# Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators

Authors: Kamsing Nonlaopon; Muhammad Naeem; Ahmed M. Zidan; Rasool Shah; Ahmed Alsanad; Abdu Gumaei
Journal: Complexity
 (2021)

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## Abstract

This paper aims to implement an analytical method, known as the Laplace homotopy perturbation transform technique, for the result of fractional-order Whitham–Broer–Kaup equations. The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag-Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham–Broer–Kaup equations with both derivatives to see their difference in a real-world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham–Broer–Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional-order derivatives.


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## Body

## 1. Introduction

In engineering and applied sciences and technology, fractional partial differential equations (FPDEs) containing nonlinearities define many phenomena, ranging from gravitation to dynamics. The nonlinear FPDEs are significant tools analyzed to model nonlinear dynamical behaviour in many areas such as plasma physics, mathematical biology, fluid dynamics, and solid-state physics. The widely held dynamical schemes can be denoted by an appropriate set of FPDEs. Moreover, it is well identified that FPDEs solve mathematical models, such as Poincare conjecture and Calabi conjecture models [1–12].It has been determined that the nonlinear development of shallow-water waves in fluid mechanics is described by a coupled system of Whitham–Broer–Kaup (WBK) equations [13]. Whitham [14], Broer [15], and Kaup [16] proposed the coupled scheme of the aforementioned equations. The aforementioned equations define shallow-water wave propagation with various spreading relations, as shown in [17]. The governing equations for the respective phenomena in classic order are provided by(1)Dℑβμφ,ℑ+μφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+g∂νφ,ℑ∂φ=0,Dℑβνφ,ℑ+μφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+p∂3μφ,ℑ∂φ3−g∂2νφ,ℑ∂φ2=0,0<β≤1,where μφ,ℑ and νφ,ℑ show the height and horizontal velocity that diverges from the liquid’s equilibrium position, respectively, and pg are constant, signified in different diffusion powers. Over the last few decades, there has been a lot of research into solutions to such nonlinear PDEs. So, many researchers have created a variety of mathematical methods to investigate the analytical results of nonlinear PDEs. The HPM was used by Biazar and Khah [18] to solve the coupled schemes of the Burger and Brusselator problems. Amjad et al. [19] applied the solution of standard order coupled with fractional-order Whitham–Broer–Kaup equation by Laplace decomposition technique. Noor et al. [20] used the homotopy perturbation technique to investigate the results of much classical order of PDEs. Whitham–Broer–Kaup equations are used by other scholars who implemented several numerical techniques, such as residual power series technique [4], reduced differential transformation technique [21], Adomian decomposition technique [7], homotopy perturbation technique [22, 23], Lie symmetry analysis [24, 25], exp-function technique [26], G′/G-expansion technique [27], and homotopy analysis technique [28].Fractional calculus (FC) is a new mathematical approach for describing models of nonlocal behaviour. Fractional derivatives have mathematically described many other physical problems in recent years; these representations have yielded excellent outcomes in the simulation of real-world issues. Some basic definitions of fractional operators were given by Riesz, Coimbra, Hadamard, Riemann–Liouville, Grunwald–Letnikov, Weyl, Caputo, Fabrizio, and Atangana Baleanu, among others [29–31]. To investigate the solutions of nonlinear FPDEs, some well-known techniques for finding actual results have been develop, for example, the homotopy perturbation transform technique [32, 33], the invariant subspace technique [34], the Hermite colocation technique [35], the q-homotopy analysis transform method [36], the optimal homotopy asymptotic technique [37], the homotopy analysis Sumudu transform technique [38], the Adomian decomposition technique [39], the Pade approximation and homotopy-Pade method [39], and the Sumudu transform series expansion technique [40]. The Laplace homotopy perturbation transform technique is a mixture of the homotopy perturbation technique introduced by Liao [41] and of the Laplace transformation [42].The rest of the paper is arranged as follows. Section2 discusses the basic definitions from fractional calculus. Section 3 is introduced to the fundamental methodology of HPTM. Sections 4 and 5 are the implementations of the techniques for the CF and AN operators. The conclusion of the work is written in Section 6.


## 2. Fractional Calculus

This section provides some fundamental concepts of fractional calculus.Definition 1 (see [42]).
The Liouville–Caputo operator is given by(2)Dℑβuφ,ℑ=1Γn−β∫0ℑℑ−θn−1unφ,θdθ,n−1<β<n,where unφ,θ is the derivative of integer nth order of uφ,θ,n∈ℕ and n−1<β≤n. If 0<β≤1, then we defined the Laplace transformation for the Caputo fractional derivative as follows:(3)LDℑβuφ,ℑs=sβLuφ,ℑs−sβ−1uφ,0.Definition 2 (see [42]).
The Caputo–Fabrizio operator (CF) is defined by(4)Dℑβuφ,ℑ=2−βMβ2n−β∫0ℑexp−βℑ−θn−βunφ,θdθ,where Mβ is a normalization form and M0=M1=1. The exponential law is used as the nonsingular kernel in this fractional operator.
If0<β≤1, then we define the Caputo–Fabrizio of Laplace transformation for the fractional derivative is given as(5)LDℑβuφ,ℑs=sLuφ,ℑs−uφ,0s+β1−s.Definition 3 (see [42]).
The fractional generalized Mittag-Leffler law with the sense of Atangana–Baleanu operator is defined as follows:(6)Dℑβuφ,ℑ=Bβ1−β∫0ℑEβ−βℑ−θβn−βunφ,θdθ,where Bβ is a normalization function with B0=B1=1. The Mittag-Leffler law is used as a nonsingular and nonlocal kernel in this fractional operator.
If0<β≤1, then we express the Laplace transformation for the Atangana–Baleanu operator fractional derivative as(7)LDℑβuφ,ℑs=sLuφ,ℑs−uφ,0sβ1−β+β.


## 3. Implementation of the LHPTM for the Solution of Fractional Partial Differential Equation

The LHPTM is general methodology and can be written as follows.The main procedure of this technique is defined as follows:Step 1: let us consider the following equation:(8)∂β∂ℑβFφ,ℑ=ΛFφ,ℑ+ΞFφ,ℑ+uφ,ℑ,β=1,2,3,…,under the initial condition(9)∂i∂ℑiFφ,0=ziφ,∂β−1∂ℑβ−1Fφ,0=0,i=0,1,2,3,…,β−2,whereuφ,ℑ is a known function, β is the order of the derivative, Λ represents a linear differential operator, and Ξ is the general nonlinear differential operator.Step 2: using both sides Laplace transformation operator of (8), we obtain(10)Lfφ,ℑ=sβLΛFφ,ℑ+sβLΞFφ,ℑ+sβLuφ,ℑ,Laplace transformation is applied to Caputo–Fabrizio (5) and Atangana–Baleanu (7) operators.Step 3: on both sides, using the inverse Laplace transformation of equation (10), we obtain(11)Fφ,ℑ=Hφ,ℑ+L−1sβLΛFφ,ℑ+sβLΞFφ,ℑ.Step 4: applying the homotopy producer, the result of the above equations in a series form is defined by(12)Fφ,ℑ,k=∑n=0∞knFnφ,ℑ,Fφ,ℑ=limk⟶1Fφ,ℑ,k,and the nonlinear terms can be expressed as(13)ΞFφ,ℑ=∑n=1∞knhnfφ,ℑ,where k∈0,1 is an embedding parameter and hnFx,ℑ are He’s polynomials that can be provided by(14)hnF0,…,Fn=1n!∂n∂kkΞ∑j=0nkjFjφ,ℑ,n∈ℕ.Finally, the LHPTM is achieved by coupling the decomposition technique which is defined by(15)∑j=0∞knFnφ,ℑ=Tφ,ℑ+k1m−1!∫0ℑℑ−τm−1uφ,ℑ+Λ∑n=0∞knFnφ,ℑ+∑n=0∞knhnFφ,ℑdτ,where Tφ,ℑ=Hφ,ℑ.The terms, comparing with the same powers ofk, produce results of many orders. The initial estimated of the approximation is Tφ,ℑ, which is actually the Taylor series for the exact result of order β.Using the aforementioned technique, we solved the fractional-order Whitham–Broer–Kaup equations in the Atangana–Baleanu and Caputo–Fabrizio senses using the LHPM.


## 4. Implementation of Caputo–Fabrizio Operator

Example 1.
Let us consider the coupled system of fractional-order WBKEs in the CFC sense:(16)DCFℑβμφ,ℑ+μφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ=0,DCFℑβνφ,ℑ+μφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2=0,0<β≤1,−1<ℑ≤1,−10≤φ≤10,under the initial conditions,(17)μφ,0=12−8tanh−2φ,νφ,0=16−16tanh2−2φ.
Using the Laplace transformation to equation (16), we obtain(18)sLμφ,ℑs−μφ,0s+β1−s=−Lμφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,sLνφ,ℑs−νφ,0s+β1−s=−Lμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2.
Simplify the above equation and use the initial conditions (17), and we obtain(19)Lμφ,ℑ=μφ,0s−s+β1−ssLμφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,Lνφ,ℑ=νφ,0s−s+β1−ssLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2.
The inverse Laplace transformation is implemented to (19), and we obtain(20)μφ,ℑ=μφ,0−L−1s+β1−ssLμφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,νφ,ℑ=νφ,0−L−1s+β1−ssLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2.
The LHPTM is used in (20), and we obtain(21)∑n=0∞μnφ,ℑ=μφ,0−pL−1s+β1−ssL∑n=0∞pnAn+∂∂φ∑n=0∞pnμnφ,ℑ+∂∂φ∑n=0∞pnνnφ,ℑ,∑n=0∞νnφ,ℑ=νφ,0−pL−1s+β1−ssL∑n=0∞pnBn+∑n=0∞pnCn+3∂3∂φ3∑n=0∞pnμnφ,ℑ−∂2∂φ2∑n=0∞pnνnφ,ℑ.
The nonlinear can be found with the help of He’s polynomial and can be defined as(22)A0=μ0∂μ0∂φ,A1=μ0∂μ1∂φ+μ1∂μ0∂φ,B0=μ0∂ν0∂β,B1=μ0∂ν1∂β+μ1∂ν0∂β,C0=ν0∂μ0∂φ,C1=ν0∂μ1∂φ+ν1∂μ0∂φ.
Comparing the coefficient ofp, we have(23)p0:μ0φ,ℑ=μφ,0=12−8tanh−2φ,p0:ν0φ,ℑ=νφ,0=16−16tanh2−2φ,p1:μ1φ,t=−L−1s+β1−ssLA0+∂∂φμ0φ,ℑ+∂∂φν0φ,ℑ=−8sech2−2φβℑ+1−β,p1:ν1φ,ℑ=−pL−1s+β1−ssLB0+C0+3∂3∂φ3μ0φ,ℑ−∂2∂φ2ν0φ,ℑ=−32sech2−2φtanh−2φβℑ+1−β,p2:μ2φ,ℑ=−L−1s+β1−ssLA1+∂∂φμ1φ,ℑ+∂∂φν1φ,ℑ=−16sech2−2φ4sech2−2φ−8tanh2−2φ+3tanh−2φ1−β2+2β1−βℑ+β2ℑ22,p2:ν2φ,ℑ=−pL−1s+β1−ssLB1+C1+3∂3∂φ3μ1φ,ℑ−∂2∂φ2ν1φ,ℑ=−32sech2−2φ40sech2−2φtanh−2φ+96tanh−2φ−2tanh2−2φ−32tanh3−2φ−25sech2−2φ1−β2+2β1−βℑ+β2ℑ22,⋮.
We can calculate few terms of (16) which can be written as(24)μφ,ℑ=μ0φ,ℑ+μ1φ,ℑ+μ2φ,ℑ+⋯,νφ,ℑ=ν0φ,ℑ+ν1φ,ℑ+ν2φ,ℑ+⋯,μφ,ℑ=12−8tanh−2φ−8sech2−2φβℑ+1−β−16sech2−2φ×4sech2−2φ−8tanh2−2φ+3tanh−2φ1−β2+2β1−βℑ+β2ℑ22+⋯,νφ,ℑ=16−16tanh2−2φ−8sech2−2φβℑ+1−β−32sech2−2φ40sech2−2φtanh−2φ+96tanh−2φ−2tanh2−2φ−32tanh3−2φ−25sech2−2φ×1−β2+2β1−βℑ+β2ℑ22+⋯.
The exact solution of (16) is(25)μφ,ℑ=12−8tanh−2φ−ℑ2,νφ,ℑ=16−16tanh2−2φ−ℑ2.
Figure1 shows the actual and approximate solutions of μφ,ℑ at β=1, and Figure 2 shows the actual and approximate solutions of νφ,ℑ at β=1. Figures 3 and 4 show that the first graph has a different fractional order with respect to φ and the second graph has a different fractional order with respect to ℑ of Example 1. Tables 1 and 2 show the different fractional-order β of μφ,ℑ and νφ,ℑ. Tables 3 and 4 show the comparisons with different methods.Figure 1
The exact and analytical solution with respect to the Caputo–Fabrizio operator ofμφ,ℑ at β=1 of Example 1.
(a)(b)Figure 2
The exact and analytical solution with respect to the Caputo–Fabrizio operator ofνφ,ℑ at β=1 of Example 1.
(a)(b)Figure 3
The different fractional-order solution with respect to the Caputo–Fabrizio operator ofμφ,ℑ at β of Example 1.
(a)(b)Figure 4
The different fractional-order solution with respect to the Caputo–Fabrizio operator ofνφ,ℑ at β of Example 1.
(a)(b)Table 1
HPTM for the approximate solution of Example1.
φ,ℑμφ,ℑ at β=0.5μφ,ℑ at β=0.75μφ,ℑ at β=1Exact solution(0.1, 0.1)−0.500817−0.500795−0.500782−0.500782(0.1, 0.3)−0.500853−0.500829−0.50081−0.50081(0.1, 0.5)−0.500878−0.500857−0.500837−0.500837(0.2, 0.1)−0.49812−0.498098−0.498085−0.498085(0.2, 0.3)−0.498154−0.498131−0.498112−0.498112(0.2, 0.5)−0.498178−0.498158−0.498139−0.498139(0.3, 0.1)−0.495491−0.49547−0.495458−0.495458(0.3, 0.3)−0.495525−0.495502−0.495484−0.495484(0.3, 0.5)−0.495548−0.495529−0.49551−0.49551(0.4, 0.1)−0.49293−0.492909−0.492897−0.492897(0.4, 0.3)−0.492963−0.49294−0.492922−0.492922(0.4, 0.5)−0.492985−0.492966−0.492948−0.492948(0.5, 0.1)−0.490433−0.490413−0.490401−0.490401(0.5, 0.3)−0.490465−0.490443−0.490426−0.490426(0.5, 0.5)−0.490487−0.490469−0.490451−0.490451Table 2
HPTM for the approximate solution of Example1.
φ,ℑνφ,ℑ at β=0.5νφ,ℑ at β=0.75νφ,ℑ at β=1Exact solution(0.1, 0.1)−0.0939215−0.0939015−0.09389−0.09389(0.1, 0.3)−0.0939536−0.0939319−0.0939146−0.0939146(0.1, 0.5)−0.0939757−0.0939571−0.0939391−0.0939391(0.2, 0.1)−0.0915064−0.091487−0.0914759−0.0914759(0.2, 0.3)−0.0915375−0.0915165−0.0914997−0.0914997(0.2, 0.5)−0.0915589−0.0915409−0.0915235−0.0915235(0.3, 0.1)−0.0891657−0.0891469−0.0891361−0.0891361(0.3, 0.3)−0.0891958−0.0891754−0.0891592−0.0891592(0.3, 0.5)−0.0892166−0.0891992−0.0891822−0.0891822(0.4, 0.1)−0.0868965−0.0868782−0.0868678−0.0868678(0.4, 0.3)−0.0869257−0.0869059−0.0868901−0.08688901(0.4, 0.5)−0.0869458−0.0869289−0.0869125−0.0869125(0.5, 0.1)−0.0846961−0.0846784−0.0846683−0.0846683(0.5, 0.3)−0.0847244−0.0847052−0.0846899−0.0846899(0.5, 0.5)−0.0847439−0.0847275−0.0847116−0.0847116Table 3
HPTM for the approximate solution of Example1.
φ,ℑAbsolute error of ADMAbsolute error of VIMAbsolute error of OHAMAbsolute error HPTM(0.1, 0.1)1.04892 × 10−41.23033 × 10−41.07078 × 10−41.67111 × 10−12(0.1, 0.3)9.64474 × 10−53.69597 × 10−43.04565 × 10−44.51196 × 10−11(0.1, 0.5)8.88312 × 10−56.16873 × 10−44.81303 × 10−42.08888 × 10−10(0.2, 0.1)4.25408 × 10−41.19869 × 10−41.04388 × 10−41.57879 × 10−12(0.2, 0.3)3.91098 × 10−43.60098 × 10−42.97260 × 10−44.26227 × 10−11(0.2, 0.5)3.60161 × 10−46.01006 × 10−44.70138 × 10−41.97328 × 10−10(0.3, 0.1)9.71922 × 10−41.16789 × 10−41.01776 × 10−41.49181 × 10−12(0.3, 0.3)8.93309 × 10−43.50866 × 10−42.90150 × 10−44.02799 × 10−11(0.3, 0.5)8.22452 × 10−45.85610 × 10−44.59590 × 10−41.86481 × 10−10(0.4, 0.1)1.75596 × 10−31.13829 × 10−49.92418 × 10−51.41043 × 10−12(0.4, 0.3)1.61430 × 10−33.41948 × 10−42.83229 × 10−43.80803 × 10−11(0.4, 0.5)1.48578 × 10−35.70710 × 10−44.49118 × 10−41.76298 × 10−10(0.5, 0.1)2.79519 × 10−31.10936 × 10−49.67808 × 10−41.33388 × 10−12(0.5, 0.3)2.56714 × 10−33.33274 × 10−42.76492 × 10−43.60145 × 10−11(0.5, 0.5)2.36184 × 10−35.56235 × 10−44.38895 × 10−41.66734 × 10−10Table 4
Comparison of different methods for the approximate solution of Example1.
φ,ℑAbsolute error of ADMAbsolute error of VIMAbsolute error of OHAMAbsolute error HPTM(0.1, 0.1)6.41419 × 10−31.10430 × 10−45.86860 × 10−53.28081 × 10−12(0.1, 0.3)5.99783 × 10−33.31865 × 10−43.04565 × 10−48.85812 × 10−11(0.1, 0.5)5.61507 × 10−35.54071 × 10−43.08812 × 10−44.10099 × 10−10(0.2, 0.1)1.33181 × 10−21.07016 × 10−45.56884 × 10−53.07768 × 10−12(0.2, 0.3)1.24441 × 10−23.21601 × 10−42.97260 × 10−48.30963 × 10−11(0.2, 0.5)1.16416 × 10−25.36927 × 10−42.92626 × 10−43.84706 × 10−10(0.3, 0.1)2.07641 × 10−21.03737 × 10−45.28609 × 10−52.88849 × 10−12(0.3, 0.3)1.93852 × 10−23.11737 × 10−42.90150 × 10−47.79908 × 10−11(0.3, 0.5)1.81209 × 10−25.20447 × 10−42.77382 × 10−43.6107 × 10−10(0.4, 0.1)2.88100 × 10−21.00579 × 10−45.01929 × 10−52.71246 × 10−12(0.4, 0.3)2.68724 × 10−23.02245 × 10−42.83229 × 10−47.32356 × 10−11(0.4, 0.5)2.50985 × 10−25.04593 × 10−42.63019 × 10−43.39055 × 10−10(0.5, 0.1)3.75193 × 10−29.75385 × 10−54.76741 × 10−52.54828 × 10−12(0.5, 0.3)3.49617 × 10−22.93107 × 10−42.76492 × 10−46.88039 × 10−11(0.5, 0.5)3.26239 × 10−24.89335 × 10−42.49480 × 10−43.18537 × 10−10Example 2.
Let us consider the coupled system of fractional-order WBKEs in the CFC sense:(26)DCFℑβμφ,ℑ+μφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ=0,DCFℑβνφ,ℑ+μφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2=0,0<β≤1,0<ℑ≤1,−100≤φ≤100,under the initial conditions,(27)μφ,0=ξ−κcothκφ+θ,νφ,0=−κ2cosech2κφ+θ.
Using the Laplace transformation to equation (26), we obtain(28)sLμφ,ℑs−μφ,0s+β1−s=−Lμφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,sLνφ,ℑs−νφ,0s+β1−s=−Lμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2.
Simplify the above equation and use the initial conditions (27), and we obtain(29)Lμφ,ℑ=μφ,0s−s+β1−ssLμφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,Lνφ,ℑ=νφ,0s−s+β1−ssLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2.
The inverse Laplace transformation is implemented to (29), and we obtain(30)μφ,ℑ=μφ,0−L−1s+β1−ssLμφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,νφ,ℑ=νφ,0−L−1s+β1−ssLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2.
The LHPTM is used in (30), and we obtain(31)∑n=0∞μnφ,ℑ=μφ,0−pL−1s+β1−ssL∑n=0∞pnAn+12∂∂φ∑n=0∞pnμnφ,ℑ+∂∂φ∑n=0∞pνnφ,ℑ,∑n=0∞νnφ,ℑ=νφ,0−pL−1s+β1−ssL∑n=0∞pnBm+∑n=0∞pnCn−12∂2∂φ2∑n=0∞pnνnφ,ℑ.
The nonlinear can be found with the help of He’s polynomial and can be defined as(32)A0=μ0∂μ0∂φ,A1=μ0∂μ1∂φ+μ1∂μ0∂φ,B0=μ0∂ν0∂β,B1=μ0∂ν1∂β+μ1∂ν0∂β,C0=ν0∂μ0∂φ,C1=ν0∂μ1∂φ+ν1∂μ0∂φ.
Comparing the coefficient ofp, we have(33)p0:μ0φ,ℑ=μφ,0=ξ−κcothκφ+θ,p0:ν0φ,ℑ=νφ,0=−κ2cosech2κφ+θ,p1:μ1φ,ℑ=−L−1s+β1−ssLA0+12∂∂φμ0φ,ℑ+∂∂φν0φ,ℑ=−ξκ2cosech2κφ+θβℑ+1−β,p1:ν1φ,ℑ=−pL−1s+β1−ssLB0+C0−12∂2∂φ2ν0φ,ℑ=−ξκ2cosech2κφ+θcothκφ+θβℑ+1−β,p2:μ2φ,ℑ=−L−1s+β1−ssLA1+12∂∂φμ1φ,ℑ+∂∂φν1φ,ℑ=−ξκ4cosech2κφ+θ3coth2κφ+θ−11−β2+2β1−βℑ+β2ℑ22+2ξ2κ5cosech2κφ+θ3β−2β+1+β2ℑ+β3ℑ36−3β2β−1ℑ22+3β2−3β+1−β3,p2:ν2φ,ℑ=−pL−1s+β1−ssLB1+C1−12∂2∂φ2ν1φ,ℑ=2ξκ5cosech2κφ+θξκcosech23coth2κφ+θ−1+2ξκcosech2coth2κφ+θ×3β−2β+1+β2ℑ+β3ℑ36−3β2β−1ℑ22+3β2−3β+1−β3−2ξcoth3cosech2κφ+θ−11−β2+2β1−βℑ+β2ℑ22,⋮
We can calculate few terms of (26) which can be written as(34)μφ,ℑ=μ0φ,ℑ+μ1φ,ℑ+μ2φ,ℑ+⋯,νφ,ℑ=ν0φ,ℑ+ν1φ,ℑ+ν2φ,ℑ+⋯,μφ,ℑ=ξ−κcothκφ+θ−ξκ2cosech2κφ+θβℑ+1−β−ξκ4cosech2κφ+θ3coth2κφ+θ−11−β2+2β1−βℑ+β2ℑ22+2ξ2κ5cosech2κφ+θ3β−2β+1+β2ℑ+β3ℑ36−3β2β−1ℑ22+3β2−3β+1−β3+⋯,νφ,ℑ=−κ2cosech2κφ+θ−ξκ2cosech2κφ+θcothκφ+θβℑ+1−β+2ξκ5cosech2κφ+θξκcosech23coth2κφ+θ−1+2ξκcosech2coth2κφ+θ×3β−2β+1+β2ℑ+β3ℑ36−3β2β−1ℑ22+3β2−3β+1−β3−2ξcoth3cosech2κφ+θ−11−β2+2β1−βℑ+β2ℑ22+⋯.
The exact solution of (26) is(35)μφ,ℑ=ξ−κcothκφ+θ−ξℑ,νφ,ℑ=−κ2cosech2κφ+θ−ξℑ.


## 5. Implementation of Atangana–Baleanu Operator

Example 3.
Let us consider the coupled system of fractional-order WBKEs in the ABC sense:(36)DABℑβμφ,ℑ+μφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ=0,DABℑβνφ,ℑ+μφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2=0,0<β≤1,−1<ℑ≤1,−10≤φ≤10,under the initial conditions,(37)μφ,0=12−8tanh−2φ,νφ,0=16−16tanh2−2φ.
Using the Laplace transformation to (36), we obtain(38)Bβ1−βsLμφ,ℑs−sβ−1μφ,0s+β/1−β=−Lμφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,Bβ1−βsLνφ,ℑs−sβ−1νφ,0s+β/1−β=−Lμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2.
Simplify the above equation and use the initial conditions (37), and we obtain(39)Lμφ,ℑ=μφ,0s−1−βsβ+βBβsβLμφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,Lνφ,ℑ=νφ,0s−1−βsβ+βBβsβLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2.
The inverse Laplace transformation implement to (39), we obtain(40)μφ,ℑ=μφ,0−L−11−βsβ+βBβsβLμφ,ℑ∂μφ,ℑ∂φ+∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,νφ,ℑ=νφ,0−L−11−βsβ+βBβsβLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ+3∂3μφ,ℑ∂φ3−∂2νφ,ℑ∂φ2.
The LHPTM is used in (40), and we obtain(41)∑n=0∞μnφ,ℑ=μφ,0−pL−11−βsβ+βBβsβL∑n=0∞pnAn+∂∂φ∑n=0∞pnμnφ,ℑ+∂∂φ∑n=0∞pnνnφ,ℑ,∑n=0∞νnφ,ℑ=νφ,0−pL−11−βsβ+βBβsβL∑n=0∞pnBm+∑n=0∞pnCn+3∂3∂φ3∑n=0∞pnμnφ,ℑ−∂2∂φ2∑n=0∞pnνnφ,ℑ.
The nonlinear can be found with the help of He’s polynomial and can be defined as(42)A0=μ0∂μ0∂φ,A1=μ0∂μ1∂φ+μ1∂μ0∂φ,B0=μ0∂ν0∂β,B1=μ0∂ν1∂β+μ1∂ν0∂β,C0=ν0∂μ0∂φ,C1=ν0∂μ1∂φ+ν1∂μ0∂φ.
Comparing the coefficient ofp, we have(43)p0:μ0φ,ℑ=μφ,0=12−8tanh−2φ,p0:ν0φ,ℑ=νφ,0=16−16tanh2−2φ,p1:μ1φ,ℑ=−L−11−βsβ+βBβsβLA0+∂∂φμ0φ,ℑ+∂∂φν0φ,ℑ=−8sech2−2φ1BββℑβΓβ+1+1−β,p1:ν1φ,ℑ=−pL−11−βsβ+βBβsβLB0+C0+3∂3∂φ3μ0φ,ℑ−∂2∂φ2ν0φ,ℑ=−32sech2−2φtanh−2φ1BββℑβΓβ+1+1−β,p2:μ2φ,ℑ=−L−11−βsβ+βBβsβLA1+∂∂φμ1φ,ℑ+∂∂φν1φ,ℑ=−16sech2−2φ4sech2−2φ−8tanh2−2φ+3tanh−2φ×1B2β1−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1,p2:ν2φ,ℑ=−pL−11−βsβ+βBβsβLB1+C1+3∂3∂φ3μ1φ,ℑ−∂2∂φ2ν1φ,ℑ=−32sech2−2φ40sech2−2φtanh−2φ+96tanh−2φ−2tanh2−2φ−32tanh3−2φ−25sech2−2φ1−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1,⋮
We can calculate few terms of (36) which can be written as(44)μφ,ℑ=μ0φ,ℑ+μ1φ,ℑ+μ2φ,ℑ+⋯,νφ,ℑ=ν0φ,ℑ+ν1φ,ℑ+ν2φ,ℑ+⋯,μφ,ℑ=12−8tanh−2φ−8sech2−2φβℑβΓβ+1+1−β−16sech2−2φ4sech2−2φ−8tanh2−2φ+3tanh−2φ1B2β1−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1+⋯,νφ,ℑ=16−16tanh2−2φ−8sech2−2φβℑβΓβ+1+1−β−32sech2−2φ×40sech2−2φtanh−2φ+96tanh−2φ−2tanh2−2φ−32tanh3−2φ−25sech2−2φ×1−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1+⋯.
The exact solution of (36) is(45)μφ,ℑ=12−8tanh−2φ−ℑ2,νφ,ℑ=16−16tanh2−2φ−ℑ2.
Figure5 shows the actual and approximate solutions of μφ,ℑ at β=1, and Figure 6 shows the actual and approximate solutions of νφ,ℑ at β=1. Figures 7 and 8 show that the first graph has a different fractional order with respect to φ and the second graph has a different fractional order with respect to ℑ of Example 3.Figure 5
The exact and analytical solution with respect to the Atangana–Baleanu operator ofμφ,ℑ at β=1 of Example 3.
(a)(b)Figure 6
The exact and analytical solution with respect to the Atangana–Baleanu operator ofνφ,ℑ at β=1 of Example 3.
(a)(b)Figure 7
The different fractional-order solution with respect to the Atangana–Baleanu operator ofμφ,ℑ at β of Example 3.
(a)(b)Figure 8
The different fractional-order solution with respect to the Atangana–Baleanu operator ofνφ,ℑ at β of Example 3.
(a)(b)Example 4.
Let us consider the coupled system of fractional-order WBKEs in the ABC sense:(46)DABCℑβμφ,ℑ+μφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ=0,DABCℑβνφ,ℑ+μφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2=0,0<β≤1,−1<ℑ≤1,−10≤φ≤10,under the initial conditions,(47)μφ,0=ξ−κcothκφ+θ,νφ,0=−κ2cosech2κφ+θ.
Using the Laplace transformation to (46), we obtain(48)Bβ1−βsLμφ,ℑs−sβ−1μφ,0s+β/1−β=−Lμφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,Bβ1−βsLνφ,ℑs−sβ−1νφ,0s+β/1−β=−Lμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2.
Simplifying the above equation and using the initial conditions (47), we obtain(49)Lμφ,ℑ=μφ,0s−1−βsβ+βBβsβLμφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,Lνφ,ℑ=νφ,0s−1−βsβ+βBβsβLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2.
The inverse Laplace transformation is implemented to (49), and we obtain(50)μφ,ℑ=μφ,0−L−11−βsβ+βBβsβLμφ,ℑ∂μφ,ℑ∂φ+12∂μφ,ℑ∂φ+∂νφ,ℑ∂φ,νφ,ℑ=νφ,0−L−11−βsβ+βBβsβLμφ,ℑ∂νφ,ℑ∂φ+νφ,ℑ∂μφ,ℑ∂φ−12∂2νφ,ℑ∂φ2.
The LHPTM is used in (50), and we obtain(51)∑n=0∞μnφ,ℑ=μφ,0−pL−11−βsβ+βBβsβL∑n=0∞pnAn+12∂∂φ∑n=0∞pnμnφ,ℑ+∂∂φ∑n=0∞pnνnφ,ℑ,∑n=0∞νnφ,ℑ=νφ,0−pL−11−βsβ+βBβsβL∑n=0∞pnBn+∑n=0∞pnCn−12∂2∂φ2∑n=0∞pnνnφ,ℑ.
The nonlinear can be found with the help of He’s polynomial and can be defined as(52)A0=μ0∂μ0∂φ,A1=μ0∂μ1∂φ+μ1∂μ0∂φ,B0=μ0∂ν0∂β,B1=μ0∂ν1∂β+μ1∂ν0∂β,C0=ν0∂μ0∂φ,C1=ν0∂μ1∂φ+ν1∂μ0∂φ.
Comparing the coefficient ofp, we have(53)p0:μ0φ,ℑ=μφ,0=ξ−κcothκφ+θ,p0:ν0φ,ℑ=νφ,0=−κ2cosech2κφ+θ,p1:μ1φ,ℑ=−L−11−βsβ+βBβsβLA0+12∂∂φμ0φ,ℑ+∂∂φν0φ,ℑ=−ξκ2cosech2κφ+θ1BββℑβΓβ+1+1−β,p1:ν1φ,ℑ=−pL−11−βsβ+βBβsβLB0+C0−12∂2∂φ2ν0φ,ℑ=−ξκ2cosech2κφ+θcothκφ+θ1BββℑβΓβ+1+1−β,p2:μ2φ,ℑ=−L−11−βsβ+βBβsβLA1+12∂∂φμ1φ,ℑ+∂∂φν1φ,ℑ=−ξκ4cosech2κφ+θ3coth2κφ+θ−11B21−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1+2ξ2κ5cosech2κφ+θ1B31−β3+3β1−β2ℑβΓ2β+1+3β21−βℑ2βΓ2β+1+β3ℑ3βΓ3β+1,p2:ν2φ,ℑ=−pL−11−βsβ+βBβsβLB1+C1−12∂2∂φ2ν1φ,ℑ=2ξκ5cosech2κφ+θξκcosech23coth2κφ+θ−1+2ξκcosech2coth2κφ+θ×1B31−β3+3β1−β2ℑβΓ2β+1+3β21−βℑ2βΓ2β+1+β3ℑ3βΓ3β+1−2ξcoth3cosech2κφ+θ−11B21−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1,⋮
We can calculate few terms of (46) which can be written as(54)μφ,ℑ=μ0φ,ℑ+μ1φ,ℑ+μ2φ,ℑ+⋯,νφ,ℑ=ν0φ,ℑ+ν1φ,ℑ+ν2φ,ℑ+⋯,μφ,ℑ=ξ−κcothκφ+θ−ξκ2cosech2κφ+θ1BββℑβΓβ+1+1−β−ξκ4cosech2κφ+θ3coth2κφ+θ−11B21−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1+2ξ2κ5cosech2κφ+θ1B31−β3+3β1−β2ℑβΓ2β+1+3β21−βℑ2βΓ2β+1+β3ℑ3βΓ3β+1+⋯,νφ,ℑ=−κ2cosech2κφ+θ−ξκ2cosech2κφ+θcothκφ+θ1BββℑβΓβ+1+1−β+2ξκ5cosech2κφ+θξκcosech23coth2κφ+θ−1+2ξκcosech2coth2κφ+θ×1B31−β3+3β1−β2ℑβΓ2β+1+3β21−βℑ2βΓ2β+1+β3ℑ3βΓ3β+1−2ξcoth3cosech2κφ+θ−1×1B21−β2+2β1−βℑβΓβ+1+β2ℑ2βΓ2β+1+⋯.
The exact solution of (46) is(55)μφ,ℑ=ξ−κcothκφ+θ−ξℑ,νφ,ℑ=−κ2cosech2κφ+θ−ξℑ.


## 6. Conclusions

In this paper, the LHPTM was considered to achieve an analytical result for the fractional-order Whitham–Broer–Kaup equations considering the Caputo–Fabrizio and Atangana–Baleanu operators. Analytical results were obtained for different fractional orderβ. Both suggested operators have shown to be critical mathematical tools for scientists working in numerous fields of applied sciences. The polynomial expansion well-thought-out in the LHPTM permits to achieve an infinite series result for the Whitham–Broer–Kaup equations. This technique established a general system to achieve fractional-order models’ analytical results, and the results are obtained in the series form, which converges quickly; the LHPTM is used for investigating other nonlinear fractional systems of partial differential equations.



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