# On Analytical Solution of Time-Fractional Biological Population Model by means of Generalized Integral Transform with Their Uniqueness and Convergence Analysis

Authors: Saima Rashid; Rehana Ashraf; Ebenezer Bonyah
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This research utilizes the generalized integral transform and the Adomian decomposition method to derive a fascinating explicit pattern for outcomes of the biological population model (BPM). It assists us in comprehending the dynamical technique of demographic variations in BPMs and generates significant projections. Besides that, generalized integral transforms are the unification of other existing transforms. To investigate the closed form solutions, we employed a fractional complex transform to deal with a partial differential equation of fractional order and a generalized decomposition method was applied to analyze the nonlinear equation. Several aspects of the Caputo and Atangana–Baleanu fractional derivative operators are discussed with the aid of a generalized integral transform. In mathematical terms, the variety of equations and their solutions have been discovered and identified with various novel features of the projected model. To provide additional context for these ideas, numerous sorts of illustrations and tabulations are presented. The precision and efficacy of the proposed technique suggest that it can be used for a variety of nonlinear evolutionary problems.


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## Body

## 1. Introduction

Historically, a framework of nonlinear developmental equations was designed to model the proportion of a demographic in particular domains [1, 2]. Several scholars have examined analytical, semiautomatic, and numerical solutions to fractional systems in a variety of fields, including virology, chaos, bifurcation, thermodynamics, neural networks, random walks, image processing, aquifer and anomalous spreading, and so on [3–5]. These behaviors have been described by expounding fractional numerical simulations compared with experimental findings in order to assert their nonlocal features, when this sort of characteristic [6–10] cannot be articulated using nonlinear PDEs of integer order.Numerous efficient and comprehensive approaches, such as the tan-cot function method [11], the Adomian decomposition method [12], the homotopy perturbation method [13], the homotopy analysis method [14], the wavelet method [15], and the Lie symmetry analysis [16], have been constructed determined by the flexibility to form complex nonlinear phenomena in diversified disciplines such as diseases, optical fibers, fluid flow, thermodynamics, electrostatics, reaction-diffusion, and plasma physics.Population dynamics simulations are being used to comprehend, interpret, and forecast the movements and permanence of biodiversity. Such frameworks are used to evaluate a population’s welfare, evaluate the reasons for population decreases or rapid expansion, recommend corporate agendas, and assess the forecasting of a population’s anticipated reactions to diverse organizational processes from a managerial perspective.Leading up to delving into the step-by-step methodologies for formulating and interpreting continuously BPMs, we will take a look at population genetic sculpting in the past, granting us an overview of the key figures in the field of ecology and evolution as well as the techniques they formed to comprehend population systems from a physics perspective.In this study, we investigate a fundamental model in biology. The degenerate parabolic equation appears in the spatial diffusion of biological populations [17, 18]:(1)Qς=Qw1w12+Qw2w22+σQ,ς≥0,w1,w2∈ℝ,subject to initial condition (IC) Qw1,w2,0, where Q signifies the population density and σ denotes the population supply due to births and deaths.Certain specific features of (1), for instance, Hölder estimates of its solutions, are investigated in [19]. Furthermore, two basic examples of constitutive formulations for σ are the Malthusian law [17],(2)σ=νQ,ν=constant,and the Verhulst law [19],(3)σ=νQ−γQ2,ν,γ=constant.Thus, we assume the more generic version ofσ as σQ=ℏQα1−r1Qβ which is held for porous media [20, 21] as follows:(4)Qς=Qw1w12+Qw2w22+ℏQα1−r1Qβ,ς≥0,w1,w2∈ℝ,where α,β,ℏ, and r1 are real constants.It is noted that Malthusian and Verhulst laws are the particular cases that can be attained by insertingℏ=ν,α=1,andr1=0 and ℏ=ν,α=β=1,andr1=γ/ν, respectively.FDEs are remarkably appropriate for simulating biochemical mechanisms because they are specifically apprehensive about biological analytical models’ memory, which appears to be a substantial advancement over traditional integer-order mathematical methodologies, and it is associated with chaotic systems, which are prevalent in BPMs. Rashid et al. [22] recently considered the fractional spatial diffusion of a BPM via a new integral transform in the singular and nonsingular kernel settings. Zellal and Belghaba [23] reported the variational iteration method to find an accurate algorithm for solving BPM. Singh [24] expounded the analysis of the fractional blood alcohol model with a composite fractional derivative. An epidemic model SEI1I2R for the transmission of HIV epidemics by the mean value theorem was investigated by Naik et al. [25]. A homotopy decomposition method was employed by investigating the HIV infection of CD4+ by Atangana and Alabaraoye [26]. For more details on BPM, see [27, 28] and the citations therein.Amidst George Adomian’s massive boost in 1980, the Adomian decomposition method introduced a well-noted terminology. It has been intensively implemented for a diverse set of nonlinear PDEs, for instance, the Korteweg–De Vries model [29], Fisher’s model [30], Zakharov–Kuznetsov equation [31], and so on. The ADM was determined to be significantly related to a variety of integral transforms, including Laplace, Swai, Mohand, Aboodh, Elzaki, and others. Very recently, Jafari [32] propounded a well-known integral transform which is known to generalized integral transform. The dominant feature of this transformation is that it has the ability to recapture several existing transformations (see Remark 1).Motivated by the above propensity, we aim to establish a semianalytical approach by mingling the generalized integral transform with the Adomian decomposition method. With the assistance of fractional derivative operators, we constructed the approximate analytical solution to BPM. Furthermore, the convergence and uniqueness analysis is carried out in the Caputo fractional derivative framework. The proposed findings are in close harmony with the exact solutions. Sketching and comparison analysis solutions are drafted with a powerful and pragmatic approach. Both operators consistently behave according to the projected method.


## 2. Preliminaries

In this section, we evoke some essential concepts, notions, and definitions concerning fractional derivative operators depending on power and Mittag-Leffler as a kernel, along with the detailed consequences of the generalized integral transform.Definition 1.
(see [6]). The Caputo fractional derivative CFD is described as follows:(5)Dςχ0c=1Γr−χ∫0ςQrw1ς−w1χ+1−rdw1,r−1<χ<r,drdςrQς,χ=r.Definition 2.
(see [7]). The Atangana–Baleanu fractional derivative operator in the Caputo form ABC is stated as follows:(6)Dςχη1ABCQς=Aχ1−χ∫η1ςQ′ςEχ−χς−w1χ1−χdw1,where Q∈ℋ1a1,a2Sobolev space,a1<a2,χ∈0,1, and Aχ signifies a normalization function as Aχ=A0=A1=1.Definition 3.
(see [7]). The fractional integral of the ABC operator is described as follows:(7)ℐςχη1ABCQς=1−χAχQς+χΓχAχ∫η1ςQw1ς−w1χ−1dw1.Definition 4.
(see [32]). Consider an integrable mapping Qς defined on a set P; then,(8)P=Qς:∃M>0,κ>0,Qς<Mexpκς,ifς≥0.Definition 5.
(see [32]). Suppose the mappings ϕs,ψs:ℝ+↦ℝ+ such that φs≠0∀s∈ℝ+. The generalized integral transform of the mapping Qς presented by Qs is described as(9)JQς,s=Qs=ϕs1∫0∞Qςexp−ψsςdς.Theorem 1 (see [32]) (convolution property).
For generalized integral transform, the subsequent holds true:(10)JQ1∗Q2=1ϕsQ1s∗Q2s.Definition 6.
The generalized integral transform of the CFD operator is stated as follows:(11)JDςχ0cQς,s=ψχsQs1−ϕs∑κ=0χ−1ψχ−κ−1s1Qκ0,r−1<χ<r,ϕ,ψ>0.Remark 1.
Definition6 leads to the following conclusions:(1)
Takingϕs=1 and ψs=s, then we acquire the Laplace transform [33](2)
Takingϕs1/s and ψs=1/s, then we acquire the α-Laplace transform [34](3)
Takingϕs=1/s and ψs=1/s, then we acquire the Sumudu transform [35](4)
Takingϕs=1/s and ψs=1, then we acquire the Aboodh transform [36](5)
Takingϕs=s and ψs=s2, then we acquire the Pourreza transform [37, 38](6)
Takingϕs=s and ψs=1/s, then we acquire the Elzaki transform [39](7)
Takingϕs=w2 and ψs=s/w2, then we acquire the natural transform [40](8)
Takingϕs=s2 and ψs=s, then we acquire the Mohand transform [41](9)
Takingϕs=1/s2 and ψs=1/s, then we acquire the Swai transform [42](10)
Takingϕs=1 and ψs=1/s, then we get the Kamal transform [43](11)
Takingϕs=sα and ψs=1/s, then we acquire the G− transform [44, 45]Definition 7.
(see [46]). The generalized integral transform of the ABC fractional derivative operator is described as(12)JDςχη1ABCQς,sχ=Aχψχsχ+1−χψχsQs−ϕsψsQ0.Remark 2.
Definition7 leads to the following conclusions:(1)
Takingϕs=1 and ψs=s, then we acquire the Laplace transform of ABC fractional derivative operator [7, 47](2)
Takingϕs=s and ψs=1/s, then we acquire the Elzaki transform of ABC fractional derivative operator [48](3)
Takingϕs=ψs=1/s, then we get the Sumudu transform of ABC fractional derivative operator [49](4)
Takingϕs=1 and ψs=s/w2, then we get the Shehu transform of ABC fractional derivative operator [49]Definition 8.
(see [50]). The Mittag-Leffler function for single parameter is described as(13)Eχz=∑κ=0∞z1κΓκχ+1,χ,z1∈ℂ,ℜχ≥0.


## 3. Description of the Generalized Decomposition Method

Consider the generic fractional form of PDE:(14)DςχQw1,ς+LQw1,ς+N˜Qw1,ς=ℱw1,ς,ς>0,0<χ≤1,with ICs(15)Qw1,0=Gw1,where Dςχ=∂χQw1,ς/∂ςχ symbolizes the Caputo and ABC fractional derivative of order χ∈0,1, while L and N˜ denote the linear and nonlinear factors, respectively. Also, ℱw1,ς represents the source term.Taking into account the generalized integral transform to (14), we acquire(16)JDςχQw1,ς+LQw1,ς+N˜Qw1,ς=Jℱw1,ς.First, by applying the differentiation rule of generalized integral transform with respect to CFD, we apply the ABC fractional derivative operator as follows:(17)ψχsUw1,s=ϕs∑κ=0ℓ−1ψχ−1−κsQκ0+JLQw1,ς+N˜Qw1,ς+Jℱw1,ς,(18)ψχsAχχ+1−χψχsUw1,s=ϕsψsψχsAχχ+1−χψχsQ0+JLQw1,ς+N˜Qw1,ς+Jℱw1,ς.The inverse generalized integral transform of (17) and (18) yields(19)Qw1,ς=J−1ϕs∑κ=0ℓ−1ψsχ−κ−1Qκ0+1ψχsJℱw1,ς−J−11ψχsJLQw1,ς+N˜Qw1,ς,(20)Qw1,ς=J−1ϕsψsQ0+χ+1−χψχsψχsAχJℱw1,ς−J−1χ+1−χψχsψχsAχJLQw1,ς+N˜Qw1,ς.The generalized decomposition method solutionQw1,ς is represented by the following infinite series:(21)Qw1,ς=∑ℓ=0∞Qℓw1,ς.Thus, the nonlinear termN˜w1,ς can be evaluated by the Adomian decomposition method prescribed as(22)N˜Qw1,ς=∑ℓ=0∞A˜ℓQ0,Q1,…,ℓ=0,1,…,where(23)A˜ℓQ0,Q1,…=1ℓ!dℓdχℓN˜∑j=0∞χjQjχ=0,ℓ>0.Inserting (21) and (22) into (19) and (20), respectively, we have(24)∑ℓ=0∞Qℓw1,ς=Gw1+G˜w1−J−11ψχsJLQw1,ς+∑ℓ=0∞A˜ℓ,(25)∑ℓ=0∞Qℓw1,ς=Gw1+G˜w1−J−1χ+1−χψχsAχψχsJLQw1,ς+∑ℓ=0∞A˜ℓ.Consequently, the recursive technique for (24) and (25) is established as follows:(26)Q0w1,ς=Gw1+G˜w1,ℓ=0,Qℓ+1w1,ς=−J−11ψχsJLQℓw1,ς+∑ℓ=0∞A˜ℓ,ℓ≥1,Qℓ+1w1,ς=−J−1χ+1−χψχsAχψχsJLQℓw1,ς+∑ℓ=0∞A˜ℓ,ℓ≥1.


## 4. Some New Mathematical Aspects of Generalized Decomposition Method

The subsequent subsections will highlight how the sufficient requirements guarantee the emergence of a unique solution. Our anticipated existence of solutions in the case of GDM is followed by [51].Theorem 2 (uniqueness theorem).
Equation (26) has a unique solution whenever 0<ϵ<1, where ϵ=K1+K2+K3ςχ/Γχ+1.Proof.
Assume all continuous functions on the Banach space are denoted byΩ=ℂℐ,⋅. Also, suppose that ℐ=0,T has the norm ⋅. Now, we define a function U:Ω↦Ω such that(27)Qℓ+1w1,ς=Qw1,ς+J−11ψχsJLQℓw1,ς+P¯Qℓw1,ς+N˜Qℓw1,ς,ℓ≥0,where LQw1,ς≡∂3Qw1,ς/∂w12 and P¯Qw1,ς≡∂Qw1,ς/∂w1. Here, suppose that LQw1,ς and MQw1,ς are also Lipschitzian with P¯Q−P¯Q∧<K1Q−Q∧ and LQ−LQ∧<K2Q−Q∧, where K1 and K2 are Lipschitz constant, respectively, and QandQ∧ are distinct functional values.(28)UQ−UQ∧=maxς∈ℐJ−11ψχsJLQw1,ς+P¯Qw1,ς+N˜Qw1,ς−J−11ψχsJLQ∧w1,ς+P¯Q∧w1,ς+N˜Q∧w1,ς≤maxς∈ℐJ−11ψχsJLQw1,ς−LQ∧w1,ς+J−11ψχsJP¯Qw1,ς−P¯Q∧w1,ς+J−11ψχsJN˜Qw1,ς−N˜Q∧w1,ς≤maxς∈ℐK1J−11ψχsJQw1,ς−Q∧w1,ς+K2J−11ψχsJQw1,ς−Q∧w1,ς+K3J−11ψχsJQw1,ς−Q∧w1,ς≤maxς∈ℐK1+K2+K3J−11ψχsJQw1,ς−Q∧w1,ς≤K1+K2+K3J−11ψχ+1sJQw1,ς−Q∧w1,ς=K1+K2+K3J−1ϕsψχsQw1,ς−Q∧w1,ς=K1+K2+K3ςχΓχ+1Qw1,ς−Q∧w1,ς.
Since0<ϵ<1, the mapping is contraction. Consequently, by Banach contraction fixed point theorem, (14) has a unique solution. This gives the desired result.Theorem 3 (convergence analysis).
The general form solution of (14) will be convergent.Proof.
Assume thatW∧ℓ be the nth partial sum, i.e., W∧ℓ=∑m=0ℓQℓw1,ς. Here, we prove a Cauchy sequence W∧ℓ in Banach space U.
We acquire by considering a new form of Adomian polynomials:(29)R¯W∧ℓ=ℋ˜ℓ+∑p=0ℓ−1ℋ˜p,N˜W∧ℓ=ℋ˜ℓ+∑c=0ℓ−1ℋ˜c.
Now,(30)W∧ℓ−W∧q=maxς∈ℐW∧ℓ−W∧q=maxς∈ℐ∑m=q+1ℓQ∧w1,ς,m=1,2,3,…≤maxς∈ℐJ−11ψχsJ∑m=q+1ℓLQℓ−1w1,ς+J−11ψχsJ∑m=q+1ℓP¯Qℓ−1w1,ς+J−11ψχsJ∑m=q+1ℓℋ˜ℓ−1w1,ς=maxς∈ℐJ−11ψχsJ∑m=qℓ−1LQℓw1,ς+J−11ψχsJ∑m=qℓ−1P¯Qℓw1,ς+J−11ψχsJ∑m=qℓ−1ℋ˜ℓw1,ς≤maxς∈ℐJ−11ψχsJ∑m=qℓ−1LW∧ℓ−1−LW∧q−1+J−11ψχsJ∑m=qℓ−1P¯W∧ℓ−1−P¯W∧q−1+J−11ψχsJ∑m=qℓ−1N˜W∧ℓ−1−N˜W∧q−1≤maxς∈ℐJ−11ψχsJLW∧ℓ−1−LW∧q−1+J−11ψχsJP¯W∧ℓ−1−P¯W∧q−1+J−11ψχsJN˜W∧ℓ−1−N˜W∧q−1≤K1maxς∈ℐJ−11ψχsJW∧ℓ−1−W∧q−1+K2maxς∈ℐJ−11ψχsJW∧ℓ−1−W∧q−1+K3maxς∈ℐJ−11ψχsJW∧ℓ−1−W∧q−1=K1+K2+K3J−1ϕsψχ+1sW∧ℓ−1−W∧q−1=K1+K2+K3ςχΓχ+1W∧ℓ−1−W∧q−1.
Considern=q+1; then,(31)W∧q+1−W∧q≤ϵW∧q−W∧q−1≤W∧2W∧q−1−W∧q−2≤⋯≤ϵqW∧1−W∧0,where K1+K2+K3ςχ/Γχ+1. Now, from triangular inequality, we have(32)W∧ℓ−W∧q≤W∧q+1−W∧q+W∧q+2−W∧q+1+⋯+W∧ℓ−W∧ℓ−1≤ϵq+ϵq+1+⋯+ϵℓ−1W∧1−W∧0≤ϵq1−ϵℓ−qϵQ1.
Since0<ϵ<1, we have 1−ϵℓ−q<1; then,(33)W∧ℓ−W∧q≤ϵq1−ϵmaxς∈ℐQ1.
Therefore,Q1<∞ (since Qw1,ς is bounded). Furthermore, as q↦∞, then W∧ℓ−W∧q↦0. Thus, W∧1 is a Cauchy sequence in K. Consequently, the series ∑n=0∞Qℓ is convergent and this yields the immediate consequence.Theorem 4 (see [51]) (error estimate).
The absolute error of the series solution (14)–(26) is calculated as(34)maxς∈ℐQw1,ς−∑ℓ=1qQℓw1,ς≤ϵq1−ϵmaxς∈ℐQ1.


## 5. Application of Generalized Integral Transform for Biological Population Model

In what follows, we illustrate the technique described in Section3 by considering three numerical tests to validate the supremacy and efficacy of the generalized decomposition method.Example 1 (Malthusian law [17]).
Assume the time-fractional BPM (4) having α=1andr1=0; then,(35)DςαQw1,w2,ς=Qw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς,with IC(36)Q0w1,w2,0=w1+w2+w1w2.Proof.
Foremost, we provide the solution of (35) in two general cases.
Case 1First, we apply the Caputo fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 1,(37)ψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς.
Taking into consideration the IC given in (36), we have(38)Uw1,s=ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς.
Employing the inverse generalized integral transform, we obtain(39)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς.
Thanks to the generalized decomposition method, we find(40)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψsϕsw1+w2+w1w2=w1+w2+w1w2.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(41)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱQ can be decomposed by an infinite series of polynomials represented by(42)ℱQ=Qw1w12=∑ℓ=0∞Aℓ,ℱQ=Qw2w22=∑ℓ=0∞ℬℓ,where Qℓw1,w2,ς will be evaluated recurrently and Aℓ and ℬℓ are the so-called polynomials of Q0,Q1,…,Qℓ established by [52].(43)∑ℓ=0∞Qℓ+1w1,w2,ς=J−11ψχsJ∑ℓ=0∞Aℓ+∑ℓ=0∞ℬℓ+ℏ∑ℓ=0∞Qℓ,ℓ=0,1,2,….
The first few Adomian polynomials are presented as follows:(44)AℓQ2w1w1=Q02w1w1,ℓ=0,2Q0Q1w1w1,ℓ=1,Q12+2Q0Q1w1w1,ℓ=2,ℬℓQ2w2w2=Q02w2w2,ℓ=0,2Q0Q1w2w2,ℓ=1,Q12+2Q0Q1w2w2,ℓ=2.
Forℓ=0,1,2,3,…,(45)Q1w1,w2,ς=J−11ψχsJA0+ℬ0+ℏQ0=ℏw1+w2+w1w2ςχΓχ+1,Q2w1,w2,ς=J−11ψχsJA1+ℬ1+ℏQ1=ℏ2w1+w2+w1w2ς2χΓ2χ+1,Q3w1,w2,ς=J−11ψχsJA2+ℬ2+ℏQ2=ℏ3w1+w2+w1w2ς3χΓ3χ+1,⋮.
The approximate solution for Example1 is expressed as follows:(46)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=w1+w2+w1w21+ℏςχΓχ+1+ℏ2ς2χΓ2χ+1+ℏ3ς3χΓ3χ+1+⋯.
Case 2. Here, we surmise ABC fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 1,(47)ψχsAχχ+1−χψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς.
Taking into consideration the IC given in (36), we have(48)Uw1,s=ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς.
Employing the inverse generalized integral transform, we obtain(49)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς.
Thanks to the generalized decomposition method, we find(50)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψs1ϕsw1+w2+w1w2=w1+w2+w1w2.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(51)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱjQ,j=1,2, can be decomposed by an infinite series of polynomials represented by(52)ℱ1Q=Qw1w12=∑ℓ=0∞Aℓ,ℱ2Q=Qw2w22=∑ℓ=0∞ℬℓ,where Qℓw1,w2,ς will be evaluated recurrently and Aℓ and ℬℓ are the so-called polynomials of Q0,Q1,…,Qℓ defined in (44).
Forℓ=0,1,2,3,…,(53)Q1w1,w2,ς=J−1χ+1−χψχsψχsAχJA0+ℬ0+ℏQ0=ℏw1+w2+w1w2AχχςχΓχ+1+1−χ,Q2w1,w2,ς=J−1χ+1−χψχsψχsAχJA1+ℬ1+ℏQ1=ℏ2w1+w2+w1w2A2χχ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2,Q3w1,w2,ς=J−1χ+1−χψχsψχsAχJA2+ℬ2+ℏQ2=ℏ3w1+w2+w1w2A3χχ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3,⋮.
The approximate solution for Example1 is expressed as follows:(54)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=w1+w2+w1w21+ℏAχςχΓχ+1+1−χ+ℏ2A2χχ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2+ℏ3A3χχ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3+⋯.
Forχ=1, we obtained the exact solution of Example 1 as(55)Qw1,w2,ς=w1+w2+w1w2expℏς.
The analytical approximate solutions including certain random initialization produced by the proposed methodology are shown in Table1. The VIMHP is employed to perform the comparative analysis, which forecasts the precision of the proposed methodology based on its lower error. The findings in this study are tremendously helpful in comprehending the internal components of natural disasters. We will describe the scientific clarification of the solutions for the BP model in this paragraph of the article. The exact and numerical solution for α=1,r1=0,ς=0.01, and χ=1 is shown in Figure 1.
Furthermore, the absolute error for the aforementioned assumptions is depicted in Figure2(a). Figure 2(b) demonstrates the behavior of the findings in three-dimensional simulation for different fractional orders by employing the Caputo fractional derivative operator.
Finally, Figure3 represents the two-dimensional behavior of exact, approximate by Caputo, and approximate solutions derived by ABC fractional operators with varying fractional orders. These solutions have a distinctive characteristic that allows them to interact with other solutions derived by [23]. The proposed findings have particle-like geometries in their solutions. The synthesized trajectory is either a success or a descent from one asymptotic state to the next. The accuracy of the proposed method can be enhanced by increasing the recursive terms.Table 1
ExactQE and approximate solutions QCFD and QABC of Qw1,w2,ς of Example 1 having absolute errors E1=QE−QCFD and E2=QE−QABC when χ=1,ς=0.2,r1=0, and ℏ=1/2 for different values of w1 and w2.
w1,w2QE sol.QCFD sol.QABC sol.VIMHP sol. [23]Error=E1Error=E2(−10, 10)11.05170911.05166611.05166611.0513444.2510e − 54.2510e − 5(−8, 8)8.8413678.8413338.8413338.8416003.4008e − 53.4008e − 5(−6, 6)6.6310256.6310006.6310006.6315552.5506e − 52.5506e − 5(−4, 4)6.6310256.6310006.6310006.6678902.5506e − 52.5506e − 5(−2, 2)2.2103412.2103332.2103332.2266998.5020e − 68.5020e − 6(0, 0)0.0000000.0000000.0000001.0089750.0000000.000000(2, 2)3.1258003.1258003.1258003.9985001.2024e − 51.2024e − 5(4, 4)5.4141005.4141005.4141006.8892002.0826e − 52.0826e − 5(6, 6)7.6571007.6571007.6571008.1006502.9453e − 52.9453e − 5(8, 8)9.8851009.8851009.8851009.9878903.8023e − 53.8023e − 5(10, 10)1.2106001.2106001.2106001.98234.6567e − 54.6567e − 5Figure 1
Three-dimensional illustration of the exact and approximate solution of Example1 when α=1,r1=0,ς=0.01, and χ=1.
(a)(b)Figure 2
Three-dimensional illustration of the absolute error and multiple surface of various fractional orders for Example1 when α=1,r1=0, and ς=0.01.
(a)(b)Figure 3
Two-dimensional illustration of Example1 for various fractional orders when α=1,r1=0, and ς=0.01.Remark 3.
Example1 leads to the conclusion that(i)
If we replace the ICQw1,w2,ς=α1w1+β1w2+γ1w1w2 in Example 1, then the approximate solution can be achieved as(56)Qw1,w2,ς=α1w1+β1w2+γ1w1w21+ℏςχΓχ+1+ℏ2ς2χΓ2χ+1+ℏ3ς3χΓ3χ+1+⋯.(ii)
The closed form solution in the stated case will beQw1,w2,ς=α1w1+β1w2+γ1w1w2expℏς, where α1 and β1 are the real constants.(iii)
Lettingα1=β1=0 and γ1=1 along with ℏ=1/2, we get the result proposed independently by Roul [53] and Shakeri and Dehgan [54], respectively.Example 2.
Assume the time-fractional BPM (4) having α=1,r1=0, and ℏ=1; then,(57)DςαQw1,w2,ς=Qw1w12w1,w2,ς+Qw2w22w1,w2,ς+Qw1,w2,ς,with IC(58)Q0w1,w2,0=sinθw1coshθw2,θ∈ℝ.Proof.
. Foremost, we provide the solution of (57) in two general cases.
Case 1. First, we apply the Caputo fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 2,(59)ψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+Qw1,w2,ς.
Taking into consideration the IC given in (58), we have(60)Uw1,s=ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+Qw1,w2,ς.
Employing the inverse generalized integral transform, we obtain(61)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+Qw1,w2,ς.
Thanks to the generalized decomposition method, we find(62)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψsϕssinθw1coshθw2=sinθw1coshθw2.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(63)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱjQ,j=1,2, can be decomposed by an infinite series of polynomials represented by ℱ1Q=Qw1w12=∑ℓ=0∞Aℓandℱ2Q=Qw2w22=∑ℓ=0∞ℬℓ defined in (44).(64)∑ℓ=0∞Qℓ+1w1,w2,ς=J−11ψχsJ∑ℓ=0∞Aℓ+∑ℓ=0∞ℬℓ+∑ℓ=0∞Qℓ,ℓ=0,1,2,….
Forℓ=0,1,2,3,…,(65)Q1w1,w2,ς=J−11ψχsJA0+ℬ0+ℏQ0=sinθw1coshθw2ςχΓχ+1,Q2w1,w2,ς=J−11ψχsJA1+ℬ1+ℏQ1=sinθw1coshθw2ς2χΓ2χ+1,Q3w1,w2,ς=J−11ψχsJA2+ℬ2+ℏQ2=sinθw1coshθw2ς3χΓ3χ+1,⋮.
The approximate solution for Example2 is expressed as follows:(66)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=sinθw1coshθw21+ςχΓχ+1+ς2χΓ2χ+1+ς3χΓ3χ+1+⋯.
Case 2. Here, we surmise ABC fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 2,(67)ψχsAχχ+1−χψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+Qw1,w2,ς.
Taking into consideration the IC given in (58), we have(68)Uw1,s=ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+Qw1,w2,ς.
Employing the inverse generalized integral transform, we obtain(69)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+Qw1,w2,ς.
Thanks to the generalized decomposition method, we find(70)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψsϕssinθw1coshθw2=sinθw1coshθw2.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(71)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱjQ,j=1,2, can be decomposed by an infinite series of polynomials represented by(72)ℱ1Q=Qw1w12=∑ℓ=0∞Aℓ,ℱ2Q=Qw2w22=∑ℓ=0∞ℬℓ,where Qℓw1,w2,ς will be evaluated recurrently and Aℓ and ℬℓ are the so-called polynomials of Q0,Q1,…,Qℓ defined in (44).
Forℓ=0,1,2,3,…,(73)Q1w1,w2,ς=J−1χ+1−χψχsψχsAχJA0+ℬ0+Q0=sinθw1coshθw2AχχςχΓχ+1+1−χ,Q2w1,w2,ς=J−1χ+1−χψχsψχsAχJA1+ℬ1+Q1=sinθw1coshθw2A2χχ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2,Q3w1,w2,ς=J−1χ+1−χψχsψχsAχJA2+ℬ2+Q2=sinθw1coshθw2A3χχ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3,⋮.
The approximate solution for Example2 is expressed as follows:(74)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=sinθw1coshθw21+1AχςχΓχ+1+1−χ+1A2χχ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2+1A3χχ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3+⋯.
Forχ=1, we obtained the exact solution of Example 2 as(75)Qw1,w2,ς=sinθw1coshθw2expς.
The analytical approximate solutions including certain random initialization produced by the proposed methodology are shown in Table2. The VIMHP is employed to perform the comparative analysis, which forecasts the precision of the proposed methodology based on its lower error. The findings in this study are tremendously helpful in comprehending the internal components of natural disasters. We will describe the scientific clarification of the solutions for the BP model in this paragraph of the article. The exact and numerical solution for α=1,r1=0,ς=0.01, and ℏ=χ=1 is shown in Figure 4.
Furthermore, the absolute error for the aforementioned assumptions is depicted in Figure5(a). Figure 5(b) demonstrates the behavior of the findings in three-dimensional simulation for different fractional orders by employing the Caputo fractional derivative operator.
Finally, Figure6 represents the two-dimensional behavior of exact, approximate by Caputo, and approximate solutions derived by ABC fractional operators with varying fractional orders. These solutions have a distinctive characteristic that allows them to interact with other solutions derived by [23]. The proposed findings have particle-like geometries in their solutions. The synthesized trajectory is either a success or a descent from one asymptotic state to the next. The accuracy of the proposed method can be enhanced by increasing the recursive terms.Table 2
ExactQE and approximate solution QA of Qw1,w2,ς of Example 2 having absolute errors E1=QE−QCFD and E2=QE−QABC when χ=1,ς=0.2,r1=−8/9, and ℏ=−1 for different values of w1 and w2.
w1,w2QE sol.QCFD sol.QABC sol.VIMHP sol. [23]Error=E1Error=E2(−10, 10)10.30343610.30300010.30285110.9934515.8565e − 45.8565e − 4(−8, 8)5.5526075.5522925.5522925.9900913.1561e − 43.1561e − 4(−6, 6)1.4558531.4557701.4557701.9950998.2751e − 58.2751e − 5(−4, 4)2.2590852.2589572.2589572.8793421.2841e − 41.2841e − 4(−2, 2)1.3917871.3917081.3917082.6008347.9109e − 57.9109e − 5(0, 0)0.0000000.0000000.0000000.9900870.0000000.000000(2, 2)1.3917871.3917001.3917001.9909807.9109e − 57.9109e − 5(4, 4)2.2590852.2589572.2589572.9984561.2841e − 41.2841e − 4(6, 6)1.4558531.4557701.4557702.0099878.2751e − 58.2751e − 5(8, 8)5.5526075.55232925.55232925.9689013.1561e − 43.1561e − 4(10, 10)10.30300010.30304410.30304410.9902345.8565e − 45.8565e − 4Figure 4
Three-dimensional illustration of exact and approximate solution of Example2 when α=1,r1=0,ς=0.01,ℏ=1, and χ=1.
(a)(b)Figure 5
Three-dimensional illustration of the absolute error and multiple surface of various fractional orders for Example2 when α=1,r1=0,ℏ=1, and ς=0.01.
(a)(b)Figure 6
Two-dimensional illustration of Example2 for various fractional orders when α=1,r1=0,ℏ=1, and ς=0.01.Example 3 (Verhulst law [19]).
Assume the time-fractional BPM (4) having α=β=1; then,(76)DςαQw1,w2,ς=Qw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς1−r1Qw1,w2,ς,with IC(77)Q0w1,w2,0=expℏr18w1+w2.Proof.
Foremost, we provide the solution of (76) in two general cases.
Case 1. First, we apply the Caputo fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 3,(78)ψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς1−r1Qw1,w2,ς.
Taking into consideration the IC given in (77), we have(79)Uw1,s=ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς1−r1Qw1,w2,ς.
Employing the inverse generalized integral transform, we obtain(80)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς1−r1Qw1,w2,ς.
Thanks to the generalized decomposition method, we find(81)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψsϕsexpℏr18w1+w2=expℏr18w1+w2.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(82)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱjQ,j=1,2,3, can be decomposed by an infinite series of polynomials represented by ℱ1Q=Qw1w12=∑ℓ=0∞Aℓandℱ2Q=Qw2w22=∑ℓ=0∞ℬℓ defined in (44) along with ℱ3Q=Q1−r1Q=∑ℓ=0∞Cℓ. Thus, we have(83)CℓQ−r1Q2=Q0−r1Q02,ℓ=0,Q2−r1Q0Q1,ℓ=1,Q2−2r1Q0Q2−r1Q12,ℓ=2,∑ℓ=0∞Qℓ+1w1,w2,ς=J−11ψχsJ∑ℓ=0∞Aℓ+∑ℓ=0∞ℬℓ+∑ℓ=0∞Cℓ,ℓ=0,1,2,….
Forℓ=0,1,2,3,…,(84)Q1w1,w2,ς=J−11ψχsJA0+ℬ0+C0=expℏr18w1+w2ςχΓχ+1,Q2w1,w2,ς=J−11ψχsJA1+ℬ1+C1=expℏr18w1+w2ς2χΓ2χ+1,Q3w1,w2,ς=J−11ψχsJA2+ℬ2+C2=expℏr18w1+w2ς3χΓ3χ+1,⋮.
The approximate solution for Example3 is expressed as follows:(85)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=expℏr18w1+w21+ςχΓχ+1+ς2χΓ2χ+1+ς3χΓ3χ+1+⋯.
Case 2. Here, we surmise ABC fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 3,(86)ψχsAχχ+1−χψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς1−r1Qw1,w2,ς.
Taking into consideration the IC given in (77), we have(87)Uw1,s=ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς1−r1Qw1,w2,ς.
Employing the inverse generalized integral transform, we obtain(88)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+ℏQw1,w2,ς1−r1Qw1,w2,ς.
Thanks to the generalized decomposition method, we find(89)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψsϕsexpℏr18w1+w2=expℏr18w1+w2.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(90)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱjQ,j=1,2,3, can be decomposed by an infinite series of polynomials represented by ℱ1Q=Qw1w12=∑ℓ=0∞Aℓandℱ2Q=Qw2w22=∑ℓ=0∞ℬℓ along with ℱ3Q=Q1−r1Q=∑ℓ=0∞Cℓ defined in (44) and (83), respectively.
Forℓ=0,1,2,3,…,(91)Q1w1,w2,ς=J−1χ+1−χψχsψχsAχJA0+ℬ0+C0=1Aχexpℏr18w1+w2χςχΓχ+1+1−χ,Q2w1,w2,ς=J−1χ+1−χψχsψχsAχJA1+ℬ1+C1=1A2χexpℏr18w1+w2χ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2,Q3w1,w2,ς=J−1χ+1−χψχsψχsAχJA2+ℬ2+C2=1A3χexpℏr18w1+w2χ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3,⋮.
The approximate solution for Example3 is expressed as follows:(92)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=expℏr18w1+w21+1AχςχΓχ+1+1−χ+1A2χχ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2+1A3χχ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3+⋯.
Forχ=1, we obtained the exact solution of Example 3 as(93)Qw1,w2,ς=expℏr18w1+w2+ς.
Table3 shows the analytical approximate solutions with some free parameters that are provided by the proposed technique. The comparison analysis is conducted with the VIMHP that predicts the preciseness of the suggested scheme due to their lower error. The analytical findings are extremely useful in deciphering the internal components of acts of nature. The exact and numerical solution for α=1=β,ς=0.01, and χ=1 is shown in Figure 7.
Furthermore, the absolute error for the aforementioned assumptions is depicted in Figure8(a). Figure 8(b) demonstrates the behavior of the findings in three-dimensional simulation for different fractional orders by employing the Caputo fractional derivative operator.
Finally, Figure9 represents the two-dimensional behavior of exact, approximate by Caputo, and approximate solutions derived by ABC fractional operators with varying fractional orders. These solutions have a distinctive characteristic that allows them to interact with other solutions derived by [23]. The proposed findings have particle-like geometries in their solutions. The synthesized trajectory is either a success or a descent from one asymptotic state to the next. The accuracy of the proposed method can be enhanced by increasing the recursive terms.Table 3
ExactQE and approximate solution QA of Qw1,w2,ς of Example 3 having absolute errors E1=QE−QCFD and E2=QE−QABC when χ=1,ς=0.2,r1=−8/9, and ℏ=−1 for different values of w1 and w2.
w1,w2QE sol.QCFD sol.QABC sol.VIMHP sol. [23]Error=E1Error=E2(−10, 10)1.04194e − 31.04185e − 31.041824e − 31.04186e − 38.1559e − 88.1599e − 8(−8, 8)3.95279e − 33.95249e − 33.95243e − 33.95248e − 33.0941e − 73.0939e − 7(−6, 6)1.49956e − 21.49940e − 21.49939e − 21.49944e − 21.1732e − 61.1730e − 6(−4, 4)5.68882e − 25.68832e − 25.68830e − 25.68838e − 24.4529e − 64.4527e − 6(−2, 2)2.15815e − 12.15792e − 12.15790e − 12.15798e − 11.6893e − 51.6890e − 5(0, 0)8.18731e − 18.186677e − 18.186670e − 18.186667e − 16.4086e − 56.4082e − 5(2, 2)3.105990003.105753.105703.105662.4312e − 42.4310e − 4(4, 4)11.783100011.7821211.7821011.782189.2233e − 49.2226e − 4(6, 6)44.701180044.6975644.6975044.697693.4989e − 33.4982e − 3(8, 8)169.581450169.56803169.56810169.568171.3274e − 21.3278e − 2(10, 10)643.335670643.28526643.28519643.285345.0357e − 25.0350e − 2Figure 7
Three-dimensional illustration of exact and approximate solution of Example3 when α=1=β,r1=0,ς=0.01,ℏ=1, and χ=1.
(a)(b)Figure 8
Three-dimensional illustration of the absolute error and multiple surface of various fractional orders for Example3 when α=1=β,r1=0,ℏ=1, and ς=0.01.
(a)(b)Figure 9
Two-dimensional illustration of Example3 for various fractional orders when α=1=β,r1=0,ℏ=1, and ς=0.01.Example 4.
Assume the time-fractional BPM (4) having α=−1,β=1,ℏ=1/96, and r1=48; then,(94)DςαQw1,w2,ς=Qw1w12w1,w2,ς+Qw2w22w1,w2,ς+196Q−1w1,w2,ς−12,with IC(95)Q0w1,w2,0=142w12+w22+w2+5.Proof.
Foremost, we provide the solution of (76) in two general cases.
Case 1. First, we apply the Caputo fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 4,(96)ψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+196Q0−12.
Taking into consideration the IC given in (95), we have(97)Uw1,s=ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+196Q0−12.
Employing the inverse generalized integral transform, we obtain(98)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+1ψχsJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+196Q0−12.
Thanks to the generalized decomposition method, we find(99)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψsϕs142w12+w22+w2+5=142w12+w22+w2+5.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(100)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱjQ,j=1,2,3, can be decomposed by an infinite series of polynomials represented by ℱ1Q=Qw1w12=∑ℓ=0∞Aℓandℱ2Q=Qw2w22=∑ℓ=0∞ℬℓ defined in (44) along with ℱ3Q=1/96Q0−1/2=∑ℓ=0∞Dℓ. Thus, we have(101)Dℓ196Q−12=196Q0−12,ℓ=0,−196Q1Q02,ℓ=1,+196−Q2Q02+Q12Q03,ℓ=2,∑ℓ=0∞Qℓ+1w1,w2,ς=J−11ψχsJ∑ℓ=0∞Aℓ+∑ℓ=0∞ℬℓ+∑ℓ=0∞Dℓ,ℓ=0,1,2,….
Forℓ=0,1,2,3,…,(102)Q1w1,w2,ς=J−11ψχsJA0+ℬ0+D0=1242w12+w22+w2+5−1/2ςχΓχ+1,Q2w1,w2,ς=J−11ψχsJA1+ℬ1+D1=−11442w12+w22+w2+5−3/2ς2χΓ2χ+1,Q3w1,w2,ς=J−11ψχsJA2+ℬ2+D2=12882w12+w22+w2+5−5/2ς3χΓ3χ+1,⋮.
The approximate solution for Example3 is expressed as follows:(103)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=142w12+w22+w2+5+1242w12+w22+w2+5−1/2ςχΓχ+1−11442w12+w22+w2+5−3/2ς2χΓ2χ+1+12882w12+w22+w2+5−5/2ς3χΓ3χ+1+⋯.
Case 2. Here, we surmise ABC fractional derivative operator coupled with the generalized integral transform and Adomian decomposition method. Applying the generalized integral transform for Example 4,(104)ψχsAχχ+1−χψχsUw1,s−ϕs∑κ=0m−1ψχ−κ−1sQκ0=JQw1w12w1,w2,ς+Qw2w22w1,w2,ς+196Q0−12.
Taking into consideration the IC given in (95), we have(105)Uw1,s=ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+196Q0−12.
Employing the inverse generalized integral transform, we obtain(106)Qw1,w2,ς=J−1ψsϕsQw1,w2,0+χ+1−χψχsψχsAχJQw1w12w1,w2,ς+Qw2w22w1,w2,ς+196Q0−12.
Thanks to the generalized decomposition method, we find(107)Q0w1,w2,ς=J−1ψsϕsQw1,w2,0=J−1ψsϕs142w12+w22+w2+5=142w12+w22+w2+5.
Here, we surmise that the unknown functionQw1,w2,ς can be written by an infinite series of the form(108)Qw1,w2,ς=∑ℓ=0∞Qℓw1,w2,ς.
Also, the nonlinearityℱjQ,j=1,2,3, can be decomposed by an infinite series of polynomials represented by ℱ1Q=Qw1w12=∑ℓ=0∞Aℓandℱ2Q=Qw2w22=∑ℓ=0∞ℬℓ along with ℱ3Q=1/96Q−1/2=∑ℓ=0∞Cℓ defined in (44) and (101), respectively.
Forℓ=0,1,2,3,…,(109)Q1w1,w2,ς=J−1χ+1−χψχsψχsAχJA0+ℬ0+D0=124Aχ2w12+w22+w2+5−1/2χςχΓχ+1+1−χ,Q2w1,w2,ς=J−1χ+1−χψχsψχsAχJA1+ℬ1+D1=−1144A2χ2w12+w22+w2+5−3/2χ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2,Q3w1,w2,ς=J−1χ+1−χψχsψχsAχJA2+ℬ2+D2=1288A3χ2w12+w22+w2+5−5/2χ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3,⋮.
The approximate solution for Example4 is expressed as follows:(110)Qw1,w2,ς=Φ0w1,w2,ς+Φ1w1,w2,ς+Φ2w1,w2,ς+Φ3w1,w2,ς+⋯=142w12+w22+w2+5+124Aχ2w12+w22+w2+5−1/2ςχΓχ+1+1−χ−1144A2χ2w12+w22+w2+5−3/2χ2ς2χΓ2χ+1+2χ1−χςχΓχ+1+1−χ2+1288A3χ2w12+w22+w2+5−5/2χ3ς3χΓ3χ+1+3χ21−χς2χΓ2χ+1+3χ1−χ2ςχΓχ+1+1−χ3+⋯.
Forχ=1, we obtained the exact solution of Example 4 as(111)Qw1,w2,ς=142w12+w22+w2+ς3+5.
Table4 shows the analytical approximate solutions with some free parameters that are provided by the proposed technique. The comparison analysis is conducted with the VIMHP that predicts the preciseness of the suggested scheme due to their lower error. The analytical findings are extremely useful in deciphering the internal components of acts of nature. The exact and numerical solution for α=−1,β=1,r1=48,ς=0.01,ℏ=1/96, and χ=1 is shown in Figure 10.
Furthermore, the absolute error for the aforementioned assumptions is depicted in Figure11(a).
Finally, Figure11(b) represents the two-dimensional behavior of exact, approximate by Caputo, and approximate solutions derived by ABC fractional operators with varying fractional orders. These solutions have a distinctive characteristic that allows them to interact with other solutions derived by [23]. The proposed findings have particle-like geometries in their solutions. The synthesized trajectory is either a success or a descent from one asymptotic state to the next. The accuracy of the proposed method can be enhanced by increasing the recursive terms.Table 4
ExactQE and approximate solution QA of Qw1,w2,ς of Example 3 having absolute error when χ=1,ς=10,r1=−8/9, and ℏ=−1 for different values of w1 and w2.
w1,w2QE sol.QCFD sol.QABC sol.VIMHP sol. [23]Error=E1Error=E2(−450, 450)224.939112224.939000224.938855224.9391129.8139e − 169.8127e − 16(−400, 400)199.939313199.939304199.939295199.9393161.7688e − 151.7679e − 15(−300, 300)149.939918149.939822149.939800149.9399257.4574e − 157.4570e − 15(−250, 250)124.940402124.940400124.940398124.9404457.4465e − 157.4456e − 15(0, 0)0.85312560.85311190.85311080.85312607.3883e − 37.3880e − 3(50, 50)25.0769648625.0769646725.0769640225.076965085.7070e − 115.7004e − 11(100, 100)50.0696486150.0696478950.0696470250.069649401.7965e − 121.7953e − 12(200, 200)100.0661239100.0661130100.0661009100.06614005.6333e − 145.6300e − 14(350, 350)175.06457142175.06457103175.06457100175.064573003.4370e − 153.4355e − 15(500, 500)250.0639500250.0635573250.0634435250.06399465.7707e − 165.7700e − 16Figure 10
Three-dimensional illustration of exact and approximate solution of Example4 when α=−1,β=1,r1=48,ς=0.01,ℏ=1/96, and χ=1.
(a)(b)Figure 11
Three-dimensional illustration of the absolute error and two-dimensional view of multiple fractional orders for Example4 when α=−1,β=1,r1=48,ℏ=1/96, and ς=0.01.
(a)(b)


## 6. Conclusion

This article investigated the more general integral transform with the Adomian decomposition method. The Caputo and ABC fractional derivative operators have been implemented to deal with the biological population model. Several distinct solutions have been proposed with the assumptions of Malthusian law, Verhulst law, and porous media. Various representations were used to elucidate these solutions, which clarified the significant properties of the fractional models in consideration. Without any restrictive assumptions, discretization, or linearization, the proposed methodology locates the solutions. Elegance and originality have been invoked to describe our trajectory. Contrasting proposed findings to those acquired in earlier scholarly articles demonstrates the peculiarity of our solutions. The strategy’s powerful and successful implementation is explored and validated in order to demonstrate its applicability to additional nonlinear evolution equations.



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