# Analytical Investigation of Some Dynamical Systems by ZZ Transform with Mittag–Leffler Kernel

Authors: Mounirah Areshi; Muhammad Naeem; Noorolhuda Wyal
Journal: Complexity
 (2022)

---

## Abstract

In this work, ZZ transformation is combined with the Adomian decomposition method to solve the dynamical system of fractional order. The derivative of fractional order is represented in the Atangana–Baleanu derivative. The numerical examples are combined for their approximate-analytical solution. It is explored using graphs that indicate that the actual and approximation results are close to each other, demonstrating the method’s usefulness. Fractional-order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series form solution that is quickly convergent to exact solutions. The acquired results suggest that the novel analytical solution technique is simple to use and very successful at assessing complicated problems that arise in related fields of research and technology.


---

## Body

## 1. Introduction

Because of their extensive applications in many science and engineering disciplines, fractional differential equations have sparked much attention in recent years. Critical phenomena well characterize differential equations of fractional order in electromagnetics, finance, viscoelasticity, acoustics, material science, and electrochemistry. Barkai et al. [1], Mainardi [2], Tadjeran and Meerschaert [3], Meerschaert et al. [4], and Magin et al. [5] just released a review article on fractional signals and systems, including control theory applications. The edited volume of Machado contains several applications of fractional calculus, such as image processing [6]. The importance and necessity of fractional calculus can be seen in several applications in transdisciplinary disciplines. Miller and Ross [7], Oldham and Spanier [8], Podlubny [9], Kilbas et al. [10], Samko et al. [11], Caponetto [12], and Diethelm [13] have all authored essential studies on the fractional derivative and fractional differential equations. A review study on the recent history of fractional calculus was written by Machado et al. [14]. An article on recent developments in the theory of abstract differential equations with fractional derivatives was published by Hernandez et al. [15]. These publications provide a systematic explanation of fractional calculus, including the existence and uniqueness of solutions and various analytical methods for solving fractional differential equations, such as Green’s function method, power series approach, Mellin transform method, and others. No method in the literature produces a precise solution for nonlinear fractional differential equations (16) and (17). Using linearization or perturbation approaches, only approximate answers can be obtained. All of these push us to develop a numerical approach for fractional differential equations that is both efficient and accurate [18–21]. Chaos theory, heat transfer, variational issues, and other fields have used the Atangana–Baleanu fractional differential extensively. Recently, a fractional-differential mask based on a fractional Gaussian kernel with Atangana–Baleanu fractional differential has been published in the literature for the detection of blood vessels in retinal pictures, with the suggested method’s efficacy compared to other well-known approaches. Furthermore, it discusses the underlying differences between power-law, exponential-law, and Mittag–Leffler kernels, as well as their potential applications in diverse domains.This paper establishes a connection between the Aboodh transformation (AT), ZZ transformation (ZZT), and Laplace transformation (LT), with several applications mentioned in [22–24]. The ZZT was then employed to define fractional Atangana–Baleanu Caputo operators and characterize Riemann–Liouville senses using theorems. Later, we solved several test problems stated in the Atangana–Baleanu sense using this ZZ transform. The current author’s contributions to this study are (i) applying the ZZ transform to solve fractional differential equations expressed in the Atangana–Baleanu derivative and (ii) establishing the connection between the Laplace, Aboodh, and ZZ transformations. A few well-known transforms that the ZZ transform generalizes can be related to other well-known transforms. Divide the ZZ transform by the adjusted variable to get the natural transform. Relationships with other integral transformations are also included in this work in terms of theorems. This transformation has the advantage of converging to the Sumudu transformation, which is advantageous when solving fractional differential equations with variable coefficients, such as [25–27].Adomian (1980) established the Adomian decomposition technique (ADM), an efficient method for finding explicit and numerical solutions to a larger and more general class of differential systems representing real-world issues [28–30]. This strategy effectively addresses initial and boundary value problems, linear and nonlinear, ordinary and partial differential equations, and stochastic systems. Furthermore, this approach does not require any linearization or perturbation. ADM has been used extensively in the last two decades since it yields approximate analytical solutions for nonlinear problems, and there has been much interest in utilizing it to solve fractional differential equations (31)–(33).The ZZ decomposition method has the following advantages with respect to Adomian decomposition method:(i)
ZZ decomposition method required small calculations as compared to the Adomian decomposition method(ii)
The fractional derivative is simplified by using the ZZ transformation first and then applying decomposition method while it is not the case if we use the Adomian decomposition method directly(iii)
The initial conditions/boundary conditions are used directly in the ZZ decomposition method, and it mostly avoids the extra calculations of Adomian polynomialsThe novelty of the present work is to deal with the analytical solutions of important fractional-order some dynamical systems. The fractional-order of some dynamical systems have many applications in physical sciences, and therefore, different graphs are presented to show various dynamics of fractional parabolic equations. The solutions are obtained in rather simpler way as compared to other techniques. The current study has been structured as follows: in Section2, some basic notions of basic definitions of ZZ transformation are described. In Section 3, we give an analysis of the suggested technique. In Section 4, we provide current solutions which suggested equations explaining how to implemented the suggested technique. Finally, the conclusion is provided.


## 2. Preliminaries

Definition 1.
The function set of the Aboodh transform (AT) is defined as(1)B=hρ:∃M,n1,n2>0,hρ<Me−sρ,and is given as [22, 23](2)Ahρ=1ς∫0∞hρe−ςρdρ,ρ>0andn1≤ς≤n.Theorem 1.
Now, we considerG and F as the Aboodh and Laplace transforms of hρ∈B; then [24, 25],(3)Gς=Fςς.
Zafar [26] was the first to develop the ZZ transform. It is mixture of the Laplace and Aboodh integral transforms. The ZZ transform is expressed in the following.Definition 2.
(ZZ transform). Suppose thathρ∀ρ≥0 is a function, then the ZZ transformation Zϱ,ς of hρ is defined as [26](4)ZZhρ=Zϱ,ς=ς∫0∞hϱρe−ςρdρ.
The ZZ transformation is linear, just as the Aboodh and Laplace transforms. The MLF is a function that is given as an extension of the exponential term.(5)Eδz=∑m=0∞zmΓ1+mδ,Reδ>0.Definition 3.
The Atangana–Baleanu Caputo derivative of a functionνφ,ρ∈H1a,b; then, for δ∈0,1, it is defined as [27](6)ABCaDρδνφ,ρ=ψδ1−δ∫aρν′φ,ρEδ−δρ−ηδ1−δdη.Definition 4.
Let the Riemann–Liouville Atangana–Baleanu derivativeνφ,ρ∈H1a,b; then, for δ∈0,1, it is given as [27](7)DρδaABRνφ,η=ψδ1−δddρ∫aρνφ,ηEδ−δρ−ηδ1−δdη,with the condition ψ0=ψ1=1, ψδ is a function, and b>a.Theorem 2.
The Laplace transform of Riemann–Liouville Atangana–Baleanu derivative and Atangana–Baleanu Caputo are, respectively, defined as [27](8)LDρδaABCνφ,ρς=ψδ1−δςδLνφ,ρ−ςδ−1νφ,0ςδ+δ/1−δand(9)LDρδaABRνφ,ρς=ψδ1−δςδLνφ,ρςδ+δ/1−δ.
The theorems that follow are based on the idea thathρ∈H1a,b,b>a, and δ∈0,1.Theorem 3.
The Aboodh transformation of Atangana–Baleanu Riemann–Liouville derivative is defined as [25](10)Gς=ADρδaABRνφ,ρς=1ςψδ1−δςδLνφ,ρςδ+δ/1−δ.Proof 1.
Using Theorem1 and equation (3), we arrive to the required solution. The relationship among the transforms of ZZ and Aboodh is given in the following theorem.Theorem 4.
The Aboodh transform of Atangana–Baleanu Caputo derivative is defined as [25](11)Gς=ADρδaABCνφ,ρς=1ςψδ1−δςδLνφ,ρ−ςδ−1νφ,0ςδ+δ/1−δ.Proof 2.
Using Theorem1 and equation (2), we can discover the desired solution.Theorem 5.
IfZϱ,ς and Gς are the Aboodh and ZZ transforms of hρ∈B, then we obtain the following [25]:(12)Zϱ,ς=ς2ϱ2Gςϱ.Proof 3.
(TheZZ transform definitions). We get(13)Zϱ,ς=ς∫0∞hϱρe−ςρdρ.
Putϱρ=ρ in (13); we get(14)Zϱ,ς=ςϱ∫0∞hρe−πzϱdρ.
The right-hand side of (14) may be expressed as(15)Zϱ,ς=ςϱFςϱ,where F. expresses the Laplace transformation of hρ. Using Theorem 1, (15) can be defined as(16)Zϱ,ς=ςϱFςϱςϱ×ςϱ=ςϱ2Gςϱ,where G. defines the Aboodh transform of hρ.Theorem 6.
ZZ transformation of hρ=ρδ−1 is defined as(17)Zϱ,ς=Γδϱςδ−1Proof 4.
The Aboodh transformation ofhρ=ρδ,δ≥0, is(18)Gs=Γδςδ+1,now,Gςϱ=Γδϱδ+1ςδ+1.
Applying (17), we achieve(19)Zϱ,ς=ς2ϱ2Gςϱ=ς2ϱ2Γδϱδ+1ςδ+1=Γδϱςδ−1.Theorem 7.
Letδ,ω∈C and Reδ>0; then, the ZZ transformation of Eδωρδ is defined as [25](20)ZZEδωρδ=Zϱ,ς=1−ωϱςδ−1.Proof 5.
We know that Aboodh transform ofEδωρδ is defined as(21)Gς=Fςς=ςδ−1ςςδ−ω.
So,(22)Gςϱ=ς/ϱδ−1ς/ϱς/ϱδ−ω.
Applying Theorem9, we achieve(23)Zϱ,ς=ςϱ2Gςϱ=ςϱ2ς/ϱδ−1ς/ϱς/ϱδ−ω=ς/ϱδς/ϱδ−ω=1−ωϱ/ςδ−1.Theorem 8.
IfZϱ,ς and Gs are the Aboodh and ZZ transforms of hρ, then the Atangana–Baleanu Caputo ZZ transformation derivative is defined as [25](24)ZZDρδ0ABChρ=ψδ1−δςδ+2/ϱδ+2Gς/ϱ−ςδ/ϱδf0ςδ/ϱδ+δ/1−δ.Proof 6.
Applying equations (1) and (5), we get(25)Gςϱ==ϱςψδ1−δς/ϱδ+1Gς/ϱ−ς/ϱδ−1f0ς/ϱδ+δ/1−δ.
So, the Atangana–Baleanu Caputo ofZZ transformation is defined as(26)Zϱ,ς=ςϱ2Gςϱ=ςϱ2ϱςψδ1−δς/ϱδ+1Gς/ϱ−ς/ϱδ−1f0ς/ϱδ+δ/1−δ=ψδ1−δς/ϱδ+2Gς/ϱ−ς/ϱδf0ς/ϱδ+δ/1−δ.Theorem 9.
Let us suppose thatZϱ,ς and Gs are the Aboodh and ZZ transforms of hρ. Then, the Atangana–Baleanu Riemann–Liouville ZZ transform derivative is defined as [25](27)ZZDρδ0ABRfρ=ψδ1−δ5δ+2/ϱδ+2Gς/ϱςμ/ϱμ+δ/1−δ.Proof 7.
Applying equations (1) and (4), we get(28)Gςϱ=ϱςψδ1−δς/ϱδ+1Gς/ϱς/ϱδ+δ/1−δ.
From (16), the ZZ transform of Riemann–Liouville Atangana–Baleanu is defined as(29)Zϱ,ς=ςϱ2Gςϱ=ςϱ2ϱςψδ1−δς/ϱδ+1Gς/ϱς/ϱδ+δ/1−δ=ψδ1−δς/ϱδ+2Gς/ϱς/ϱδ+δ/1−δ.


## 3. Idea of MDM

Consider the fractional order partial differential equation by MDM:(30)Dρδνφ,ρ=ℋ¯1ν,φ+N1ν,φ,0<δ≤1,with the initial condition(31)νφ,0=ξφ,where Dρδ=∂δ/∂ρδ is the Atangana–Baleanu fractional derivative of order δ; ℋ¯1 is linear and N1 nonlinear terms, respectively. On both sides, we use ZZ transformation of (30), to achieve(32)ZDρδνφ,ρ=Zℋ¯1ν,φ+N1ν,φ.By the differentiation property of ZZ transformation, we get(33)ψδ1−δ+δϱςδνφ,ρ−ϱςνφ,0=Zℋ¯1ν,φ+N1ν,φ.(33) implies that(34)νφ,ρ=ϱςνφ,0+1−δ+δϱςδψδZℋ¯1ν,φ+N1ν,φ.Applying the ZZ inverse transformation of (34), we get(35)νφ,ρ=νφ,0+Z−11−δ+δϱςδψδZℋ¯1ν,φ+N1ν,φ.MDM determines the infinite sequence’s result ofνφ,ρ:(36)νφ,ρ=∑m=0∞νmφ,ρ.The nonlinear functions can be found with the help of Adomian polynomialsN1 which is expressed as(37)N1ν,φ=∑m=0∞Am.The Adomian polynomials can show all types of nonlinearity as(38)Am=1m!∂m∂ℓmN1∑k=0∞ℓkνk,∑k=0∞ℓkφkℓ=0.Putting (36) and (38) into (35), it gives(39)∑m=0∞νmφ,ρ=νφ,0+Z−11−δ+δς/ϱδψδZℋ¯1∑m=0∞νm,∑m=0∞φm+∑m=0∞Am.The following terms are described:(40)ν0φ,ρ=νφ,0,ν1φ,ρ=Z−11−δ+δϱ/ςδψδZℋ¯1ν0,φ0+A0.The general form form≥1 is determined as(41)νm+1φ,ρ=Z−11−δ+δϱ/ςδψδZℋ¯1νm,φm+Am.


## 4. Numerical Examples

Example 1.
Here, we take the following FPDE:(42)Dρδμ−∂ν∂ζ+ν+μ=0,Dρδν−∂μ∂ζ+ν+μ=0,δ,∈0,1,.with initial source(43)μζ,0=sinhζνζ,0=coshζ.
The exact result atδ=1 is (1) μφ,ρ=sinhφ−ρ and (2) νφ,ρ=coshφ+ρ.
Applying ZZT (42), we get(44)Z∂δμ∂ρδ=Z∂ν∂φ−ν−μ,Z∂δν∂ρδ=Z∂μ∂φ−ν−μ,ψδ1−δ+δϱ/ςδZμφ,ρ−ϱςμφ,0=Z∂ν∂φ−ν−μ,ψδ1−δ+δϱ/ςδZνφ,ρ−ϱςνφ,0=Z∂μ∂φ−ν−μ.
We get(45)Zμφ,ρ=ϱςμφ,0+1−δ+δϱ/ςδψδZ∂ν∂φ−ν−μ,Zνφ,ρ=ϱςνφ,0+1−δ+δϱ/ςδψδZ∂μ∂φ−ν−μ.
Applying the inverse ZZT to (45), we get(46)μφ,ρ=μφ,0+Z−11−δ+δϱ/ςδψδZ∂ν∂φ−ν−μ,νφ,ρ=νφ,0+Z−11−δ+δϱ/ςδψδZ∂μ∂φ−ν−μ.
Decomposition results forμφ,ρ and νφ,ρ can be expressed as(47)μφ,ρ=∑ℵ=0∞μℵφ,ρ,andνφ,ρ=∑ℵ=0∞νℵφ,ρ,∑ℵ=0∞μℵφ,ρ=μφ,0+Z−11−δ+δϱ/ςδψδZ∂∑ℵ=0∞νℵφ,ρ∂φ−∑ℵ=0∞νℵφ,ρ−∑ℵ=0∞μℵφ,ρ,∑ℵ=0∞νℵφ,φ,ρ=νφ,0+Z−11−δ+δϱ/ςδψδZ∂∑ℵ=0∞μℵφ,ρ∂φ−∑ℵ=0∞νℵφ,ρ−∑ℵ=0∞μℵφ,ρ.
Furthermore,(48)∑ℵ=0∞μℵφ,ρ=sinhφ+Z−11−δ+δϱ/ςδψδZ∂∑ℵ=0∞νℵφ,ρ∂φ−∑ℵ=0∞νℵφ,ρ−∑ℵ=0∞μℵφ,ρ,∑ℵ=0∞νℵφ,ρ=coshφ+Z−11−δ+δϱ/ςδψδZ∂∑ℵ=0∞μℵφ,ρ∂φ−∑ℵ=0∞νℵφ,ρ−∑ℵ=0∞μℵφ,ρ.
The component comparison in (48) provides the following recursive MDM algorithm:(49)μ0φ,ρ=sinhφ,ν0φ,ρ=coshφ.
Forℵ=0,(50)μ1φ,ρ=−coshφ1ψδ1−δ+δρδΓδ+1,ν1φ,ρ=−sinhφ1ψδ1−δ+δρδΓδ+1.
Forℵ=1,(51)μ2φ,ρ=−coshφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+sinhφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+coshφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1,ν2φ,ρ=−sinhφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+coshφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+sinhφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1.
Forℵ=2,(52)μ3φ,ρ=−coshφρ3δΓ3δ+1,ν3φ,ρ=sinhφρ3δΓ3δ+1.⋮
Similar toℵ>2, MDM can be used to determine the remaining terms of μm and νm. In general, MDM’s solution is as follows:(53)μφ,ρ=∑ℵ=0∞μℵφ,ρ=μ0φ+μ1φ+μ2φ+μ3φ+⋯,νφ,ρ=∑ℵ=0∞νℵφ,ρ=ν0φ+ν1φ+ν2φ+ν3φ+⋯,μφ,ρ=∑ℵ=0∞μℵφ=sinhφ−coshφ1ψδ1−δ+δρδΓδ+1−coshφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+sinhφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+coshφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1⋯,νφ,ρ=∑ℵ=0∞νℵφ=coshφ−sinφ1ψδ1−δ+δρδΓδ+1−sinhφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+coshφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+sinhφ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1⋯,μφ,ρ=sinhφ1+1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+⋯−coshφ1ψδ1−δ+δρδΓδ+1+1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1−1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+⋯,νφ,ρ=coshφ1+1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+⋯−sinhφ1ψδ1−δ+δρδΓδ+1+1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1−1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+⋯.
Setδ=1 in (42); we get(54)μφ,ρ=sinhφ1+ρ22!+ρ44!+⋯−coshφρ1!+ρ33!+ρ55!+⋯=sinhφ−ρ,νφ,ρ=coshφ1+ρ22!+ρ44!+⋯−sinhφρ1!+ρ33!+ρ55!+⋯=coshφ+ρ.
The exact results are atδ=1.(55)μφ,ρ=sinhφ−ρ,νφ,ρ=coshφ+ρ.
We analyze the solution figures of the problem, which have been investigated by applying the ZZ decomposition method in the sense of the Atangana–Baleanu operator. Figure1 represents the three-dimensional solution-figures for variables μ of example 1 at fractional order δ=1 and 0.8, respectively; Figure 2 represents different fractional order of δ=0.6 and 0.4; and Figure 3 represents that at δ. In Figure 4, different fractional order with respect to φ and ρ. It is observed that the ZZ decomposition method solution-figures are identical and in close contact with each other. In the same way, Figures 5–8 show different fractional order graphs of δ at ν of Example 1.Figure 1
(a) The exact and approximate solution atδ=1 and (b) second fractional order at δ=0.8.
(a)(b)Figure 2
The graph shows the fractional order atδ=0.6 and 0.4.
(a)(b)Figure 3
The graph shows different fractional orders ofδ.Figure 4
The graph of the two dimensions of different fractional orders atδ with respect to φ and ρ.Figure 5
(a) The exact and approximate solution atδ=1 and (b) second fractional order at δ=0.8.
(a)(b)Figure 6
The graph shows the fractional order atδ=0.6 and 0.4.
(a)(b)Figure 7
The graph shows different fractional orders ofδ.Figure 8
The graph of the two dimensions of different fractional orders atδ with respect to φ and ρ.Example 2.
Here, we take the following FPDE:(56)Dρδμ+νφωχ−νχωφ=−μ,Dρδν+μχωφ+μχωφ=ν,Dρδω+μφνχ+μχνφ=ω,δ∈0,1,.with initial sources(57)μφ,χ,0=expφ+χ,νφ,χ,0=expφ−χ,ωφ,χ,0=exp−φ+χ.
The exact solution atδ=1 is(58)μφ,χ,ρ=expφ+χ−ρ,νφ,χ,ρ=expφ−χ+ρ,ωφ,χ,ρ=exp−φ+χ+ρ.
Using ZZT equation (33), it can be written as(59)Z∂δμ∂ρδ=Z−μ+νφωχ−νχωφ,Z∂δν∂ρδ=Zν−μχωφ−μχωφ,Z∂δω∂ρδ=Zω−μφνχ−μχνφ,ψδ1−δ+δϱ/ςδZμφ,χ,ρ−ϱςμφ,χ,0=Z−μ+νφωχ−νχωφ,ψδ1−δ+δϱ/ςδZνφ,χ,ρ−ϱςνφ,χ,0=Zν−μχωφ−μχωφ,ψδ1−δ+δϱ/ςδZωφ,φ,ρ−ϱςωφ,χ,0=Zω−μφνχ−μχνφ.
After simplification, we obtain(60)ψδ1−δ+δϱ/ςδZμφ,χ,ρ=ϱςμφ,χ,0+Z−μ+νφωχ−νχωφ,ψδ1−δ+δϱ/ςδZνφ,χ,ρ=ϱςνφ,χ,0+Zν−μχωφ−μχωφ,ψδ1−δ+δϱ/ςδZωφ,φ,ρ=ϱςωφ,χ,0+Zω−μφνχ−μχνφ,Zμφ,χ,ρ=ϱςμφ,χ,0+1−δ+δϱ/ςδψδZ−μ+νφωχ−νχωφ,Zνφ,χ,ρ=ϱςνφ,χ,0+1−δ+δϱ/ςδψδZν−μχωφ−μχωφ,Zωφ,φ,ρ=ϱςωφ,χ,0+1−δ+δϱ/ςδψδZω−μφνχ−μχνφ.
Taking inverse ZZT of (60), we obtain(61)μφ,χ,ρ=μφ,χ,0+Z−11−δ+δϱ/ςδψδZ−μ+νφωχ−νχωφ,νφ,χ,ρ=νφ,χ,0+Z−11−δ+δϱ/ςδψδZν−μχωφ−μχωφ,ωφ,χ,ρ=ωφ,χ,0+Z−11−δ+δϱ/ςδψδZω−μφνχ−μχνφ.
Assume decomposition solutions for variablesμφ,χ,ρ, νφ,χ,ρ, and ωφ,χ,ρ, it can be written as(62)μφ,χ,ρ=∑ℵ=0∞μℵφ,χ,ρ,νφ,χ,ρ=∑ℵ=0∞νℵφ,χ,ρ,andωφ,χ,ρ=∑ℵ=0∞ωℵφ,χ,ρ.
Remember thatνφωχ=∑ℵ=0∞Am, νχωφ=∑ℵ=0∞ℬℵ, μφωχ=∑ℵ=0∞Cℵ, μχωφ=∑ℵ=0∞Dℵ,μφνχ=∑ℵ=0∞Eℵ, and μχνφ=∑ℵ=0∞ℱℵ are the Adomian polynomials and the nonlinear terms were characterized, which can be further simplified as(63)∑ℵ=0∞μℵφ,χ,ρ=μφ,χ,0+Z−11−δ+δϱ/ςδψδZ−∑ℵ=0∞μℵφ,χ,ρ+∑ℵ=0∞Aℵ−∑ℵ=0∞ℬℵ,∑ℵ=0∞νℵφ,χ,ρ=νφ,χ,0+Z−11−δ+δϱ/ςδψδZ∑ℵ=0∞νℵφ,χ,ρ−∑ℵ=0∞Cℵ+∑ℵ=0∞Dℵ,∑ℵ=0∞ωℵφ,χ,ρ=ωφ,φ,δ,0+Z−11−δ+δϱ/ςδψδZ∑ℵ=0∞ωℵφ,χ,ρ−∑ℵ=0∞Zℵ+∑ℵ=0∞ℱℵ.
Using (38), the nonlinearity in the given problem can be expressed as(64)A0=∂ν0∂φ∂ω0∂χ,A1=∂ν0∂φ∂ω1∂χ+∂ν1∂φ∂ω0∂χ,ℬ0=∂ν0∂χ∂ω0∂φ,ℬ1=∂ν0∂χ∂ω1∂φ+∂ν0∂φ∂ω1∂χ,C0=∂μ0∂φ∂ω0∂χ,C1==∂μ1∂φ∂ω0∂χ+∂μ0∂φ∂ω1∂χ,D0=∂μ0∂χ∂ω0∂φ,D1=∂μ0∂χ∂ω1∂φ+∂μ1∂χ∂ω0∂φ,E0=∂μ0∂φ∂ν0∂φ,E1=∂μ1∂φ∂ν0∂χ+∂μ0∂φ∂ν1∂χ,ℱ0=∂μ0∂χ∂ν0∂φ,ℱ1=∂μ1∂χ∂ν0∂φ+∂μ0∂φ∂ν1∂χ.
The component comparison provides the following recursive MDM algorithm:(65)μ0φ,χ,ρ=μφ,χ,0,ν0φ,χ,ρ=νφ,χ,0,ω0φ,χ,ρ=ωφ,χ,0,μ1φ,χ,ρ=Z−11−δ+δϱ/ςδψδZ−μ0φ,χ,ρ+A0−ℬ0,ν1φ,χ,ρ=Z−11−δ+δϱ/ςδψδZν0φ,χ,ρ−C0+D0,ω1φ,χ,ρ=Z−11−δ+δϱ/ςδψδZω0φ,χ,ρ−E0+ℱ0,μℵ+1φ,χ,ρ=Z−11−δ+δϱ/ςδψδZ−μℵφ,χ,ρ+Aℵ−ℬℵ,νℵ+1φ,χ,ρ=Z−11−δ+δϱ/ςδψδZνℵφ,χ,ρ−Cℵ+Dℵ,ωℵ+1φ,χ,ρ=Z−11−δ+δϱ/ςδψδZωℵφ,χ,ρ−Eℵ+ℱℵ,μ0φ,χ,ρ=expφ+χ,ν0φ,χ,ρ=expφ−χ,ω0φ,χ,ρ=exp−φ+χ.
Forℵ=0,(66)μ1φ,χ,ρ=−expφ+χ1ψδ1−δ+δρδΓδ+1,ν1φ,χ,ρ=expφ−χ1ψδ1−δ+δρδΓδ+1,ω1φ,χ,ρ=exp−φ+χ1ψδ1−δ+δρδΓδ+1.
Forℵ=1,(67)μ2φ,χ,ρ=expφ+χ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1,ν2φ,χ,ρ=expφ−χ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1,ω2φ,χ,ρ=exp−φ+χ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1.
Forℵ=2,(68)μ3φ,χ,ρ=−expφ+χ1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2,ν3φ,χ,ρ=expφ−χ1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2,ω3φ,χ,ρ=exp−φ+χ1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2.⋮
In same manner, the remaining terms ofμℵ,νℵ, and ωℵ for ℵ>3 can be calculated easily by using MDM. The general solution of MDM is given by(69)μφ,χ,ρ=∑ℵ=0∞μℵφ,χ,ρ=μ0φ,χ,ρ+μ1φ,χ,ρ+μ2φ,χ,ρ+μ3φ,χ,ρ+⋯,νφ,χ,ρ=∑ℵ=0∞νℵφ,χ,ρ=ν0φ,χ,ρ+ν1φ,χ,ρ+ν2φ,χ,ρ+ν3φ,χ,ρ+⋯,ωφ,χ,ρ=∑ℵ=0∞ωℵφ,χ,ρ=ω0φ,χ,ρ+ω1φ,χ,ρ+ω2φ,χ,ρ+ω3φ,χ,ρ+⋯,μφ,χ,ρ=∑ℵ=0∞μℵφ,χ,ρ=expφ+χ−expφ+χ1ψδ1−δ+δρδΓδ+1+expφ+χ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1−expφ+χ1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2⋯,νφ,χ,ρ=∑ℵ=0∞μℵφ,χ,ρ=expφ−χ+expφ−χ1ψδ1−δ+δρδΓδ+1+expφ−χ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+expφ−χ1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2+⋯,ωφ,χ,ρ=∑ℵ=0∞μℵφ,χ,ρ=exp−φ+χ+exp−φ+χ1ψδ1−δ+δρδΓδ+1+exp−φ+χ1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+exp−φ+χ1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2+⋯,μφ,χ,ρ=expφ+χ1−1ψδ1−δ+δρδΓδ+1+1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1−1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2+⋯,νφ,χ,ρ=expφ−χ1+1ψδ1−δ+δρδΓδ+1+1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2+⋯,ωφ,χ,ρ=exp−φ+χ1+1ψδ1−δ+δρδΓδ+1+1Bδ21−δ2+2δ1−δρδΓδ+1+δ2ρ2δΓ2δ+1+1Bδ31−δ3+3δ1−δ2ρδΓδ+1+δ21−δρ2δ+1Γ2δ+2+2δ21−δρ2δΓ2δ+1+δ3ρ2δ+1Γ2δ+2⋯.
Settingδ=1 in (69), we get(70)μφ,χ,ρ=expφ+χ1−ρΓ2+ρ2Γ3−ρ3Γ4⋯,νφ,χ,ρ=expφ−χ1+ρΓ2+ρ2Γ3+ρ3Γ4⋯,ωφ,χ,ρ=exp−φ+χ1+ρΓ2+ρ2Γ3+ρ3Γ4⋯.μφ,χ,ρ=expφ+χ1−ρ1!+ρ22!−ρ33!⋯,νφ,χ,ρ=expφ−χ1+ρ1!+ρ22!+ρ33!⋯,ωφ,χ,ρ=exp−φ+χ1+ρ1!+ρ22!+ρ33!⋯,which is the MDM solution in closed form of equation (34). When δ=1,(71)μφ,χ,ρ=expφ+χ−ρ,νφ,χ,ρ=expφ−χ+ρ,ωφ,χ,ρ=exp−φ+χ+ρ.
We analyze the solution-figures of the problem, which have been investigated by applying the ZZ decomposition method in the sense of the Atangana–Baleanu operator. Figure9 represents the two-dimensional solution-figures for variables μ of example 2 and second graph of different fractional order δ. Figure 10 represents the two-dimensional solution-figures for variables ν of example 2 and second graph of different fractional-order δ. Figure 11 represents the two-dimensional solution-figures for variables ω of example 2 and second graph of different fractional-order δ. It is observed that the ZZ decomposition method solution-figures are identical and in close contact with each other.Figure 9
(a) The exact and approximate solution atδ=1 and (b) the different fractional order of δ.
(a)(b)Figure 10
(a) The exact and approximate solution atδ=1 and (b) the different fractional order of δ.
(a)(b)Figure 11
(a) The exact and approximate solution atδ=1 and (b) the different fractional order of δ.
(a)(b)


## 5. Conclusion

In this paper, some important system of fractional partial differential equations is considered for its analytical solution using the ZZ decomposition method. It has been demonstrated from the figures that the present techniques have the greater tendency to analyze the results of the given models. The problems results at different time fractional are investigated which cover the various aspects of the proposed models and proposed method. The results at different fractional orders are suggested and shown a very closed convergence phenomena of the fractional results towards integer order solutions. The graph has shown a very consistent relation between the integer and fractional orders results. It is noted that the effective and straight-forward solution of the ZZ decomposition method implies its applicability to solve other fractional partial differential equations.



---

*Source: 7635696-2022-09-24.xml*