective system provides amazing correctness in respect to the method mentioned in the research [19, 21, 26]. Figures 1(a) and 1(b) compare the results of Θx,ζ¯ for Example 1 when compared to the exact and analytical solution. The responses to the found formulations for the modified Boussinesq problem involving various arbitrary functions, fractional-order, and integer-order are shown in Figures 2(a) and 2(b). The significance of the asymptotic component n in the q˜-HATM approach is illustrated in Figures 3(a) and 3(b).Figure 1
Nature of the solutions ofΘx,ζ¯ (a) exact solution (b) approximate solution via the q˜-HATM of Example 1 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 2
Nature of the solutions ofΘx,ζ¯ (a) multiple ℏ (b) multiple fractional-order via the q˜-HATM of Example 1 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 3
Nature of the solutions ofΘx,ζ¯ peaks (a) exact solution (b) approximate solution via the q˜-HATM of Example 1 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Analogously, Figures4(a) and 4(b) compare the results of Υx,ζ¯ for Example 1 when compared to the exact and analytical solution. The responses to the found formulations for the modified Boussinesq problem involving various arbitrary functions, fractional-order, and integer-order are shown in Figures 5(a) and 5(b). The significance of the asymptotic component n in the q˜-HATM approach is illustrated in Figures 6(a) and 6(b). Furthermore, Figures 6(a) and 6(b) return a vector with the local maxima (peaks) of the input signal vector, data. The peaks are output in the order of occurrence. If a peak is flat, the function returns only the point with the lowest index. The comparison analysis of the fractional operators reveals that both solution profiles are highly effective and easy to compute. This aids us in comprehending the complexity of linked equations.Figure 4
Nature of the solutions ofΥx,ζ¯ (a) exact solution (b) approximate solution via the q˜-HATM of Example 1 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 5
Nature of the solutions ofΥx,ζ¯ (a) multiple ℏ (b) multiple fractional-order via the q˜-HATM of Example 1 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 6
Nature of the solutions ofΥx,ζ¯ (a) exact solution, (b) approximate solution via the q˜-HATM of Example 2 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Example 2.
([19, 21, 26]). Assuming the fractional-order approximate long wave model is given as(93)Dζ¯δΘx,ζ¯+Θx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υ∂x2=0,0<δ≤1,Dζ¯δΥx,ζ¯+Θx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υ∂x2=0,subject to ICs(94)Θx,0=ϖ−σcothσx+λ,Υx,0=−σ2csch2σx+λ.Proof.
First and foremost, we find the solution of (102) in two different ways.Case I: here, we commence in view of the Caputo fractional derivative operator, which we comprise alongside the ET andq¯-HATM.By implementing the ET on (92), one can obtain(95)EDζ¯δΘx,ζ¯=−EΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υ∂x2,EDζ¯δΥx,ζ¯=−EΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υx,ζ¯∂x2.Consequently, we have(96)ϱ−δEΘx,ζ¯−∑j=0n−1ϱ2−δ+jΘj0=−EΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υ∂x2,ϱ−δEΥx,ζ¯−∑j=0n−1ϱ2−δ+jΥj0=−EΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υx,ζ¯∂x2.As a result, we have(97)EΘx,ζ¯=ϱ2Θx,0−ϱδEΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υ∂x2,EΥx,ζ¯=ϱ2Υx,0−ϱδEΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υx,ζ¯∂x2.The nonlinearity can be stated using (96) as follows:(98)N1ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Eψ1x,ζ¯;q˜−ϱ2ψ1x,0+ϱδEΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υ∂x2,N2ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Eψ2x,ζ¯;q˜−ϱ2ψ2x,0+ϱδEΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υx,ζ¯∂x2.Using the technique outlined in Section3, the s th order deformation equation is summarized as(99)EΘsx,ζ¯−KsΘs−1x,ζ¯=ℏℛ1,sΘ⟶s−1,Υ⟶s−1,where(100)ℛ1,sΘ⟶s−1,Υ⟶s−1=EΘs−1x,ζ¯−1−jsnϱ2ϖ−σcothσx+λ+ϱδ∑j=0s−1Θjx,ζ¯∂Θs−1−j∂x+∂Υs−1−j∂x+12∂2Υs−1−j∂x2,ℛ2,sΘ⟶s−1,Υ⟶s−1=EΘs−1x,ζ¯−1−jsnϱ2−σ2csch2σx+λ+ϱδ∑j=0s−1Θjx,ζ¯∂Υs−1−j∂x+∑j=0s−1Υjx,ζ¯∂Θs−j−1∂x−12∂2Υs−1−jx,ζ¯∂x2.Making the use of inverse ET, we acquire(101)Θsx,ζ¯=KsΘs−1x,ζ¯+ℏE−1ℛ1,sΘ⟶s−1,Υ⟶s−1,Υsx,ζ¯=KsΥs−1x,ζ¯+ℏE−1ℛ2,sΘ⟶s−1,Υ⟶s−1.Utilizing the ICs, then we have(102)Θ0x,ζ¯=ϖ−σcothσx+λ,Υ0x,ζ¯=−σ2csch2σx+λ.Inserting theΘ0x,ζ¯ and Υ0x,ζ¯, letting s=1 in (99), then one can find(103)Θ1x,ζ¯=K1Θ0x,ζ¯+ℏE−1ℛ1,1Θ⟶0,Υ⟶0,Υ1x,ζ¯=K1Υ0x,ζ¯+ℏE−1ℛ2,1Θ⟶0,Υ⟶0.Whens=1, then (101) reduces to(104)ℛ1,1Θ⟶0,Υ⟶0=ΕΘ0x,ζ¯−1−j1nϱ2ϖ−σcothσx+λ+ϱδΕΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x+12∂2Υ0x,ζ¯∂x2,ℛ2,1Θ⟶0,Υ⟶0=ΕΥ0x,ζ¯−1−j1nϱ2−σ2csch2σx+λ+ϱδΕΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x−12∂2Υ0x,ζ¯∂x2.Inserting (22) and (104) in (103), then one can obtain(105)Θ1x,ζ¯=ℏE−1ϱ2ϖ−σcothσx+λ−1−0nϱ2ϖ−σcothσx+λ+ϱδEΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x+12∂2Υ0x,ζ¯∂x2,Υ1x,ζ¯=ℏE−1ϱ2−σ2csch2σx+λ−1−0nϱ2−σ2csch2σx+λ+ϱδEΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x−12∂2Υ0x,ζ¯∂x2.Therefore, we have(106)Θ1x,ζ¯=ℏσ2ϖcsch2σx+λζ¯δΓδ+1,Υ1x,ζ¯=ℏσ3ϖcothσx+λcsch2σx+λζ¯δΓδ+1.Similarly, fors=2, then (103) and (104) produce(107)Θ2x,ζ¯=nΘ1x,ζ¯+ℏE−1EΘ1x,ζ¯−1−nnϱ2ϖ−σcothσx+λ+ϱδEΘ0x,ζ¯∂Θ1x,ζ¯∂x+Θ1x,ζ¯∂Θ0x,ζ¯∂x+∂Υ1x,ζ¯∂x+12∂2Υ1x,ζ¯∂x2,Υ2x,ζ¯=nΥ1x,ζ¯+ℏE−1EΥ1x,ζ¯−1−nnϱ2−σ2csch2σx+λ+ϱδEΘ0x,ζ¯∂Υ1x,ζ¯∂x+Θ1x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ1x,ζ¯∂x+Υ1x,ζ¯∂Θ0x,ζ¯∂x−12∂2Υ1x,ζ¯∂x2.The subsequent process is simplified after modification, as previously stated:(108)Θ2x,ζ¯=n+ℏℏσ2ϖcsch2σx+λζ¯δΓδ+1−2ℏ2σ3ϖ2cothσx+λcsch2σx+λζ¯2δΓ2δ+1,Υ2x,ζ¯=2n+ℏℏσ3ϖcothσx+λcsch2σx+λζ¯δΓδ+1−2ℏ2σ4ϖ22+coshσx+λcsch4σx+λζ¯2δΓ2δ+1,Θ3x,ζ¯=n+ℏ2ℏσ2ϖcsch2σx+λζ¯δΓδ+1−4ℏ+nℏ2σ3ϖ2cothσx+λcsch2σx+λζ¯2δΓ2δ+1+ℏ3σ4ϖ2csch5σx+λζ¯3δΓα+12Γ3δ+1−2σcoshσx+λΓ2δ+1+Γα+124σcoshσx+λ+ϖ3sinhσx+λ+sinh3σx+λ,Υ3x,ζ¯=2n+ℏ2ℏσ3ϖcsch2σx+λcothσx+λζ¯δΓδ+1−4ℏ+nℏ2σ5ϖ2+cosh2σx+λcsch4σx+λζ¯2δΓ2δ+1+ℏ3σ5ϖ2csch6σx+λζ¯3δΓα+12Γ3δ+1−2σ3+2cosh2σx+λΓ2δ+1+Γα+1212σ+8σcosh2σx+λ+10ϖsinh2σx+λ+ϖsinh4σx+λ.Analogously, in view of the iterative scheme (3.36), we derive the various iterates as(109)Θx,ζ¯=Θ0x,ζ¯+∑s=1∞Θsx,ζ¯1ns,Υx,ζ¯=Υ0x,ζ¯+∑s=1∞Υsx,ζ¯1ns.Case II: now, we construct the solution scheme via the Atangana–Baleanu fractional derivative operator in the Caputo context coupled alongside the ET andq¯-HATM.
By implementing the ET on (92), one can obtain(110)Mδδϱδ+1−δEΘx,ζ¯−ϱ2Mδδϱδ+1−δΘj0=−EΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υx,ζ¯∂x2,Mδδϱδ+1−δEΥx,ζ¯−ϱ2Mδδϱδ+1−δΥj0=−EΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υx,ζ¯∂x2.
As a result, we have(111)EΘx,ζ¯=ϱ2Θx,0−δϱδ+1−δMδEΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υ0x,ζ¯∂x2,EΥx,ζ¯=ϱ2Υx,0−δϱδ+1−δMδδEΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υx,ζ¯∂x2.
The nonlinearity can be stated using (110) as follows:(112)N1ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Eψ1x,ζ¯;q˜−ϱ2ψ1x,0+δϱδ+1−δMδEΘx,ζ¯∂Θx,ζ¯∂x+∂Υx,ζ¯∂x+12∂2Υx,ζ¯∂x2,N2ψ1x,ζ¯;q˜,ψ2x,ζ¯;q˜=Eψ2x,ζ¯;q˜−ϱ2ψ2x,0+δϱδ+1−δMδEΘx,ζ¯∂Υx,ζ¯∂x+Υx,ζ¯∂Θx,ζ¯∂x−12∂2Υx,ζ¯∂x2.
Using the technique outlined in Section3, the s th order deformation equation is summarized as(113)EΘsx,ζ¯−KsΘs−1x,ζ¯=ℏℛ1,sΘ⟶s−1,Υ⟶s−1,where(114)ℛ1,sΘ⟶s−1,Υ⟶s−1=EΘs−1x,ζ¯−1−jsnϱ2ϖ−σcothσx+λ+δϱδ+1−δMδ∑j=0s−1Θjx,ζ¯∂Θs−1−j∂x+∂Υs−1−j∂x+12∂2Υs−1−jx,ζ¯∂x2,ℛ2,sΘ⟶s−1,Υ⟶s−1=EΘs−1x,ζ¯−1−jsnϱ2−σ2csch2σx+λ+δϱδ+1−δMδ∑j=0s−1Θjx,ζ¯∂Υs−1−j∂x+∑j=0s−1Υjx,ζ¯∂Θs−j−1∂x−12∂2Υs−1−j∂x2.
Making use of inverse ET, we acquire(115)Θsx,ζ¯=KsΘs−1x,ζ¯+ℏE−1ℛ1,sΘ⟶s−1,Υ⟶s−1,Υsx,ζ¯=KsΥs−1x,ζ¯+ℏE−1ℛ2,sΘ⟶s−1,Υ⟶s−1.
Utilizing the ICs, then we have(116)Θ0x,ζ¯=ϖ−σcothσx+λ,Υ0x,ζ¯=−σ2csch2σx+λ.
Inserting theΘ0x,ζ¯ and Υ0x,ζ¯, letting s=1 in (113), then one can find(117)Θ1x,ζ¯=K1Θ0x,ζ¯+ℏE−1ℛ1,1Θ⟶0,Υ⟶0,Υ1x,ζ¯=K1Υ0x,ζ¯+ℏE−1ℛ2,1Θ⟶0,Υ⟶0.
Whens=1, then (5.54) reduces to(118)ℛ1,1Θ⟶0,Υ⟶0=EΘ0x,ζ¯−1−j1nϱ2ϖ−σcothσx+λ+δϱδ+1−δMδEΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x+12∂2Υ0x,ζ¯∂x2,ℛ2,1Θ⟶0,Υ⟶0=EΥ0x,ζ¯−1−j1nϱ2−σ2csch2σx+λ+δϱδ+1−δMδEΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x−12∂2Υ0x,ζ¯∂x2.
Inserting (22) and (5.58) in (5.57), then one can obtain(119)Θ1x,ζ¯=ℏE−1ϱ2ϖ−σcothσx+λ−1−0nϱ2ϖ−σcothσx,λ¯+δϱδ+1−δMδEΘ0x,ζ¯∂Θ0x,ζ¯∂x+∂Υ0x,ζ¯∂x+12∂2Υ0x,ζ¯∂x2,Υ1x,ζ¯=ℏE−1ϱ2−σ2csch2σx+λ−1−0nϱ2−σ2csch2σx+λ+δϱδ+1−δMδEΘ0x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ0x,ζ¯∂x−12∂2Υ0x,ζ¯∂x2.
Therefore, we have(120)Θ1x,ζ¯=ℏσ2ϖcsch2σx+λδζ¯δΓδ+1+1−δ,Υ1x,ζ¯=2ℏσ3ϖcothσx+λcsch2σx+λδζ¯δΓδ+1+1−δ.
Similarly, fors=2, then (5.57) and (5.58) produce(121)Θ2x,ζ¯=nΘ1x,ζ¯+ℏΕ−1ΕΘ1x,ζ¯−1−nnϱ2ϖ−σcothσx+λ+δϱδ+1−δMδΕΘ0x,ζ¯∂Θ1x,ζ¯∂x+Θ1x,ζ¯∂Θ0x,ζ¯∂x+∂Υ1x,ζ¯∂x+12∂2Υ1x,ζ¯∂x2,Υ2x,ζ¯=nΥ1x,ζ¯+ℏΕ−1ΕΥ1x,ζ¯−1−nnϱ2−σ2csch2σx+λ+δϱδ+1−δMδΕΘ0x,ζ¯∂Υ1x,ζ¯∂x+Θ1x,ζ¯∂Υ0x,ζ¯∂x+Υ0x,ζ¯∂Θ1x,ζ¯∂x+Υ1x,ζ¯∂Θ0x,ζ¯∂x−12∂2Υ1x,ζ¯∂x2.
The subsequent process is simplified after modification, as previously stated:(122)Θ2x,ζ¯=n+ℏℏσ2ϖcsch2σx+λδζ¯δΓδ+1+1−δ−2ℏ2σ3ϖ2cothσx+λcsch2σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2,ϒ2x,ζ¯=2n+ℏℏσ3ϖcothσx+λcsch2σx+λδζ¯δΓδ+1+1−δ−2ℏ2σ4ϖ22+coshσx+λcsch4σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2,Θ3x,ζ¯=n+ℏ2ℏσ2ϖcsch2σx+λδζ¯δΓδ+1+1−δ−4ℏ+nℏ2σ3ϖ2cothσx+λsch2σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2+ℏ3σ4ϖ2csch5σx+λΓα+12−2σcoshσx+λΓ2δ+1+Γα+124σcoshσx+λ+ϖ3sinhσx+λ+sinh3σx+λ×δ3ζ¯3δΓ3δ+1+3δ21−δζ¯2δΓ2δ+1+3δ1−δ2ζ¯δΓδ+1+1−δ3,ϒ3x,ζ¯=2n+ℏ2ℏσ3ϖcsch2σx+λcothσx+λδζ¯δΓδ+1+1−δ−4ℏ+nℏ2σ5ϖ2+cosh2σx+λcsch4σx+λδ2ζ¯2δΓ2δ+1+2δ1−δζ¯δΓδ+1+1−δ2+ℏ3σ5ϖ2csch6σx+λΓα+12−2σ3+2cosh2σx+λΓ2δ+1+Γα+1212σ+8σcosh2σx+λ+10ϖsinh2σx+λ+ϖsinh4σx+λ×δ3ζ¯3δΓ3δ+1+3δ21−δζ¯2δΓ2δ+1+3δ1−δ2ζ¯δΓδ+1+1−δ3.
Analogously, in view of the iterative scheme (3.36), we derive the various iterates as(123)Θx,ζ¯=Θ0x,ζ¯+∑s=1∞Θsx,ζ¯1ns,Υx,ζ¯=Υ0x,ζ¯+∑s=1∞Υsx,ζ¯1ns.
Whenδ=1, then the closed form solution approximate long wave model (5.33) is(124)Θx,ζ¯=ϖ−σcothσx+λ−ϖt1,Yx,ζ¯=−σ2csch2σx+λ−ϖt1.The numerical modelling was conducted in order to illustrate that using the Caputo and Atangana–Baleanu fractional derivative operators, the proposed strategy enhances dependability. The results reveal that the proposed system is incredibly accurate in comparison to the way pertaining to the research [19, 21, 26]. Figures 7(a) and 7(b) make a comparison of Θx,ζ¯ for Example 2 to the precise and approximate solution. Figures 8(a) and 8(b) demonstrate the responses to the established approaches for the approximate regular wave problem concerning multiple arbitrary functions, fractional-order, and integer-order, respectively. Figures 9(a) and 9(b) show the importance of the asymptotic element n via the q˜-HATM technique. Figures 10(a) and 10(b) analyze the performance of Υx,ζ¯ for Example 2 to the precise analytical solution. Figures 11(a) and 11(b) demonstrate the responses to the established interpretations for the approximate regular wave issue including multiple arbitrary functions, fractional-order, and integer-order, respectively. Figures 12(a) and 12(b) show the importance of the asymptotic component n in the q˜-HATM technique. Furthermore, Figures 12(a) and 12(b) return a vector with the local maxima (peaks) of the input signal vector, data. The peaks are output in the order of occurrence. Since a peak is scattered, the function returns only the point with the highest index. The fractional operators’ comparison study demonstrates that both solution profiles are very effective and simple to compute. This makes it easier to grasp the complexity of linked equations.Figure 7
Nature of the solutions ofΘx,ζ¯ (a) exact solution, (b) approximate solution via the q˜-HATM of Example 2 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 8
Nature of the solutions ofΘx,ζ¯ (a) multiple ℏ, (b) multiple fractional-order via the q˜-HATM of Example 2 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 9
Nature of the solutions ofΘx,ζ¯ peaks (a) exact solution, (b) approximate solution via the q˜-HATM of Example 2 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 10
Nature of the solutions ofΥx,ζ¯ (a) exact solution, (b) approximate solution via the q˜-HATM of Example 1 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 11
Nature of the solutions ofΥx,ζ¯ (a) multiple ℏ, (b) multiple fractional-order via the q˜-HATM of Example 2 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)Figure 12
Nature of the solutions ofΥx,ζ¯ peaks (a) exact solution, (b) approximate solution via the q˜-HATM of Example 1 when ϖ=0.001,σ=0.2, and λ=12.
(a)(b)


## 6. Conclusion

Theq˜-HATM is applied to formulate the approximation for coupled modified Boussinesq and approximate long wave models of fractional-order in this paper. Two cases are used to demonstrate and assess the effectiveness of the proposed method under consideration pertaining to the Caputo and Atangana–Baleanu fractional derivative operators. The current methodology includes convergence and error analysis to demonstrate coherence and usefulness. In terms of absolute inaccuracy, simulations are presented for both fractional governing equations. We can observe from the figures and plots that the suggested strategy is extremely successful and precise compared to previous approaches and that it includes CFRDTM data as a particular scenario (n=1 and ℏ=−1). Furthermore, the methodology tries to influence and regulate the numerical solutions, which interact swiftly with the exact model in a narrow accessible region. As a result, we can indicate that the model technique is enormously effective and well suited for studying associated systems emerging in physical systems and that both fractional and integer order derivatives mathematically and computationally characterize real-world issues in an effective and constructive fashion.



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