# The Analysis of Fractional-Order Proportional Delay Physical Models via a Novel Transform

Authors: Meshari Alesemi; Naveed Iqbal; Ahmed A. Hamoud
Journal: Complexity
 (2022)

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## Abstract

In this paper, we deal with an alternative analytical analysis of fractional-order partial differential equations with proportional delay, achieved by applying Yang decomposition method, where the fractional derivative is taken in Caputo sense. The suggested series results are discovered to quickly converge to an exact solution. The computation of three test problems of fractional-order with proportional delay partial differential equations was presented to confirm the validity and efficiency of suggested method. The system appears to be a very dependable, effective, and powerful method for solving a variety of physical problems that arise in engineering and science.


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## Body

## 1. Introduction

Because of its numerous applications in modelling,μ0,η=β0η,μ1,η=β1η,η∈0,T, the study of fractional calculus has recently attracted a lot of attention of fluid dynamics, electrodynamics, biological mathematics, signal processing, and numerous different fields of science. The researchers have written a number of books on fractional calculus such as Ross and Miller [1], Zhou et al. [2], Kilbas et al. [3], Podlubny [4], and Agarwal et al. [5]. Furthermore, delay differential problems have various uses in transportation networks, biological and chemical, which can be identified in [6–11]. In this paper, we suggest with time delay coefficient variable partial differential equation (PDE), provided by(1)Dηδμζ,η−νζμζζ,ηζ+μpζ,ημζζ,η=gζ,η,μζ,η−η,ζ,η∈0,1×0,T,the boundaries condition and initial condition(2)μζ,η=φζ,η,ζ,η∈0,1×−η,0,where μζ,η∈C2,10,1×−η,T is the unknown term with ζ,η space and time variables. When νζ is variable coefficient of space which provides 0<c0≤νζ≤c1, p is any positive integer and φζ,η is sufficiently smooth prehistory function.Forδ=1, (1) becomes classical semilinear with time delay convection reaction diffusion equation (CRDE). Numerous numeric systems have been investigated at δ=1. For example, Zhang and Zhang [12] described a linearize splitting multicompact system to overcome nonlinear PDEs with delay time. Pao [13] suggested a monotone iterative algorithm for the numeric solution of a delay CRDE. For nonlinear CRDEs with time delay, Zhang et al. [14] introduced explicit implicit multistep finite element technique. For variable coefficient time delay PDEs, Ran and He [15] suggested a linearized Crank–Nicolson method.In the meantime, fractional with delay differential equations have attracted the interest of investigators due to its wide implementation in dynamics population, finance, automatic control, etc. [16–18]. Furthermore, by using variable coefficients, more complex natural phenomena can be introduced in [19–21]. It is not straightforward to solve fractional delay PDEs effectively and accurately. The evolution of the dependent variable of fractional delay partial differential equations is at time η, but also on all earlier findings, due to the nature of history dependence of a fractional derivative. In a few cases, analytical solutions to fractional differential equations with delay can be found. Ouyang [22], for example, investigated the uniqueness and existence of results for nonlinear fractional-order delay PDEs. To analyze time fractional delay PDEs, Rihan [23] suggest a different approach which is unconditionally implicit stable. For fractional nonlinear delay diffusion equations, Pimenov and Ahmed [24] suggested a numerical solution for a class of time fractional diffusion equations with delay. For semilinear fractional PDEs with time delay, Zang et al. [25] suggested a compact finite difference approach. For the numeric solution of semilinear fractional delay partial differential equations, Mohebbi [26] suggested a spectral collocation and finite difference methods. For the mathematical investigation of variable order delay Burgers equation, Sweilam et al. [27] presented a nonstandard weighted average finite difference approach. To explore the propagation of the population growth model, Jaradat et al. [28] suggested two numerical schemes based on fractional homotopy perturbation and power series methods.Adomian decomposition method and Yang transform are two well-known techniques that have been applied to have introduced Yang decomposition method. Several physical phenomena which are modeled by partial differential equations and fractional partial differential equations are solved by applying Adomian transform decomposition method, such as the analytical investigation of fractional system partial differential equations are proposed in [29–31], the analysis of nonlinear ordinary differential equations is successfully shown in [32], nonlinear partial differential equations [33–35], fractional unsteady flow of fractional telegraph equations [36], polytropic gas model [37], fractional Schrodinger equation [38], and Fokker–Plank equation [39–41].


## 2. Basic Definitions

In this portion, we described a few basic concepts of fractional calculus along with properties of Laplace transformation.


### 2.1. Definition

The Caputo fractional derivative is define as(3)Dηδνψ,η=1Γk−δ∫0ηη−ρk−δ−1νkψ,ρdρ,k−1<δ≤k,k∈N.


### 2.2. Definition

Xiao Jun Yang introduced the Yang Laplace transform in 2018. The Yang transformation for a termνη is determined by Yνη or M(u) and is given as(4)Yνη=Mu=∫0∞e−η/uνηdη,η>0,u∈−η1,η2.The inverse Yang transformation is expressed as(5)Y−1Mu=νη,where Y−1 is the inverse Yang operator.


### 2.3. Definition

Thenth derivatives of Yang transformation are given as(6)Yνnη=Muun−∑k=0n−1νk0un−k−1,∀n=1,2,3,….


### 2.4. Definition

The fractional-order derivatives of Yang transformation are define as(7)Yνδη=Muuγ−∑k=0n−1νk0uδ−k+1,0<δ≤n.


## 2.1. Definition

The Caputo fractional derivative is define as(3)Dηδνψ,η=1Γk−δ∫0ηη−ρk−δ−1νkψ,ρdρ,k−1<δ≤k,k∈N.


## 2.2. Definition

Xiao Jun Yang introduced the Yang Laplace transform in 2018. The Yang transformation for a termνη is determined by Yνη or M(u) and is given as(4)Yνη=Mu=∫0∞e−η/uνηdη,η>0,u∈−η1,η2.The inverse Yang transformation is expressed as(5)Y−1Mu=νη,where Y−1 is the inverse Yang operator.


## 2.3. Definition

Thenth derivatives of Yang transformation are given as(6)Yνnη=Muun−∑k=0n−1νk0un−k−1,∀n=1,2,3,….


## 2.4. Definition

The fractional-order derivatives of Yang transformation are define as(7)Yνδη=Muuγ−∑k=0n−1νk0uδ−k+1,0<δ≤n.


## 3. Idea of Yang Decomposition Technique

In this portion, the YDM to investigate the general solution of fractional delay partial differential equations:(8)Dηδμζ,η+Lμζ,η+Nμζ2,η2=qζ,η,ζ,η≥0,ℓ−1<δ<ℓ,μζ,η=kζ,where Dδ=∂δ/∂ηδ is the Caputo operator δ,ℓ∈ℕ, where L is linear and N is the nonlinear term, and q is the source term. Using the Yang transformation to (8), we have(9)YDδμζ,η+YLμζ,η+Nμζ2,η2=Yqζ,η,and applying the Yang transformation of differentiation property, we get(10)1sδYμζ,η−sμζ,0=Yqζ,η−YLμζ,η+Nμζ2,η2,Yμζ,η=skζ+sδYqζ,η−sδYLμζ,η+Nμζ2,η2.The YDM solutionμζ,η is represented by the following infinite series:(11)μζ,η=∑ℓ=0∞μℓζ,η,and the nonlinear functions are expressed by the Adomian polynomials,(12)Nμζ2,η2=∑ℓ=0∞Aℓ,Aℓ=1ℓ!dℓdλℓN∑ℓ=0∞λℓμℓλ=0,ℓ=0,1,2,…,and substituting equations (10) and (11) in (10), we get(13)Y∑ℓ=0∞μζ,η=skζ+sδYqζ,η−sδYL∑ℓ=0∞μℓζ,η+∑ℓ=0∞Aℓ,Yμ0ζ,η=sμζ,0+sδYqζ,η,Yμ1ζ,η=−sδYLμ0ζ,η+A0.Generally, we can write(14)Yμℓ+1ζ,η=−sδYLμℓζ,η+Aℓ,ℓ≥1.Applying the inverse Yang transform in (14), we get(15)μ0ζ,η=kζ,η,μℓ+1ζ,η=−Y−1sδYLμℓζ,η+Aℓ.


## 4. Numerical Results

Example 1.
Consider the generalized Burgers equation with proportional delay as defined by [42]:(16)∂δμ∂ηδ−∂2μζ,η∂ζ2−μζ2,η2∂μζ,η/2∂ζ−12μζ,η=0,with initial condition(17)μζ,0=ζ.
Taking Yang transform of (16), we get(18)1sδYμζ,η−sμζ,0=Y∂2μζ,η∂ζ2+μζ2,η2∂μζ,η/2∂ζ+12μζ,η.
Applying inverse Yang transform,(19)μζ,η=Y−1sμζ,0−sδY∂2μζ,η∂ζ2+μζ2,η2∂μζ,η/2∂ζ+12μζ,η.
Using ADM procedure, we get(20)μ0ζ,η=Y−1sμζ,0=ζ,∑ℓ=0∞μℓ+1ζ,η=Y−1sδY∑ℓ=0∞μζζζ,ηℓ+∑ℓ=0∞Aℓμμζ+12∑ℓ=0∞μℓζ,η,ℓ=0,1,2,…,A0μμζ=μ0∂μ0∂ζ,A1μμζ=μ0∂μ1∂ζ+μ1∂μ0∂ζ,A2μμζ=μ0∂μ2∂ζ+μ1∂μ1∂ζ+μ2∂μ0∂ζ,for ℓ=0,(21)μ1ζ,η=Y−1sδY∂2μ0ζ,η∂ζ2+μ0ζ2,η2∂μ0ζ,η/2∂ζ+12μ0ζ,η,μ1ζ,η=ζηδΓδ+1.
The subsequent terms are(22)μ2ζ,η=Y−1sδY∂2μ1ζ,η∂ζ2+μ0ζ2,η2∂μ1ζ,η/2∂ζ+μ1ζ2,η2∂μ0ζ,η/2∂ζ+12μ1ζ,η,μ2ζ,η=ζ2+2δη2δ2δ2Γ2δ+1,μ3ζ,η=Y−1sδY∂2μ2ζ,η∂ζ2+μ0ζ2,η2∂μ2ζ,η/2∂ζ+μ1ζ2,η2∂μ1ζ,η/2∂ζ+μ2ζ2,η2∂μ0ζ,η/2∂ζ+12μ2ζ,η,μ3ζ,η=ζη3δ4Γ3δ+11+22δ+222δ+2223δ+2Γ1+2δ2δΓ1+δ2,μ4ζ,η=Y−1sδY∂2μ3ζ,η∂ζ2+μ0ζ2,η2∂μ3ζ,η/2∂ζ+μ1ζ2,η2∂μ2ζ,η/2∂ζ+μ2ζ2,η2∂μ1ζ,η/2∂ζ+μ3ζ2,η2∂μ0ζ,η/2∂ζ+12μ3ζ,η,μ4ζ,η=ζη4δ8Γ4δ+11+2926δ+2825δ+3×2723δ+2722δ+272δ+2824δ+2825δ+2722δΓ2δ+1Γδ+12+2924δ+2823δΓ3δ+1Γδ+1Γ2δ+1.
The YDM series form solution for Examples 1–3 is(23)μζ,η=μ0ζ,η+μ1ζ,η+μ2ζ,η+μ3ζ,η+μ4ζ,η…,μζ,η=ζ+ζηδΓδ+1+ζ2+2δη2δ2δ2Γ2δ+1+ζη3δ4Γ3δ+11+22δ+222δ+2223δ+2Γ1+2δ2δΓ1+δ2+ζη4δ8Γ4δ+11+2926δ+2825δ+3×2723δ+2722δ+272δ+2824δ+2825δ+2722δΓ2δ+1Γδ+12+2924δ+2823δΓ3δ+1Γδ+1Γ2δ+1+…,μζ,η=ζ+ζηδΓδ+1+ζ2+2δη2δ2δ2Γ2δ+1+ζη3δ4Γ3δ+11+22δ+222δ+2223δ+2Γ1+2δ2δΓ1+δ2+ζη4δ8Γ4δ+11+2926δ+2825δ+3×2723δ+2722δ+272δ+2824δ+2825δ+2722δΓ2δ+1Γδ+12+2924δ+2823δΓ3δ+1Γδ+1Γ2δ+1+…,when δ=1, then YDM series form result is(24)μζ,η=ζ1+η+η22!+η33!+η44!+….
The exact result is(25)μζ,η=ζeη.
In Figure1, the approximate solution graph of problem 4.1, at δ=1 and 0.8, which shows the close contact with each other. Figure 2 shows the approximate result graph of problem 4.1, at δ=0.6 and 0.4. Figure 3 shows the analytical solution graph of different fractional-order of δ.Figure 1
The approximate solution graph of problem 4.1, atδ=1 and 0.8.Figure 2
The approximate result graph of problem 4.1, atδ=0.6 and 0.4.Figure 3
The approximate result graph of problem 4.1, at different fractional-order ofδ.Example 2.
Consider the fractional partial differential equation with proportional delay as defined in [42]:(26)∂δμζ,t∂ηδ−μζ,η2∂2μζ,η/2∂ζ2+μζ,η=0,with the initial condition(27)μζ,0=ζ2.
Taking Yang transform of (26), we get(28)1sδYμζ,η−sμζ,0=Yμζ,η2∂2μζ,η/2∂ζ2−μζ,η.
Applying inverse Yang transform,(29)μζ,η=Y−1sμζ,0=Yμζ,η2∂2μζ,η/2∂ζ2−μζ,η.
Using ADM procedure, we get(30)μ0ζ,η=Y−1sμζ,0=ζ2,∑ℓ=0∞μℓ+1ζ,η=Y−1sδY∑ℓ=0∞Bℓμμζζ−∑ℓ=0∞μℓζ,η,ℓ=0,1,2,B0μμζζ=μ0∂2μ0∂ζ2,B1μμζζ=μ0∂2μ1∂ζ2+μ1∂2μ0∂ζ2,B2μμζζ=μ0∂2μ2∂ζ2+μ1∂2μ1∂ζ2+μ2∂2μ0∂ζ2,for ℓ=0,(31)μ1ζ,η=Y−1sδYμ0ζ,η2∂2μ0ζ,η/2∂ζ2+μ0ζ,η,μ1ζ,η=ζ2ηδΓδ+1.
The subsequent terms are(32)μ2ζ,η=Y−1sδYμ0ζ,η2∂2μ1ζ,η/2∂ζ2+μ1ζ,η2∂2μ0ζ,η/2∂ζ2−μ1ζ,η,μ2ζ,η=ζ22−2δη2δ2δΓ2δ+1,μ3ζ,η=Y−1sδYμ0ζ,η2∂μ2ζ,η/2∂ζ+μ1ζ,η2∂μ1ζ,η/2∂ζ+μ2ζ,η2∂μ0ζ,η/2∂ζ−μ2ζ,η,μ3ζ,η=ζ2η3δΓ3δ+11−22δ−2222δ+2423δ+2Γ2δ+12δΓδ+12.
The YDM result for problem 4.2 is(33)μζ,η=μ0ζ,η+μ1ζ,η+μ2ζ,η+μ3ζ,η+μ4ζ,η…,μζ,η=ζ2+ζ2ηδΓδ+1+ζ22−2δη2δ2δΓ2δ+1+ζ2η3δΓ3δ+11−22δ−2222δ+2423δ+2Γ2δ+12δΓδ+12…,when δ=1, then YDM series form result is(34)μζ,η=ζ21+η+η22!+η33!+η44!+….
The exact solution(35)μζ,η=ζ2eη.
In Figure4, approximate and exact result graph of problem 4.2, at δ=1, which shows the close contact with each other. Figure 5 shows the approximate solution graph of problem 4.1, at different fractional order of δ=1 with respect to ζ and η. Figure 6 shows the different fractional-order graphs of problem 4.2, with respect to time.Figure 4
The approximate and exact result graph of problem 4.2, atδ=1.Figure 5
The approximate solution graph of problem 4.1, at different fractional order ofδ=1 with respect to ζ and η.Figure 6
The different fractional-order graphs of problem 4.2, with respect to time.Example 3.
Consider the fractional partial differential equation with proportional delay as given by [42]:(36)∂δμζ,η∂ηδ−∂2μζ/2,η/2∂ζ2∂μζ/2,η/2∂ζ+18μζ,η+μζ,η=0,with initial condition(37)μζ,0=ζ2.
Using Yang transformation of (36), we get(38)1sδYμζ,η−sμζ,0=Y∂2μζ/2,η/2∂ζ2∂μζ/2,η/2∂ζ+18μζ,η+μζ,η.
Applying inverse Yang transform,(39)μζ,η=Y−1sμζ,0−sδY∂2μζ/2,η/2∂ζ2∂μζ/2,η/2∂ζ+18μζ,η+μζ,η.
Using ADM procedure, we get(40)μ0ζ,η=Y−1sμζ,0=ζ2,∑ℓ=0∞μℓ+1ζ,η=Y−1sδY∑ℓ=0∞Cℓμζζμζ−18∑ℓ=0∞μℓζ,η−∑ℓ=0∞μℓζ,η,ℓ=0,1,2,C0μζζμζ=∂2μ0∂ζ2∂μ0∂ζ,C1μζζμζ=∂2μ0∂ζ2∂μ1∂ζ+∂2μ1∂ζ2∂μ0∂ζ,C2μζζμζ=∂2μ0∂ζ2∂μ2∂ζ+∂2μ1∂ζ2∂μ1∂ζ+∂2μ2∂ζ2∂μ0∂ζ,for ℓ=0,|No Image for this Article(41)μ1ζ,η=Y−1sδY∂2μζ/2,η/2∂ζ2∂μζ/2,η/2∂ζ+18μζ,η+μζ,η,μ1ζ,η=−ζ2ηδΓδ+1,μ2ζ,η=Y−1sδY∂2μζ/2,η/2∂ζ2∂μζ/2,η/2∂ζ+18μζ,η+μζ,η,μ2ζ,η=ζ21−δ+22ζ+1η2δ2Γ2δ+1,μ3ζ,η=Y−1ssδY∂2μζ/2,η/2∂ζ2∂μζ/2,η/2∂ζ+18μζ,η+μζ,η,μ3ζ,η=η3δ2Γ3δ+1−1−2ζ2−24+12δ+122δ+2−32δ+2−223δ+ζ2−122δ2Γ1+2δ2δΓ1+δ2.
The YDM result of problem 4.3 is(42)μζ,η=μ0ζ,η+μ1ζ,η+μ2ζ,η+μ3ζ,η+μ4ζ,η…,μζ,η=ζ2−ζ2ηδΓδ+1+ζ21−δ+22ζ+1η2δ2Γ2δ+1+η3δ2Γ3δ+1−1−2ζ2−24+12δ+122δ+2−32δ+2−223δ+ζ2−122δ2Γ1+2δ2δΓ1+δ2+…,when δ=1, then YDM result is(43)μζ,η=ζ21−η+η22!−η33!+η44!−η55!+….
The exact result is(44)μζ,η=ζ2e−η.
Figure7 shows the approximate result graph of problem 4.3, at δ=1 and 0.8. Figure 8 shows the approximate result graphs of problem 4.3, at δ=0.6 and 0.4. Figure 9 shows the approximate solution graph of problem 4.3, at different fractional-order of δ. Figure 10 shows the approximate solution graph of problem 4.3, at different fractional-order of δ with respect to η.Figure 7
The approximate result graph of problem 4.3, atδ=1 and 0.8.Figure 8
The approximate result graphs of problem 4.3, atδ=0.6 and 0.4.Figure 9
The approximate solution graph of problem 4.3, at different fractional-order ofδ.Figure 10
The approximate solution graph of problem 4.3, at different fractional-order ofδ with respect to η.


## 5. Conclusions

In this article, the approximate results of fractional-order delay partial differential equations are calculated, applying Yang decomposition method. The Yang decomposition method solutions are calculated for both fractional and integer order models. The suggested results are in close contact with homotopy perturbation technique [42], reduced differential transform technique [43], and homotopy perturbation transformation technique. The Yang decomposition method results have seen the highest concurrence with the actual results of the models. Moreover, the applicability and validity of the suggested technique confirm with the aid of three numeric problems. Yang decomposition method, results for fractional-order models, will prove the better understanding of the real world models represented by fractional partial differential equations.



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