# On the Analytical Treatment for the Fractional-Order Coupled Partial Differential Equations via Fixed Point Formulation and Generalized Fractional Derivative Operators

Authors: Saima Rashid; Sobia Sultana; Nazeran Idrees; Ebenezer Bonyah
Journal: Journal of Function Spaces
 (2022)

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## Abstract

High-dimensional fractional equation investigation is a cutting-edge discipline with considerable pragmatic and speculative consequences in engineering, epidemiology, and other scientific disciplines. In this study, a hybrid Jafari transform mixed with the Adomian decomposition method for obtaining the analytical solution to Burgers’ problem is provided. Burgers’ equation is a vital mathematical expression that appears in a variety of computational modelling fields, including fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. By considering a hybrid transform, semianalytical techniques are constructed for the Caputo and Atangana-Baleanu fractional derivative operators. Besides that, existence and uniqueness analyses are carried out with the aid of the Banach contraction-fixed point theory. To obtain the models’ findings, we employed the Jafari transform on fractional-order Burger equations (BEs), supplemented by the inverse Jafari transform. The projected findings for the fractional BEs have been depicted visually. Ultimately, numerical figures are provided to validate the practicality and efficacy. The solution obtained by employing the supplied methodologies has been validated to have the appropriate rate of convergence to the precise solution. The main advantage of the suggested method is the relatively small number of computations performed. It can also be used to address fractional-order scientific issues in a multitude of fields.


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## Body

## 1. Introduction

Fractional calculus (FC) is a novel scientific and technical area of investigation that is broadly utilized in applied mathematics, pharmacology, information theory, fluid, gas turbulence, mathematical biology, and related domains [1, 2]. The topic of FC has subsequently gained a lot of attention. Numerous researchers have made significant contributions to this topic by developing multiple fractional expressions in diverse publications. The findings of advanced calculus are frequently far more comprehensive than those of classical calculus. Furthermore, fractional differential formulations have greater characteristics than integer-order ones, including Caputo [3], Caputo and Fabrizio [4], and Atangana and Baleanu [5]. The noted Caputo order has significant difficulties as well, including the fact that their kernel is singular. This flaw has an impact on simulating major challenges. To address the aforementioned issues, the authors [5] proposed an innovative formulation that has fractional order relying on the Mittag-Leffler (ML) function. It is worth noting that their fractional integral is the fractional average of the supplied function’s Riemann-Liouville fractional integral and the function themselves [6–9]. In regard to the improvements listed previously, the derivative has been shown to be particularly effective in heat-like research and structural research [10, 11]. This special generation of derivatives containing fractional orders is both a filtration derivative and a fractional derivative.Recently, the research design of fractional derivatives [3–5] has become increasingly appropriate for simulation, permitting fractional differential equations to be progressively implemented in all disciplines of science, with increasing feasibility, effectiveness, and precision. Researchers are continuously exploring different ways of evaluating the credibility and validity of fractional differential equations (FDEs). Some prevalent numerical/analytical methodologies are the Adomian decomposition method (ADM) [12], Adams-Bashforth Moulton (ABM)[13], Hirota bilinear method (HBM) [14], Exp-function method (EFM)[15], variation iteration method (VIM) [16], Newton polynomial approach (NPA) [17], finite volume method (FVM) [18], finite difference method (FDM) [19], residue power series method (RPM) [20], and so on.The purpose of this paper is at employing the Jafari decomposition approach to nonlinear fractional coupled BEs as follows:(1)Dtδ1f+f∂f∂u1+g∂f∂u2=1Re∂2f∂u12+∂2f∂u22,Dtδ2g+f∂g∂u1+g∂g∂u2=1Re∂2g∂u12+∂2g∂u22,0<δ1,δ2<1.Model (1) of dissipation seems to be a very prominent unsteady flow system. Various scholars have contemplated investigating this notion as a framework of vibration propagation in order to gain scientific understanding. The distinctive functionality of model (1) is the simplest computational description of the rivalry between viscosity propagation and nonlinear convection. It comprises the simplest fundamental versions of the decomposition factor g∂f/∂u2 and the nonlinear convection-diffusion factor f∂f/∂u1, where V=f,g and Re is the Reynolds number employed to imitate the practical properties of signal oscillations and hence influence the behaviour of the system. Cole [21] examined BE algebraic features. Dynamical processes are crucial in science and numerical methods. In applied sciences, the significance of attaining the exact or estimated outcomes of partial differential equations is in order to examine innovative strategies; this is currently a contentious issue for accomplishing the exact or analytical results [22–24]. For this goal, several approaches for getting the varied reported performance of diverse scientific models that represented using nonlinear PDEs have been presented. Bateman [25] created a prominent framework and established consistent conclusions that are applicable to various viscous dissipations. Burgers [26] subsequently proposed it as the most distinguished framework for addressing computational instability challenges.During Gorge Adomian’s meteoric rise in 1980, the Adomian decomposition approach established a well-known methodology. This has been increasingly incorporated into a broad range of nonlinear systems, including the Black-Scholes model [27] and the Swift-Hohenberg [28]. The ADM has been shown to be strongly related to a wide range of integral transforms [27, 28]. Jafari [29] recently proposed a hybrid integral transform known as the Jafari transform. The capacity to recapitulate numerous prior transforms is the transformation’s key characteristic (see Remark 8).Due to the aforesaid tendency, we employ the Jafari transform decomposition methodology (JTDM) to determine the required clarification of the fractional-order BEs. To generate a new algorithmic strategy, the Jafari transform incorporated the ADM in an effective way. The Jafari transform is a modification of many previous formulae; see Remark8. Both the recommended techniques produce analytical findings in the format of a convergent series. The Atangana-Baleanu fractional derivative operator in the Caputo interpretation is used to explain the quantitative categorizations of the BEs. The proffered methodologies are well represented in modelling and compilation investigations. The obtained method is a valuable tool for evaluating the behaviour of systems that are difficult to numerically analyze, notably for fractional PDEs. Fractal-fractional phenomena can be investigated using the approximate expression.


## 2. Preliminaries

In this part, we revisit certain key concepts, ideas, and terminologies connected to fractional derivative formulations involving the power law and ML as a kernel, as well as the Jafari transform’s specific ramifications.Definition 1 (see [3]).
The fractional derivative of Caputo (CFD) is specifically defined as follows:(2)0cDtδft=1Γr−δ∫0tfru1t−u1δ+1−rdu1,r−1<δ<r,drdtrft,δ=r.Definition 2 (see [5]).
TheABC is specifically defined as follows:
(3)η1ABCDtδft=ABCδ1−δ∫η1tf′tEδ−δt−u1δ1−δdu1,where f∈Δ1a1,a2Sobolevspace,a1<a2,δ∈0,1, and ABCδ indicates the normalization function as ABCδ=ABC0=ABC1=1.Definition 3 (see [5]).
TheABC fractional integral operator is expressed in the following form:
(4)η1ABCI¯tδft=1−δABCδft+δΓδABCδ∫η1tfu1t−u1δ−1du1.Definition 4 (see [29]).
Consider an integrable mappingft defined on a set P, and then,
(5)P=ft:∃M>0,κ>0,ft<Mexpκt,ift≥0.Definition 5 (see [29]).
Assume that the functionsΦρ,Ψρ:ℝ+↦ℝ+ such that φρ≠0∀ρ∈ℝ+. The Jafari integral transform of the function ft expressed by Qρ is stated as
(6)Jft,ρ=Qρ=Φρ∫0∞ftexp−Ψρtdt.Theorem 6 (see [29]) (convolution property).
The following is valid for the Jafari integral transform:(7)Jf1∗f2=1ΦρQ1ρ∗Q2ρ.Definition 7.
The following is the Jafari transform of the CFD operator:(8)J0cDtδft,ρ=ΨδρQρ−Φρ∑κ=0δ−1Ψδ−κ−1ρfκ0,r−1<δ<r,Φ,Ψ>0.Remark 8.
Definition7 refers to the following resulting assumptions:
(1)
ChoosingΦρ=1 and Ψρ=ρ, then, this leads to the Laplace transform [30](2)
ChoosingΦρ1/ρ and Ψρ=1/ρ, then, this leads to the α-Laplace transform [31](3)
ChoosingΦρ=1/ρ and Ψρ=1/ρ, then, this leads to the Sumudu transform [32](4)
ChoosingΦρ=1/ρ and Ψρ=1, then, this leads to the Aboodh transform [33](5)
ChoosingΦρ=ρ and Ψρ=ρ2, then, this leads to the Pourreza transform [34, 35](6)
ChoosingΦρ=ρ and Ψρ=1/ρ, then, this leads to the Elzaki transform [36](7)
ChoosingΦρ=u1 and Ψρ=ρ/u1, then, this leads to the natural transform [37](8)
ChoosingΦρ=ρ2 and Ψρ=ρ, then, this leads to the Mohand transform [38](9)
ChoosingΦρ=1/ρ2 and Ψρ=1/ρ, then, this leads to the Swai transform [39](10)
ChoosingΦρ=1 and Ψρ=1/ρ, then, we get the Kamal transform [40](11)
ChoosingΦρ=ρα and Ψρ=1/ρ, then, this leads to the G−transform [41, 42]Definition 9 (see [43]).
TheABC fractional derivative operator has the following Jafari transform:
(9)J0ABCDtδft,ρδ=ABCδΨδρδ+1−δΨδρQρ−ΦρΨρf0.Remark 10.
Definition9 leads to the following conclusions:
(1)
ChoosingΦρ=1 and Ψρ=ρ, then, this leads to the Laplace transform of the ABC fractional derivative operator [44](2)
ChoosingΦρ=ρ and Ψρ=1/ρ, then, this leads to the Elzaki transform of the ABC fractional derivative operator [45](3)
ChoosingΦρ=Ψρ=1/ρ, then, we get the Sumudu transform of the ABC fractional derivative operator [46](4)
ChoosingΦρ=1 and Ψρ=ρ/u1, then, we get the Shehu transform of the ABC fractional derivative operator [46]Definition 11 (see [47]).
The ML function for a single parameter is defined as(10)Eδz=∑κ=0∞z1κΓκδ+1,δ,z1∈ℂ,Rδ≥0.


## 3. Analysis of Semianalytical Techniques

In this part, we illustrate an explanation of the generic methodology for the subsequent system via the Jafari transform.(11)Dtδfu1,t+Lfu1,t+N~fu1,t=Fu1,t,t>0,0<δ≤1,with ICs
(12)fu1,0=Gu1,where L denotes being linear and N is nonlinear, whilst ℏu1,τ represents the source terms.Considering the Jafari transform to (11), we obtain
(13)JDtδfu1,t+Lfu1,t+N~fu1,t=JFu1,t.Now, implement the differentiation property of Jafari transform in regard of CFD and then utilize the ABC fractional derivative operator as follows:(14)ΨδρUu1,ρ=Φρ∑κ=0r−1Ψδ−1−κρfκ0+JLfu1,t+N~fu1,t+JFu1,t,and
(15)ΨδρABCδδ+1−δΨδρUu1,ρ=ΦρΨρΨδρABCδδ+1−δΨδρf0+JLfu1,t+N~fu1,t+JFu1,t.The inverse Jafari transform of (14) and (15) gives
(16)fu1,t=J−1Φρ∑κ=0r−1Ψρδ−κ−1fκ0+1ΨδρJFu1,t−J−11ΨδρJLfu1,t+N~fu1,t,and
(17)fu1,t=J−1ΦρΨρf0+δ+1−δΨδρΨδρABCδJFu1,t−J−1δ+1−δΨδρΨδρABCδJLfu1,t+N~fu1,t.The Jafari decomposition method solutionfu1,t is described by the subsequent infinite series
(18)fu1,t=∑r=0∞fru1,t.Consequently, the nonlinear componentN~u1,t can be analyzed utilizing the Adomian decomposition approach
(19)N~fu1,t=∑r=0∞A~rf0,f1,⋯,,r=0,1,⋯,where
(20)A~rf0,f1,⋯,=1r!drdδrN~∑j=0∞δjfjδ=0,r>0.Plugging (17) and (18) into (15) and (16), respectively, we get
(21)∑r=0∞fru1,t=Gu1+G~u1−J−11ΨδρJLfu1,t+∑r=0∞A~r,and
(22)∑r=0∞fru1,t=Gu1+G~u1−J−1δ+1−δΨδρABCδΨδρJLfu1,t+∑r=0∞A~r.As a result, the iterative approach for (19) and (21) is constructed as
(23)f0u1,t=Gu1+G~u1,r=0,fr+1u1,t=−J−11ΨδρJLfru1,t+∑r=0∞A~r,r≥1,fr+1u1,t=−J−1δ+1−δΨδρABCδΨδρJLfru1,t+∑r=0∞A~r,r≥1.


## 4. Mathematical Formulation of the Jafari Transform Decomposition Approach

The following classifications will reveal how the essential prerequisites ensure the development of a unique solution. In the case of JTDM, we predict the existence of solutions that are accompanied by [48].


### 4.1. Uniqueness Results

In this part, we will present the uniqueness analysis for theJTDMC and JTDMABC fractional operator.Theorem 12.
TheJTDMC solution of (23) is unique when 0<ε<1, where ε=Y1+Y2+Y3tδ/Γδ+1.Proof.
Given that all continuous mappings on the Banach space are represented byΩ=ℂI¯,.. Further, surmise that I¯=0,T¯ have the norm .. Presently, we specify a mapping U:Ω↦℧ as
(24)fr+1u1,t=fu1,t+J−11ΨδρJLfru1,t+P¯fru1,t+N~fru1,t,r≥0,where Lfu1,t≡∂3fu1,t/∂u12 and P¯fu1,t≡∂fu1,t/∂u1. Either, surmise that Lfu1,t and Mfu1,t are also Lipschitzian with P¯f−P¯f^<Y1f−f^ and Lf−Lf^<Y2f−f^, where Y1 and Y2 are Lipschitz constants and f,f^ are distinguishable functional variables.
(25)Uf−Uf^=maxt∈I¯J−11ΨδρJLfu1,t+P¯fu1,t+N~fu1,t−J−11ΨδρJLf^u1,t+P¯f^u1,t+N~f^u1,t≤maxt∈I¯J−11ΨδρJLfu1,t−Lf^u1,t+J−11ΨδρJP¯fu1,t−P¯f^u1,t+J−11ΨδρJN~fu1,t−N~f^u1,t≤maxt∈I¯Y1J−11ΨδρJfu1,t−f^u1,t+Y2J−11ΨδρJfu1,t−f^u1,t+Y3J−11ΨδρJfu1,t−f^u1,t≤maxt∈I¯Y1+Y2+Y3J−11ΨδρJfu1,t−f^u1,t≤Y1+Y2+Y3J−11ΨδρJfu1,t−f^u1,t=Y1+Y2+Y3J−1ΦρΨδ+1ρfu1,t−f^u1,t=Y1+Y2+Y3tδΓδ+1fu1,t−f^u1,t.
As0<ε<1, so, U is contraction. As a conclusion of the Banach contraction fixed point theorem, (11) is unique. This yields the intended outcome.Theorem 13.
TheJTDMABC solution of (23) is unique when 0<ε<1, where ε=Y1+Y2+Y3δtδ/Γδ+1+1−δ.Proof.
Given that all continuous mappings on the Banach space are represented by℧=ℂI¯,., further, surmise that I¯=0,T¯ have the norm .. Presently, we specify a mapping U:℧↦℧ as
(26)fr+1u1,t=fu1,t+J−1δ+1−δΨδρABCδΨδρJLfru1,t+P¯fru1,t+N~fru1,t,r≥0,where Lfu1,t≡∂3fu1,t/∂u12 and P¯fu1,t≡∂fu1,t/∂u1. Either, surmise that Lfu1,t and Mfu1,t are also Lipschitzian with P¯f−P¯f^<Y1f−f^ and Lf−Lf^<Y2f−f^, where Y1 and Y2 are the Lipschitz constant, respectively, and f,f^ are distinguishable functional variables.
(27)Uf−Uf^=maxt∈I¯J−1δ+1−δΨδρABCδΨδρJLfu1,t+P¯fu1,t+N~fu1,t−J−1δ+1−δΨδρABCδΨδρJLf^u1,t+P¯f^u1,t+N~f^u1,t≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJLfu1,t−Lf^u1,t+J−1δ+1−δΨδρABCδΨδρJP¯fu1,t−P¯f^u1,t+J−1δ+1−δΨδρABCδΨδρJN~fu1,t−N~f^u1,t≤maxt∈I¯Y1J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t+Y2J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t+Y3J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t≤maxt∈I¯Y1+Y2+Y3J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t≤Y1+Y2+Y3J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t=Y1+Y2+Y3J−1ΦρΨρδ+1−δΨδρABCδΨδρfu1,t−f^u1,t=Y1+Y2+Y3δtδΓδ+1+1−δfu1,t−f^u1,t.
As0<ε<1, so, U is contraction. As a conclusion of the Banach contraction fixed point theorem, (11) is unique. This yields the intended outcome.


### 4.2. Convergence Analysis

This section consists of the convergence analysis based onJTDMC and JTDMABC.Theorem 14.
TheJTDMC solution of (11) is convergent.Proof.
Surmise thatQ^r=∑m=0rfru1,t. Furthermore, in order to show that Q^r is Cauchy sequence in U, by analyzing a model consisting of Adomian polynomials, we obtain
(28)R¯Q^r=Δ~r+∑p=0r−1Δ~p,N~Q^r=Δ~r+∑c=0r−1Δ~c.
Now,(29)Q^r−Q^q=maxt∈I¯Q^r−Q^q=maxt∈I¯∑m=q+1rf^u1,t,m=1,2,3,⋯,≤maxt∈I¯J−11ΨδρJ∑m=q+1rLfr−1u1,t+J−11ΨδρJ∑m=q+1rP¯fr−1u1,t+J−11ΨδρJ∑m=q+1rΔ~r−1u1,t=maxt∈I¯J−11ΨδρJ∑m=qr−1Lfru1,t+J−11ΨδρJ∑m=qr−1P¯fru1,t+J−11ΨδρJ∑m=qr−1Δ~ru1,t≤maxt∈I¯J−11ΨδρJ∑m=qr−1LQ^r−1−LQ^q−1+J−11ΨδρJ∑m=qr−1P¯Q^r−1−P¯Q^q−1+J−11ΨδρJ∑m=qr−1N~Q^r−1−N~Q^q−1≤maxt∈I¯J−11ΨδρJLQ^r−1−LQ^q−1+J−11ΨδρJP¯Q^r−1−P¯Q^q−1+J−11ΨδρJN~Q^r−1−N~Q^q−1≤Y1maxt∈I¯J−11ΨδρJQ^r−1−Q^q−1+Y2maxt∈I¯J−11ΨδρJQ^r−1−Q^q−1+Y3maxt∈I¯J−11ΨδρJQ^r−1−Q^q−1=Y1+Y2+Y3J−1ΦρΨδ+1ρQ^r−1−Q^q−1.=Y1+Y2+Y3tδΓδ+1Q^r−1−Q^q−1.
Taken=q+1; then,
(30)Q^q+1−Q^q≤εQ^q−Q^q−1≤ε2Q^q−1−Q^q−2≤⋯≤εqQ^1−Q^0,where ε=Y1+Y2+Y3tδ/Γδ+1. Analogously, we have
(31)Q^r−Q^q≤Q^q+1−Q^q+Q^q+2−Q^q+1+⋯+Q^r−Q^r−1≤εq+εq+1+⋯+εr−1Q^1−Q^0≤εq1−εr−qεf1.
As0<ε<1, we have 1−εr−q<1. Accordingly,
(32)Q^r−Q^q≤εq1−εmaxt∈I¯f1.
Therefore,f1<∞ (since fu1,t is bounded). Additionally, as q↦∞, then, Q^r−Q^q↦0. As a result, Q^1 is a Cauchy sequence in K. Thus, the series ∑n=0∞fr is convergent.Theorem 15.
TheJTDMABC solution of (11) is convergent.Proof.
Surmise thatQ^r=∑m=0rfru1,t. Furthermore, in order to show that Q^r is a Cauchy sequence in U, by analyzing a model consisting of Adomian polynomials, we obtain
(33)R¯Q^r=Δ~r+∑p=0r−1Δ~p,N~Q^r=Δ~r+∑c=0r−1Δ~c.
Now,(34)Q^r−Q^q=maxt∈I¯Q^r−Q^q=maxt∈I¯∑m=q+1rf^u1,t,m=1,2,3,⋯,≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJ∑m=q+1rLfr−1u1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=q+1rP¯fr−1u1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=q+1rΔ~r−1u1,t=maxt∈I¯J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1Lfru1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1P¯fru1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1Δ~ru1,t≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1LQ^r−1−LQ^q−1+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1P¯Q^r−1−P¯Q^q−1+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1N~Q^r−1−N~Q^q−1≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJLQ^r−1−LQ^q−1+J−1δ+1−δΨδρABCδΨδρJP¯Q^r−1−P¯Q^q−1+J−1δ+1−δΨδρABCδΨδρJN~Q^r−1−N~Q^q−1≤Y1maxt∈I¯J−1δ+1−δΨδρABCδΨδρJQ^r−1−Q^q−1+Y2maxt∈I¯J−1δ+1−δΨδρABCδΨδρJQ^r−1−Q^q−1+Y3maxt∈I¯J−1δ+1−δΨδρABCδΨδρJQ^r−1−Q^q−1=Y1+Y2+Y3J−1ΦρΨδ+1ρδ+1−δΨδρABCδΨδρQ^r−1−Q^q−1.=Y1+Y2+Y3δtδΓδ+1+1−δQ^r−1−Q^q−1.
Taken=q+1; then,
(35)Q^q+1−Q^q≤εQ^q−Q^q−1≤ε2Q^q−1−Q^q−2≤⋯≤εqQ^1−Q^0,where ε=Y1+Y2+Y3δtδ/Γδ+1+1−δ. Analogously, we have
(36)Q^r−Q^q≤Q^q+1−Q^q+Q^q+2−Q^q+1+⋯+Q^r−Q^r−1≤εq+εq+1+⋯+εr−1Q^1−Q^0≤εq1−εr−qεf1.
As0<ε<1, we have 1−εr−q<1. Accordingly,
(37)Q^r−Q^q≤εq1−εmaxt∈I¯f1.
Therefore,f1<∞ (since fu1,t is bounded). Additionally, as q↦∞, then, Q^r−Q^q↦0. As a result, Q^1 is a Cauchy sequence in K. Thus, the series ∑n=0∞fr is convergent.


## 4.1. Uniqueness Results

In this part, we will present the uniqueness analysis for theJTDMC and JTDMABC fractional operator.Theorem 12.
TheJTDMC solution of (23) is unique when 0<ε<1, where ε=Y1+Y2+Y3tδ/Γδ+1.Proof.
Given that all continuous mappings on the Banach space are represented byΩ=ℂI¯,.. Further, surmise that I¯=0,T¯ have the norm .. Presently, we specify a mapping U:Ω↦℧ as
(24)fr+1u1,t=fu1,t+J−11ΨδρJLfru1,t+P¯fru1,t+N~fru1,t,r≥0,where Lfu1,t≡∂3fu1,t/∂u12 and P¯fu1,t≡∂fu1,t/∂u1. Either, surmise that Lfu1,t and Mfu1,t are also Lipschitzian with P¯f−P¯f^<Y1f−f^ and Lf−Lf^<Y2f−f^, where Y1 and Y2 are Lipschitz constants and f,f^ are distinguishable functional variables.
(25)Uf−Uf^=maxt∈I¯J−11ΨδρJLfu1,t+P¯fu1,t+N~fu1,t−J−11ΨδρJLf^u1,t+P¯f^u1,t+N~f^u1,t≤maxt∈I¯J−11ΨδρJLfu1,t−Lf^u1,t+J−11ΨδρJP¯fu1,t−P¯f^u1,t+J−11ΨδρJN~fu1,t−N~f^u1,t≤maxt∈I¯Y1J−11ΨδρJfu1,t−f^u1,t+Y2J−11ΨδρJfu1,t−f^u1,t+Y3J−11ΨδρJfu1,t−f^u1,t≤maxt∈I¯Y1+Y2+Y3J−11ΨδρJfu1,t−f^u1,t≤Y1+Y2+Y3J−11ΨδρJfu1,t−f^u1,t=Y1+Y2+Y3J−1ΦρΨδ+1ρfu1,t−f^u1,t=Y1+Y2+Y3tδΓδ+1fu1,t−f^u1,t.
As0<ε<1, so, U is contraction. As a conclusion of the Banach contraction fixed point theorem, (11) is unique. This yields the intended outcome.Theorem 13.
TheJTDMABC solution of (23) is unique when 0<ε<1, where ε=Y1+Y2+Y3δtδ/Γδ+1+1−δ.Proof.
Given that all continuous mappings on the Banach space are represented by℧=ℂI¯,., further, surmise that I¯=0,T¯ have the norm .. Presently, we specify a mapping U:℧↦℧ as
(26)fr+1u1,t=fu1,t+J−1δ+1−δΨδρABCδΨδρJLfru1,t+P¯fru1,t+N~fru1,t,r≥0,where Lfu1,t≡∂3fu1,t/∂u12 and P¯fu1,t≡∂fu1,t/∂u1. Either, surmise that Lfu1,t and Mfu1,t are also Lipschitzian with P¯f−P¯f^<Y1f−f^ and Lf−Lf^<Y2f−f^, where Y1 and Y2 are the Lipschitz constant, respectively, and f,f^ are distinguishable functional variables.
(27)Uf−Uf^=maxt∈I¯J−1δ+1−δΨδρABCδΨδρJLfu1,t+P¯fu1,t+N~fu1,t−J−1δ+1−δΨδρABCδΨδρJLf^u1,t+P¯f^u1,t+N~f^u1,t≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJLfu1,t−Lf^u1,t+J−1δ+1−δΨδρABCδΨδρJP¯fu1,t−P¯f^u1,t+J−1δ+1−δΨδρABCδΨδρJN~fu1,t−N~f^u1,t≤maxt∈I¯Y1J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t+Y2J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t+Y3J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t≤maxt∈I¯Y1+Y2+Y3J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t≤Y1+Y2+Y3J−1δ+1−δΨδρABCδΨδρJfu1,t−f^u1,t=Y1+Y2+Y3J−1ΦρΨρδ+1−δΨδρABCδΨδρfu1,t−f^u1,t=Y1+Y2+Y3δtδΓδ+1+1−δfu1,t−f^u1,t.
As0<ε<1, so, U is contraction. As a conclusion of the Banach contraction fixed point theorem, (11) is unique. This yields the intended outcome.


## 4.2. Convergence Analysis

This section consists of the convergence analysis based onJTDMC and JTDMABC.Theorem 14.
TheJTDMC solution of (11) is convergent.Proof.
Surmise thatQ^r=∑m=0rfru1,t. Furthermore, in order to show that Q^r is Cauchy sequence in U, by analyzing a model consisting of Adomian polynomials, we obtain
(28)R¯Q^r=Δ~r+∑p=0r−1Δ~p,N~Q^r=Δ~r+∑c=0r−1Δ~c.
Now,(29)Q^r−Q^q=maxt∈I¯Q^r−Q^q=maxt∈I¯∑m=q+1rf^u1,t,m=1,2,3,⋯,≤maxt∈I¯J−11ΨδρJ∑m=q+1rLfr−1u1,t+J−11ΨδρJ∑m=q+1rP¯fr−1u1,t+J−11ΨδρJ∑m=q+1rΔ~r−1u1,t=maxt∈I¯J−11ΨδρJ∑m=qr−1Lfru1,t+J−11ΨδρJ∑m=qr−1P¯fru1,t+J−11ΨδρJ∑m=qr−1Δ~ru1,t≤maxt∈I¯J−11ΨδρJ∑m=qr−1LQ^r−1−LQ^q−1+J−11ΨδρJ∑m=qr−1P¯Q^r−1−P¯Q^q−1+J−11ΨδρJ∑m=qr−1N~Q^r−1−N~Q^q−1≤maxt∈I¯J−11ΨδρJLQ^r−1−LQ^q−1+J−11ΨδρJP¯Q^r−1−P¯Q^q−1+J−11ΨδρJN~Q^r−1−N~Q^q−1≤Y1maxt∈I¯J−11ΨδρJQ^r−1−Q^q−1+Y2maxt∈I¯J−11ΨδρJQ^r−1−Q^q−1+Y3maxt∈I¯J−11ΨδρJQ^r−1−Q^q−1=Y1+Y2+Y3J−1ΦρΨδ+1ρQ^r−1−Q^q−1.=Y1+Y2+Y3tδΓδ+1Q^r−1−Q^q−1.
Taken=q+1; then,
(30)Q^q+1−Q^q≤εQ^q−Q^q−1≤ε2Q^q−1−Q^q−2≤⋯≤εqQ^1−Q^0,where ε=Y1+Y2+Y3tδ/Γδ+1. Analogously, we have
(31)Q^r−Q^q≤Q^q+1−Q^q+Q^q+2−Q^q+1+⋯+Q^r−Q^r−1≤εq+εq+1+⋯+εr−1Q^1−Q^0≤εq1−εr−qεf1.
As0<ε<1, we have 1−εr−q<1. Accordingly,
(32)Q^r−Q^q≤εq1−εmaxt∈I¯f1.
Therefore,f1<∞ (since fu1,t is bounded). Additionally, as q↦∞, then, Q^r−Q^q↦0. As a result, Q^1 is a Cauchy sequence in K. Thus, the series ∑n=0∞fr is convergent.Theorem 15.
TheJTDMABC solution of (11) is convergent.Proof.
Surmise thatQ^r=∑m=0rfru1,t. Furthermore, in order to show that Q^r is a Cauchy sequence in U, by analyzing a model consisting of Adomian polynomials, we obtain
(33)R¯Q^r=Δ~r+∑p=0r−1Δ~p,N~Q^r=Δ~r+∑c=0r−1Δ~c.
Now,(34)Q^r−Q^q=maxt∈I¯Q^r−Q^q=maxt∈I¯∑m=q+1rf^u1,t,m=1,2,3,⋯,≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJ∑m=q+1rLfr−1u1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=q+1rP¯fr−1u1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=q+1rΔ~r−1u1,t=maxt∈I¯J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1Lfru1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1P¯fru1,t+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1Δ~ru1,t≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1LQ^r−1−LQ^q−1+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1P¯Q^r−1−P¯Q^q−1+J−1δ+1−δΨδρABCδΨδρJ∑m=qr−1N~Q^r−1−N~Q^q−1≤maxt∈I¯J−1δ+1−δΨδρABCδΨδρJLQ^r−1−LQ^q−1+J−1δ+1−δΨδρABCδΨδρJP¯Q^r−1−P¯Q^q−1+J−1δ+1−δΨδρABCδΨδρJN~Q^r−1−N~Q^q−1≤Y1maxt∈I¯J−1δ+1−δΨδρABCδΨδρJQ^r−1−Q^q−1+Y2maxt∈I¯J−1δ+1−δΨδρABCδΨδρJQ^r−1−Q^q−1+Y3maxt∈I¯J−1δ+1−δΨδρABCδΨδρJQ^r−1−Q^q−1=Y1+Y2+Y3J−1ΦρΨδ+1ρδ+1−δΨδρABCδΨδρQ^r−1−Q^q−1.=Y1+Y2+Y3δtδΓδ+1+1−δQ^r−1−Q^q−1.
Taken=q+1; then,
(35)Q^q+1−Q^q≤εQ^q−Q^q−1≤ε2Q^q−1−Q^q−2≤⋯≤εqQ^1−Q^0,where ε=Y1+Y2+Y3δtδ/Γδ+1+1−δ. Analogously, we have
(36)Q^r−Q^q≤Q^q+1−Q^q+Q^q+2−Q^q+1+⋯+Q^r−Q^r−1≤εq+εq+1+⋯+εr−1Q^1−Q^0≤εq1−εr−qεf1.
As0<ε<1, we have 1−εr−q<1. Accordingly,
(37)Q^r−Q^q≤εq1−εmaxt∈I¯f1.
Therefore,f1<∞ (since fu1,t is bounded). Additionally, as q↦∞, then, Q^r−Q^q↦0. As a result, Q^1 is a Cauchy sequence in K. Thus, the series ∑n=0∞fr is convergent.


## 5. Solutions of Fractional-Order Burgers Equation

In this part, we demonstrate how to use the aforementioned strategies to derive analytical findings for the Burgers equations under various initial guesses.Example 16.
Let us surmise the generic form of BEs (1) as follows:
(38)Dtδ1f+f∂f∂u1+g∂f∂u2=1Re∂2f∂u12+∂2f∂u22,Dtδ2g+f∂g∂u1+g∂g∂u2=1Re∂2g∂u12+∂2g∂u22,0<δ1,δ2<1,having ICs
(39)fu1,u2,0=34−141+exp0.03125Re−4u1+4u2,gu1,u2,0=34+141+exp0.03125Re−4u1+4u2,where Re denotes the Reynolds number.
In the exact solution of (38) when δ1=δ2=1, then,
(40)fu1,u2,t=34−141+exp0.03125Re−4u1+4u2−t,gu1,u2,t=34+141+exp0.03125Re−4u1+4u2−t.Case 17 (Caputo fractional operator).
Applying the Jafari transform on (38), we have
(41)JDtδ1f+Jf∂f∂u1+Jg∂f∂u2=1ReJ∂2f∂u12+∂2f∂u22,JDtδ2g+Jf∂g∂u1+Jg∂g∂u2=1ReJ∂2g∂u12+∂2g∂u22.
It follows that(42)Ψδ1ρJfu1,u2,ρ−Φρ∑κ=1m1−1Ψδ1−κ−1ρfκu1,u2,0=−Jf∂f∂u1−Jg∂f∂u2+1ReJ∂2f∂u12+∂2f∂u22,Ψδ2ρJgu1,u2,ρ−Φρ∑κ=1m1−1Ψδ2−κ−1ρgκu1,u2,0=−Jf∂g∂u1−Jg∂g∂u2+1ReJ∂2g∂u12+∂2g∂u22.
The aforementioned equation can be written as(43)Jfu1,u2,ρ=ΦρΨρ34−141+exp0.03125Re−4u1+4u2−1Ψδ1ρJf∂f∂u1−1Ψδ1ρJg∂f∂u2+1Re1Ψδ1ρJ∂2f∂u12+∂2f∂u22,(44)Jgu1,u2,ρ=ΦρΨρ34+141+exp0.03125Re−4u1+4u2−1Ψδ2ρJf∂g∂u1−1Ψδ2ρJg∂g∂u2+1Re1Ψδ2ρJ∂2g∂u12+∂2g∂u22.
Further, implementing the inverse Jafari transform on (43), then, it diminishes to
(45)fu1,u2,ρ=J−1ΦρΨρ34−141+exp0.03125Re−4u1+4u2−J−11Ψδ1ρJf∂f∂u1+g∂f∂u2−1Re∂2f∂u12+∂2f∂u22,(46)gu1,u2,ρ=J−1ΦρΨρ34+141+exp0.03125Re−4u1+4u2−J−11Ψδ2ρJf∂g∂u1+g∂g∂u2−1Re∂2g∂u12+∂2g∂u22.
In this case, we hypothesize that the undefined functionsfu1,u2,t and gu1,u2,t can be described as an infinite series of the mode as follows
(47)fu1,u2,t=∑r=0∞fru1,u2,t,gu1,u2,t=∑r=0∞gru1,u2,t.
It is clear thatffu1=∑r=0∞Ar,fgu1=∑r=0∞Br,gfu2=∑r=0∞Cr,ggu2=∑r=0∞Dr indicate the Adomian polynomials and are referred to as the nonlinear factors.
In view of the Adomian polynomials, (45) can be described as
(48)∑r=0∞fr+1u1,u2,t=34−141+exp0.03125Re−4u1+4u2−J−11Ψδ1ρJ∑r=0∞Ar+∑r=0∞Br−1Re∑r=0∞fru1u1+∑r=0∞fru2u2,∑r=0∞gr+1u1,u2,t=34+141+exp0.03125Re−4u1+4u2−J−11Ψδ2ρJ∑r=0∞Cr+∑r=0∞Dr+1Re∑r=0∞gru1u1+∑r=0∞gru2u2.
Analyzing term by term (48), we may immediately get the iterative terms indicated as follows:
(49)f0u1,u2,t=34−141+exp0.03125Re−4u1+4u2,g0u1,u2,t=34+141+exp0.03125Re−4u1+4u2,f1u1,u2,t=−Re.exp0.125Reu1−u21281+exp0.125Reu1−u22.tδ1Γδ1+1,g1u1,u2,t=Re.exp0.125Reu1−u21281+exp0.125Reu1−u22.tδ2Γδ2+1,f2u1,u2,t=−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24.t2δ1Γ2δ1+1×Re2−exp0.125Reu1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2,g2u1,u2,t=−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24.t2δ2Γ2δ2+1×Re2expRe4u1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2.
Proceeding in the analogous manner, the additional factors offr and gr,r≥3 of the JTDM solution can be attained effortlessly. As a result, we arrive to the mathematical formulation as
(50)fu1,u2,t=∑r=0∞fu1,u2,t=f0u1,u2,t+f1u1,u2,t+f2u1,u2,t+,⋯,fu1,u2,t=34−141+exp0.03125Re−4u1+4u2−Re.exp0.125Reu1−u21281+exp0.125Reu1−u22.tδ1Γδ1+1−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24.t2δ1Γ2δ1+1×Re2−exp0.125Reu1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2+,⋯,and
(51)gu1,u2,t=∑r=0∞gu1,u2,t=g0u1,u2,t+g1u1,u2,t+g2u1,u2,t+,⋯,gu1,u2,t=34+141+exp0.03125Re−4u1+4u2−Re.exp0.125Reu1−u21281+exp0.125Reu1−u22.tδ2Γδ2+1−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24.t2δ2Γ2δ2+1×Re2−expRe4u1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2+⋯+.Case 18 (ABC fractional operator).
Applying the Jafari transform on (38) with respect to ABC-fractional derivative operator sense as
(52)Ψδ1ρABCρδ1+1−δ1Ψδ1ρJfu1,u2,ρ−Φρ∑κ=1m1−1Ψδ−κ−1ρfκu1,u2,0=−Jf∂f∂u1−Jg∂f∂u2+1ReJ∂2f∂u12+∂2f∂u22,Ψδ2ρABCρδ2+1−δ2Ψδ2ρJgu1,u2,ρ−Φρ∑κ=1m1−1Ψδ−κ−1ρgκu1,u2,0=−Jf∂g∂u1−Jg∂g∂u2+1ReJ∂2g∂u12+∂2g∂u22.
The aforementioned equation can be written as(53)Jfu1,u2,ρ=ΦρΨρ34−141+exp0.03125Re−4u1+4u2−δ1+1−δ1Ψδ1ρΨδ1ρABCρJf∂f∂u1+g∂f∂u2−1ReJ∂2f∂u12+∂2f∂u22,(54)Jgu1,u2,ρ=ΦρΨρ34+141+exp0.03125Re−4u1+4u2−δ2+1−δ2Ψδ2ρΨδ2ρABCρJf∂g∂u1+g∂g∂u2−1ReJ∂2g∂u12+∂2g∂u22.
Further, implementing the inverse Jafari transform on (53), then, it diminishes to
(55)fu1,u2,ρ=J−1ΦρΨρ34−141+exp0.03125Re−4u1+4u2−J−1δ1+1−δ1Ψδ1ρΨδ1ρABCρJf∂f∂u1+g∂f∂u2−1Re∂2f∂u12+∂2f∂u22,(56)gu1,u2,ρ=J−1ΦρΨρ34+141+exp0.03125Re−4u1+4u2−J−1δ2+1−δ2Ψδ2ρΨδ2ρABCρJf∂g∂u1+g∂g∂u2−1Re∂2g∂u12+∂2g∂u22.
In this case, we hypothesize that the undefined functionsfu1,u2,t and gu1,u2,t can be described as an infinite series of the mode as follows:
(57)fu1,u2,t=∑r=0∞fru1,u2,t,gu1,u2,t=∑r=0∞gru1,u2,t.
It is clear thatffu1=∑r=0∞Ar,fgu1=∑r=0∞Br,gfu2=∑r=0∞Cr,ggu2=∑r=0∞Dr indicate the Adomian polynomials and were referred to as the nonlinear factors.
In view of the Adomian polynomials, (55) can be described as
(58)∑r=0∞fr+1u1,u2,t=34−141+exp0.03125Re−4u1+4u2−J−1δ1+1−δ1Ψδ1ρΨδ1ρABCρJ∑r=0∞Ar+∑r=0∞Br−1Re∑r=0∞fru1u1+∑r=0∞fru2u2,(59)∑r=0∞gr+1u1,u2,t=34+141+exp0.03125Re−4u1+4u2−J−1δ2+1−δ2Ψδ2ρΨδ2ρABCρJ∑r=0∞Cr+∑r=0∞Dr+1Re∑r=0∞gru1u1+∑r=0∞gru2u2.
Analyzing term by term (58), we may immediately get the iterative terms indicated below
(60)f0u1,u2,t=34−141+exp0.03125Re−4u1+4u2,g0u1,u2,t=34+141+exp0.03125Re−4u1+4u2,f1u1,u2,t=−Re.exp0.125Reu1−u21281+exp0.125Reu1−u22.tδ1Γδ1+1+1−δ1,g1u1,u2,t=Re.exp0.125Reu1−u21281+exp0.125Reu1−u22.tδ2Γδ2+1+1−δ2,f2u1,u2,t=−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24.δ1tδ1Γ2δ1+1+2δ11−δ1tδ1Γδ1+1+1−δ12×Re2−exp0.125Reu1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2,g2u1,u2,t=−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24.δ2tδ2Γ2δ2+1+2δ21−δ2tδ2Γδ2+1+1−δ22×Re2expRe4u1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2.
Proceeding in the analogous manner, the additional factors offr and gr,r≥3 of the JTDM solution can be attained effortlessly. As a result, we arrive to the mathematical formulation as
(61)fu1,u2,t=∑r=0∞fu1,u2,t=f0u1,u2,t+f1u1,u2,t+f2u1,u2,t+,⋯,fu1,u2,t=34−141+exp0.03125Re−4u1+4u2−Re.exp0.125Reu1−u21281+exp0.125Reu1−u22×δ1tδ1Γδ1+1+1−δ1−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24.δ1tδ1Γ2δ1+1+2δ11−δ1tδ1Γδ1+1+1−δ12×Re2−exp0.125Reu1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2+,⋯,and
(62)gu1,u2,t=∑r=0∞gu1,u2,t=g0u1,u2,t+g1u1,u2,t+g2u1,u2,t+,⋯,gu1,u2,t=34+141+exp0.03125Re−4u1+4u2−Re.exp0.125Reu1−u21281+exp0.125Reu1−u22×δ2tδ2Γδ2+1+1−δ2−Re.exp0.125Reu1−u240961+exp0.125Reu1−u24×δ2tδ2Γ2δ2+1+2δ21−δ2tδ2Γδ2+1+1−δ22×Re2−expRe4u1−u2+exp0.125Reu1−u2−1+exp0.125Reu1−u2+⋯+.Example 19.
Let us surmise the generic form of BEs (1) as
(63)Dtδ1f−2f∂f∂u1−2g∂f∂u2=∂2f∂u12+∂2f∂u22,Dtδ2g−2f∂g∂u1−2g∂g∂u2=∂2g∂u12+∂2g∂u22,0<δ1,δ2<1,having ICs
(64)fu1,u2,0=1−tanh1−u1+2u2,gu1,u2,0=1+2tanh1−u1+2u2,where Re denotes the Reynolds number.
For the exact solution of (63) when δ1=δ2=1, then,
(65)fu1,u2,t=1−tanh1−u1+2u2+2t,gu1,u2,t=1+2tanh1−u1+2u2+2t.Case 20 (Caputo fractional operator).
Applying the Jafari transform on (18), we have
(66)JDtδ1f−2Jf∂f∂u1−2Jg∂f∂u2=J∂2f∂u12+∂2f∂u22,JDtδ2g−2Jf∂g∂u1−2Jg∂g∂u2=J∂2g∂u12+∂2g∂u22.
It follows that(67)Ψδ1ρJfu1,u2,ρ−Φρ∑κ=1m1−1Ψδ1−κ−1ρfκu1,u2,0=2Jf∂f∂u1+2Jg∂f∂u2+J∂2f∂u12+∂2f∂u22,Ψδ2ρJgu1,u2,ρ−Φρ∑κ=1m1−1Ψδ2−κ−1ρgκu1,u2,0=2Jf∂g∂u1+2Jg∂g∂u2+J∂2g∂u12+∂2g∂u22.
The aforementioned equation can be written as(68)Jfu1,u2,ρ=ΦρΨρ1−tanh1−u1+2u2+2Ψδ1ρJf∂f∂u1+2Ψδ1ρJg∂f∂u2+1Ψδ1ρJ∂2f∂u12+∂2f∂u22,Jgu1,u2,ρ=ΦρΨρ1+2tanh1−u1+2u2+2Ψδ2ρJf∂g∂u1+2Ψδ2ρJg∂g∂u2+1Ψδ2ρJ∂2g∂u12+∂2g∂u22.
Further, implementing the inverse Jafari transform on (68), then, it diminishes to
(69)fu1,u2,ρ=J−1ΦρΨρ1−tanh1−u1+2u2+J−11Ψδ1ρJ2f∂f∂u1+2g∂f∂u2+∂2f∂u12+∂2f∂u22,gu1,u2,ρ=J−1ΦρΨρ1−2tanh1−u1+2u2+J−11Ψδ2ρJ2f∂g∂u1+2g∂g∂u2+∂2g∂u12+∂2g∂u22.
In this case, we hypothesize that the undefined functionsfu1,u2,t and gu1,u2,t can be described as an infinite series of the mode as follows
(70)fu1,u2,t=∑r=0∞fru1,u2,t,gu1,u2,t=∑r=0∞gru1,u2,t.
It is clear thatffu1=∑r=0∞Ar,fgu1=∑r=0∞Br,gfu2=∑r=0∞Cr,ggu2=∑r=0∞Dr indicate the Adomian polynomials and were referred to as the nonlinear factors.
In view of the Adomian polynomials, (69) can be described as
(71)∑r=0∞fr+1u1,u2,t=1−tanh1−u1+2u2+J−11Ψδ1ρJ2∑r=0∞Ar+2∑r=0∞Br+∑r=0∞fru1u1+∑r=0∞fru2u2,∑r=0∞gr+1u1,u2,t=1+2tanh1−u1+2u2+J−11Ψδ2ρJ2∑r=0∞Cr+2∑r=0∞Dr+∑r=0∞gru1u1+∑r=0∞gru2u2.
Analyzing term by term (71), we may immediately get the iterative terms indicated as follows:
(72)f0u1,u2,t=1−tanh1−u1+2u2,g0u1,u2,t=1+2tanh1−u1+2u2,f1u1,u2,t=−2sech1−u1+2u2.tδ1Γδ1+1,g1u1,u2,t=4sech1−u1+2u2.tδ2Γδ2+1,f2u1,u2,t=−8sech21−u1+2u2−2sech21−u1+2u2+2sech21−u1+2u2+tanh1−u1+2u2.t2δ1Γ2δ1+1,g2u1,u2,t=−8sech21−u1+2u2−2sech21−u1+2u2+2sech21−u1+2u2+2tanh1−u1+2u2.t2δ1Γ2δ1+1.
Proceeding in the analogous manner, the additional factors offr and gr,r≥3 of the JTDM solution can be attained effortlessly. As a result, we arrive to the mathematical formulation as
(73)fu1,u2,t=∑r=0∞fu1,u2,t=f0u1,u2,t+f1u1,u2,t+f2u1,u2,t+,⋯,fu1,u2,t=1−tanh1−u1+2u2−2sech1−u1+2u2.tδ1Γδ1+1+8sech21−u1+2u2×−2sech21−u1+2u2+2sech21−u1+2u2+tanh1−u1+2u2.t2δ1Γ2δ1+1−,⋯,and
(74)gu1,u2,t=∑r=0∞gu1,u2,t=g0u1,u2,t+g1u1,u2,t+g2u1,u2,t+,⋯,gu1,u2,t=1+2tanh1−u1+2u2+4sech1−u1+2u2.tδ1Γδ1+1−8sech21−u1+2u2×−sech21−u1+2u2+2sech21−u1+2u2+2tanh1−u1+2u2.t2δ1Γ2δ1+1−⋯−.Case 21 (ABC fractional operator).
Applying the Jafari transform on (63) with respect to the ABC-fractional derivative operator sense as
(75)Ψδ1ρABCρδ1+1−δ1Ψδ1ρJfu1,u2,ρ−Φρ∑κ=1m1−1Ψδ−κ−1ρfκu1,u2,0=2Jf∂f∂u1+2Jg∂f∂u2+J∂2f∂u12+∂2f∂u22,Ψδ2ρABCρδ2+1−δ2Ψδ2ρJgu1,u2,ρ−Φρ∑κ=1m1−1Ψδ−κ−1ρgκu1,u2,0=2Jf∂g∂u1+2Jg∂g∂u2+J∂2g∂u12+∂2g∂u22.
The aforementioned equation can be written as(76)Jfu1,u2,ρ=ΦρΨρ1−tanh1−u1+2u2+δ1+1−δ1Ψδ1ρΨδ1ρABCρJ2f∂f∂u1+2g∂f∂u2+J∂2f∂u12+∂2f∂u22,Jgu1,u2,ρ=ΦρΨρ1+2tanh1−u1+2u2+δ2+1−δ2Ψδ2ρΨδ2ρABCρJ2f∂g∂u1+2g∂g∂u2+J∂2g∂u12+∂2g∂u22.
Further, implementing the inverse Jafari transform on (76), then, it diminishes to
(77)fu1,u2,ρ=J−1ΦρΨρ1−tanh1−u1+2u2+J−1δ1+1−δ1Ψδ1ρΨδ1ρABCρJ2f∂f∂u1+2g∂f∂u2+∂2f∂u12+∂2f∂u22,gu1,u2,ρ=J−1ΦρΨρ1+2tanh1−u1+2u2+J−1δ2+1−δ2Ψδ2ρΨδ2ρABCρJ2f∂g∂u1+2g∂g∂u2+∂2g∂u12+∂2g∂u22.
In this case, we hypothesize that the undefined functionsfu1,u2,t and gu1,u2,t can be described as an infinite series of the mode as follows:
(78)fu1,u2,t=∑r=0∞fru1,u2,t,gu1,u2,t=∑r=0∞gru1,u2,t.
It is clear thatffu1=∑r=0∞Ar,fgu1=∑r=0∞Br,gfu2=∑r=0∞Cr,ggu2=∑r=0∞Dr indicate the Adomian polynomials and were referred to as the nonlinear factors.
In view of the Adomian polynomials, (77) can be described as
(79)∑r=0∞fr+1u1,u2,t=1−tanh1−u1+2u2+J−1δ1+1−δ1Ψδ1ρΨδ1ρABCρJ2∑r=0∞Ar+2∑r=0∞Br+∑r=0∞fru1u1+∑r=0∞fru2u2,∑r=0∞gr+1u1,u2,t=1+2tanh1−u1+2u2+J−1δ2+1−δ2Ψδ2ρΨδ2ρABCρJ2∑r=0∞Cr+2∑r=0∞Dr+∑r=0∞gru1u1+∑r=0∞gru2u2.
Analyzing term by term (79), we may immediately get the iterative terms indicated as follows:
(80)f0u1,u2,t=1−tanh1−u1+2u2,g0u1,u2,t=1+2tanh1−u1+2u2,f1u1,u2,t=−2sech1−u1+2u2.tδ1Γδ1+1+1−δ1,g1u1,u2,t=4sech1−u1+2u2.tδ2Γδ2+1+1−δ2,f2u1,u2,t=−8sech21−u1+2u2−2sech21−u1+2u2+2sech21−u1+2u2+tanh1−u1+2u2.δ1tδ1Γ2δ1+1+2δ11−δ1tδ1Γδ1+1+1−δ12,g2u1,u2,t=−8sech21−u1+2u2−2sech21−u1+2u2+2sech21−u1+2u2+2tanh1−u1+2u2.δ2tδ2Γ2δ2+1+2δ21−δ2tδ2Γδ2+1+1−δ22.
Proceeding in the analogous manner, the additional factors offr and gr,r≥3 of the JTDM solution can be attained effortlessly. As a result, we arrive to the mathematical formulation as
(81)fu1,u2,t=∑r=0∞fu1,u2,t=f0u1,u2,t+f1u1,u2,t+f2u1,u2,t+,⋯,fu1,u2,t=1−tanh1−u1+2u2−2sech1−u1+2u2.tδ1Γδ1+1+1−δ1−8sech21−u1+2u2−2sech21−u1+2u2+2sech21−u1+2u2+tanh1−u1+2u2.δ1tδ1Γ2δ1+1+2δ11−δ1tδ1Γδ1+1+1−δ12+,⋯,and
(82)gu1,u2,t=∑r=0∞gu1,u2,t=g0u1,u2,t+g1u1,u2,t+g2u1,u2,t+,⋯,gu1,u2,t=1+2tanh1−u1+2u2+4sech1−u1+2u2.tδ2Γδ2+1+1−δ2−8sech21−u1+2u2−2sech21−u1+2u2+2sech21−u1+2u2+2tanh1−u1+2u2.δ2tδ2Γ2δ2+1+2δ21−δ2tδ2Γδ2+1+1−δ22+⋯+.


## 6. Numerical Results and Discussion

We employed two distinctive techniques to analyze the approximate findings of fractional-order paired BEs in this research. With the aid of MATLAB 2021, one may acquire computational information for the framework of BEs in any configuration for varying parameters of spatial and temporal factors. We attempted graphical studies for multiple fractional processes in Tables1 and 2 for the model in Example 16 assuming varied components of u1 and t. Tables 1–4 illustrate a mathematical evaluation of the VIM [16] and the JTDM in perspective of absolute error for model (38). The findings of a simulation work for the interacting mechanism addressed in Example 16 are included in Tables 1–4. With relevant facts in the given data, we can deduce that the findings acquired by the JTDM are trustworthy. Figure 1 describes the performance of the JTDM result via fu1,u2,t and exact outcome for Example 16, whereas Figure 2 depicts the structure of the absolute error and varied fractional parameters of δ1. Also, Figure 3 represents the comparison analysis of the both integer and fractional order for Example 16.Table 1
The exact,JTDMC and JTDMABC solutions of fu1,u2,t of Example 16 for multiple fractional orders considering different values of u1 and t when u2=0.5 and Re=75.
u1tVIM [16]δ1=0.8δ1=0.9δ1=1JTDMCδ1=1JTDMABCExact0.20.10.72889403670.73047176940.73171829420.73269507740.73345699030.73236010100.20.72449059390.72644853710.72811955260.72953247190.72953247190.72810904010.30.71722638990.71934793620.72132522160.72314489240.72314489240.71675894000.40.69566928980.69566928980.65765492720.63806909430.63806909430.63788193740.50.71398837620.71604689840.71803280040.71991991880.71991991880.70940499080.40.10.65237815680.65879980810.66376968600.66376968600.66376968600.66722006510.20.63407718460.64246289390.64942052190.65516988370.65516988370.65377196390.30.61767997770.62711178130.63529223010.64234442460.64234442460.64234442460.40.60226487860.61221912420.62116221340.62912823090.62912823090.62500000000.50.58745863280.59756666150.60693611440.61552130250.61552130250.61041825140.60.10.54113543320.54843602520.55389861530.55800389150.55912905910.55912905910.20.51964571080.52993241140.53812497830.54465860510.54465860510.54920655090.30.49932530710.51162886630.52184897710.53031554300.53031554300.54059500910.40.47941824230.49312043680.50490636740.51497470510.51497470510.53324106000.50.45964338180.47426236710.48724060240.49863609140.49863609140.52704723860.80.10.50617741260.50822627290.50822627290.51085859780.51085859780.51133814510.21.1191927011.0771776861.0380803511.0025408191.0025408191.0025189170.30.49416132290.49780315140.50078130100.50321105830.50321105830.50721772680.41.2675920921.2333661091.1983254071.1626376931.1626376931.1622966700.50.48238858680.48686506090.49078533350.49417882240.49417882240.50456606961.00.10.50092079990.50127173270.50152972070.50172082230.50172082230.50180819090.20.49987105650.50038398000.50078471270.50109871300.50109871300.50143256160.30.49885314680.49948329160.49999709930.50041508130.50041508130.50113460790.40.49783788040.49855496810.49916116130.49966992720.49966992720.50089840070.50.49681525140.49759458370.49827542300.49886325090.49886325090.5007112274Table 2
The exactJTDMC and JTDMABC solutions of gu1,u2,t of Example 16 for multiple fractional orders considering different values of u1 and t when u2=0.5 and Re=75.
u1tVIM [16]δ2=0.8δ2=0.9δ2=1JTDMCδ2=1JTDMABCExact0.20.10.77117259190.76956475810.76830135930.76731531730.76731531730.76763989900.20.77568523990.77366219340.77194888490.77050910690.77050910690.77189095990.30.77962011660.77741925020.77546454270.77374447530.77374447530.77704723860.40.78323763850.78098773590.77891309170.77702142250.77702142250.78324106000.50.72816829040.71214116190.69543142600.67778238270.67778238270.67733155110.40.10.84887412220.84188672250.83659970020.83259075900.83259075900.83277993490.20.86922759940.85961827770.85186575580.84561158550.84561158550.84622803610.30.88815005320.87686978460.86737646520.85941388130.85941388130.86041825140.40.90645649890.89408980840.88331686550.87399764610.87399764610.87500000000.50.92446089630.91144929710.89975697680.88936288030.88936288030.88958174860.60.10.88936288030.95331709210.94704464560.94249499640.94249499640.94087094090.20.98879335100.97538204650.96515964680.96515964680.96515964680.96515964680.31.0155621980.99853841840.98496577510.98496577510.98496577510.95940499090.41.0378166281.0097947630.98110586300.95188723370.95188723370.95160802830.51.0862994401.0623897741.0374617771.0111324981.0111324981.0104599360.80.10.99493205280.99238196370.99059036030.98931448930.98931448930.98866185490.21.0028631530.99880217710.99576497070.99348443310.99348443310.99094498600.31.0110038341.0057243451.0015830480.99834672510.99834672510.99278227320.41.0194369001.0131690651.0080467951.0039013651.0039013650.99425565750.51.0281716041.0211226911.0151444571.0101483531.0101483530.99543393041.00.10.99927637530.99883636370.99852844030.99830993890.99830993890.99819180910.21.0006492920.99994370680.99941781570.99902433170.99902433170.99856743840.31.0020648091.0011436181.0004230950.99986176930.99986176930.99886539210.41.0035353271.0024383921.0015440831.0008222521.0008222520.99910159930.51.0050615141.0038250061.0027784241.0019057791.0019057790.9992887726Table 3
The comparison study among VIM [16], JTDMC, and JTDMABC of Example 16 for estimated solutions of fu1,u2,t and absolute error Eabs=Eexact−Eapprox at δ1=1 considering multiple values of u1 and t.
u1tExact−VIMExact−JTDMCExact−JTDMABC0.20.10.0003349760.00033497640.000344467720.20.00142343180.00142343180.00142343180.30.00339631540.00339631540.00339631540.40.00638595240.00638595240.00638595240.50.01051492800.01051492800.01051492800.40.10.00038454320.00038454320.00038454320.20.00139791980.00139791980.00139791980.30.00276267610.00276267600.00276267600.40.00412823090.00412823090.00412823090.50.00510305110.00510305110.00510305110.60.1-0.0011251676-0.0011251676-0.00112516760.2-0.0045479458-0.0045479458-0.00454794580.3-0.0102794660-0.0102794661-0.01027946610.4-0.0182663549-0.0182663549-0.01826635490.5-0.0284111473-0.0284111472-0.02841114720.80.1-0.0004795473-0.0004795473-0.00047954730.2-0.0018470989-0.0018470989-0.00184709890.3-0.0040066685-0.0040066685-0.00400666850.4-0.0068763152-0.0068763152-0.00687631520.5-0.0103872472-0.0103872472-0.01038724721.00.10.5018081909-0.0000873686-0.00008736860.2-0.0003338486-0.0003338486-0.00033384860.3-0.0007195266-0.0007195266-0.00071952660.4-0.0012284735-0.0012284735-0.00122847350.5-0.0018479765-0.0018479765-0.0018479765Table 4
The comparison study among VIM [16], JTDMC, and JTDMABC of Example 16 for estimated solutions of gu1,u2,t and absolute error Eabs=Eexact−Eapprox at δ2=1 considering multiple values of u1 and t.
u1tExact−VIMExact−JTDMCExact−JTDMABC0.20.1−0.0003349764−0.0003245817−0.00032458170.2−0.0014234318−0.0013818530−0.00138185300.3−0.0033963154−0.0033027633−0.00330276330.4−0.0063859524−0.0062196375−0.00621963750.5−0.0105149280−0.0102550608−0.01025506080.40.1−0.0003845432−0.0001891759−0.00018917590.2−0.0013979198−0.0006164506−0.00061645060.3−0.0027626761−0.0010043701−0.00100437010.4−0.0041282309−0.0010023539−0.00100235390.5−0.0051030511−0.0002188683−0.00021886830.60.10.00112516760.00162405550.00162405550.20.00454794580.00654349740.00654349740.30.01027946600.01476945730.01476945730.40.01826635490.02624856150.02624856150.50.02841114760.04088334460.04088334460.80.10.00047954730.00065263440.00065263440.20.00184709890.00253944710.00253944710.30.00400666850.00556445190.00556445190.40.00687631550.00964570750.00964570750.50.01038724660.01471442260.01471442261.00.10.00008736860.00011812980.00011812980.20.00033384860.00045689330.00045689330.30.00071952660.00099637720.00099637720.40.00122847370.00172065270.00172065270.50.00184797640.00261700640.0026170064Figure 1
3D view of the exact and approximate solution via JTDM of Example16 for fu1,u2,t when δ1=1.
(a)(b)Figure 2
(a) 3D view of absolute error forfu1,u2,t. (b) 2D view of multiple fractional orders via JTDM of Example 16 for fu1,u2,t when t=0.7.
(a)(b)Figure 3
3D view of the exact and approximate solution via JTDM of Example16 for gu1,u2,t when δ2=1.
(a)(b)In the analogous fashion, Figure4 displays the obtained outcome gu1,u2,t for Example 16. Figure 5 demonstrates the effect of collected information for Example 16 considering various classical and fractional orders δ2=1,0.9,0.8,0.7. Figure 6 exhibits the performance of the JTDM outcome and exact solution from fu1,u2,t for Example 19, whilst Figure 7 indicates the structure of the absolute error as well as multiple fractional orders. The various fractional orders δ1=1,0.9,0.8,0.7 are included in plot 7. Analogously, Figure 8 symbolizes the acquired solution gu1,u2,t for (39). Figure 9 represent the strength of collected information for Example 19 considering various fractional orders and absolute errors. Furthermore, the comparison of the exact and approximate solution for both compartments is compared in Figure 10. Based on modelling, we discovered that fractional-order solution trajectories incorporate integer-order solution trajectories. In a nutshell, the JTDM allows a framework for performing productively in a unified manner. Therefore, the JTDM is a well-known system that produces a reasonable approach avoiding any linearization assumptions [27].Figure 4
(a) 3D view of absolute error forgu1,u2,t. (b) 2D view of multiple fractional orders via JTDM of Example 16 for gu1,u2,t when t=0.7.
(a)(b)Figure 5
3D comparison view of the exact and approximate solution of Example16 when δ1=δ2=1.
(a)(b)Figure 6
3D view of the exact and approximate solution via JTDM of Example19 for fu1,u2,t when δ1=1.
(a)(b)Figure 7
(a) 3D view of absolute error forfu1,u2,t. (b) 2D view of multiple fractional orders via JTDM of Example 19 for fu1,u2,t when t=0.7.
(a)(b)Figure 8
3D view of the exact and approximate solution via JTDM of Example19 for gu1,u2,t when δ2=1.
(a)(b)Figure 9
(a) 3D view of absolute error forgu1,u2,t. (b) 2D view of multiple fractional orders via JTDM of Example 19 for gu1,u2,t when t=0.7.
(a)(b)Figure 10
3D comparison view of the exact and approximate solution of Example19 when δ1=δ2=1.
(a)(b)


## 7. Conclusion

The objective of this study is at putting into practice the Jafari transform decomposition approach to address Burgers’ equation by incorporating the Caputo and Antagana-Baleanu fractional derivative operators. Moreover, we present extensive conceptual evidence for the proposed strategy’s existence and uniqueness. Furthermore, we provide two mathematical formulations to demonstrate that suggested Burgers’ equation approach is viable and effective. The results for fractional problems are determined, and they are intimately associated with their realistic values. The proposed approach yields a series of outcomes of a recurrence connection with extreme precision and the fewest computations. Several numerical findings are evaluated using well-known analytical approaches, and the exact results are obtained whenδ1=δ2=1. The plots indicate that the precise and analytical findings have a clear association. The appropriateness of the specified procedures was validated by the generated figures. To better monitor the mechanisms of the provided challenges, results in various fractional orders are generated and displayed using graphs. The effectiveness of the proposed strategy has been validated by the convergence process. It was then extended by the investigators to address various scenarios using fractional partial differential equations.



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