# Novel Evaluation of Fuzzy Fractional Biological Population Model

Authors: Rabab Alyusof; Shams Alyusof; Naveed Iqbal; Sallieu Kabay Samura
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This article discusses an iterative transformation method via fuzziness that mixtures the Laplace transform with the iterative iterative method. Using Caputo derivative operator, the proposed technique demonstrates the inherent reliability of fractional fuzzy biological population equations with initial fuzzy conditions. The obtained results to the fuzzy fractional biological equations are more general and apply to a broad variety of problems. A parametric description of the solutions is obtained by translating the fuzzy fractional differential equation into an equivalent system of corresponding fractional differential equations. The proposed method is numerically tested against crisp solutions and those produced by other methods, demonstrating that it is a convenient and remarkably accurate way to solve a tool for solving a wide variety of physics and engineering problems.


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## Body

## 1. Introduction

Fuzzy set theory is a very useful technique for simulating uncertain problems. As a result, fuzzy notions have been used to represent a wide variety of natural events. The fractional fuzzy differential equation is a model that is extensively used in a range of scientific domains, including the evaluation of weapon systems, electro hydraulics, population modeling, and civil engineering problems. In fuzzy calculus, therefore, the concept of the fractional derivative is essential. Consequently, fuzzy fractional differential equations have attracted a great deal of interest in the domains of science and mathematics [1–3]. The first is an Agarwal et al. [4] work on fuzzy fractional differential equations. In order to analyze fractional fuzzy differential equations, under the Hukuhara notion, they developed the Riemann-Liouville idea. The reality is that we still reside in a world of confusion and ambiguity. Many individuals are susceptible to questioning everything around them and pondering why this is for them or others. Because their reports are poor or incorrect and lack clarity [5], assume we are in a situation where there is a great deal of incorrect information and uncertainty. Many of our reasonable questions cannot be answered because they are found on inaccurate facts. This mind set, characterized by an acceptance of ambiguity, is crucial for scientists [3, 6–8].Recent emphasis has been placed on fractional calculus as a helpful tool for getting actual answers to science and math problem including as communication systems, aerodynamic and process control, and bio mathematical problems [9, 10]. In addition, several scholars have examined fractional differential equations under fuzzy conditions and solved them utilizing a different technique [11–13]. Hoa used [14] to investigate fractional fuzzy differential equations with Caputo gH-differentiability. Concurrently, Agarwal et al. undertook research on the identical topic in [15] to illustrate its applicability to optimal control problems. Long et al. [16] demonstrated the solvability of fuzzy fractional differential equations, while Salahshour et al. [17] implemented Laplace fuzzy transform to investigate the problems and so on [18–20].Biological scientists believe that emigration and dispersion are important factors in the formation of species populations. Three independent position functionsΦ=ζ,χ in the area C with ϑ [21] are used to denote the spread of a biological species. Diffusion velocity uΦ,ϑ, population supply pζ,ϑ, and population density vζ,ϑ are the three variables. The rate at which humans are provided per unit volume by birth and death is defined as pζ,ϑ, while the quantity of individuals is provided by vζ,ϑ. Furthermore, uζ,ϑ represents the population’s and individuals’ average velocity movement from one location to the next. The v,u, and p for each D⊂C subregion must be consistent with
(1)dμdϑμ∫DvdU+∫∂Dvu·n^dA=,where n^ is the unit normal extending outward from the boundary ∂D result [22].
(2)p=pv,u=−λv∇,where λv>0 for v>0 and ∇ is the Laplace nonlinear degenerated parabolic partial differential equation can be obtain and which is presented as
(3)Dϑδv=∂2ϕv∂ζ2+∂2ϕv∂χ2+pv.In this case, the fractional order is taken into account in the Caputo sense. Furthermore, Gurney and Nisbet [23] used v as a special example in order to simulate and assess the animal population. The preparations are generally performed by young animals who want to create their own breeding area after reaching maturity and migrating adult animals who have strayed from their natal territory who are threatened by mature intruders. It is far more likely that they will be directed toward the neighbouring unoccupied land in any of these two scenarios. The size of the population density gradient between these two possibilities resolves the probability distribution on the mesh side [24–26].Now, Equation (3) with ϕv=v2 leads to
(4)Dϑδv=∂2v2∂ζ2+∂2v2∂χ2+pv,ϑ≥0,ζ,χ∈ℝ,with the initial point vζ,χ,0. For μ=1, equation (III) simplifies the conventional concept of biological population:
(5)∂v∂ϑ=∂2v2∂ζ2+∂2v2∂χ2+pv,ϑ≥0,ζ,χ∈ℝ.Forpv, the following are illustrations of governing equations (i) pv=cv,c= constant, Malthusian law [21]. (ii) pv=c1v−c2v2,c1,c2= positive constant, Verhulst law [22]. (iii) pv=cvγ,c>0,0<γ<1, porous media [27, 28]Several academics have recently developed more precise and effective strategies for finding and analyzing solutions to nonlinear and complicated issues. George Adomian, an American mathematician and aeronautical engineer, invented the Adomian decomposition technique (ADM) [29] in response to this. ADM has been successfully used to investigate the behaviour of nonlinear systems without the need of linearization or perturbation. ADM, on the other hand, needs a lot of time and computer memory for computational effort. Rawashdeh and Al-Jammal created and nurtured the natural decomposition method [30, 31], which is a hybrid of natural transform and Adomian decomposition method, to meet these needs. FNDM does not need pertubation, linearization, or discretization because it is an enhanced version of ADM. Many mathematicians and physicists have recently used FNDM to comprehend physical behaviour in a variety of complicated situations due to its dependability and efficacy [32, 33]. The considered technique is unique in that it uses a simple method to assess the result and is based on Adomian polynomials, which allows for quick convergence of the found solution for the nonlinear section of the issue. With the arbitrary external parameter, these polynomials generalise to a Maclaurin series. Many writers have solved the given biological population model using various numerical and analytical approaches in order to examine the behaviour and demonstrate the effectiveness of the algorithms [34–36].


## 2. Basic Definitions

Definition 1.
Consider a fuzzy continuous functionυ~ on 0,ω∈R; fuzzy fractional Riemann-Liouvilli integral is defined as
(6)Iϱυ~=∫0φφ−ηϱ−1υ~ηΓϱdη,ϱ,η∈0,∞.
Moreover, ifυ~∈CF0,ω∩LF0,ω, where CF0,ω represents the universes of fuzzy continue function and LF0,ω represents the continuous fuzzy space function. If the functions are Lebesgue integrable, then the fuzzy fractional integral is defined as
(7)Iϱυ~φσ=Iϱυ¯σ,Iϱυ¯σ,0≤σ≤1,such that
(8)Iϱυ¯σ=∫0φφ−nϱ−1υ¯σηΣϱη,ϱ,η∈0,∞,Iϱυ¯σ=∫0φφ−nϱ−1υ¯σηΣϱη,ϱ,η∈0,∞.Definition 2.
For a termυ~∈CF0,ω∩LF0,ω, such that υ~=wσφ,υ¯σφ,σ∈0,1, and φ0∈0,ω, then the fuzzy Caputo fractional derivative is define as
(9)Dϱυ~φ0σ=Dϱυ¯φ0,Dϱυ¯φ0,0<ϱ≤1,where
(10)Dϱυ¯σφ0=∫0φφ−nm−ϱ−1dm/dηmυ¯σηΣϱηφ=φ0,Dϱυ¯σφ0=∫0φφ−nm−ϱ−1dm/dηmυ¯σηΣρηφ=φ0,in integrating convergence occurs and m=ϱ. Since ϱ∈0,1, m=1.Definition 3.
The fuzzy Laplace transform forf℘, where f℘ is the fuzzy value function, is given as
(11)G℘=Lf℘=∫0∞exp−℘φfφdφ,φ>0.Definition 4.
In fuzzy convolution function, a Laplace fuzzy transform is define as(12)Lf1∗f2=Lf1∗Lf2,where f1∗f2; define the fuzzy convolution between f1 and f2, i.e.,
(13)f1∗f2=∫0℘f1φ∗f2℘−φdφ.Definition 5.
The “Function Mittag-Leffler”Eρp is defined as
(14)Eρφ=∑n=0∞φnΣnρ+1,where ρ>0.Definition 6.
Letκ:R⟶0,1 be a count with the appropriated fuzzy quality
(i)
κ is an upper semicontinue numbers(ii)
κμχ1+μχ2≥minκχ1,κχ2(iii)
∃χ0∈R such that κχ0=1, i.e., v is normal(iv)
clχ∈R,κχ>0 is compact
The fuzzy set number is shown by the symbolE.Definition 7.
The preceding number can be expressed in parametric representation asκ¯σ,κ¯σ, so that σ∈0,1 in addition to the values
(i)
κ¯σ from the left is a continue, and bound functions are growing across the range 0,1(ii)
κ¯σ from right is continue, and bound functions decrease over 0,1(iii)
κ¯≤κ¯.Theorem 8.
Letℏ′ψ be a fuzzy integrable value function, and ℏψ is the primitives of ℏ′ψ on 0,∞. Then, Lℏ′ψ=p⊙Lℏψ−gℏ0 where ℏ is (i)-differentiable or Lℏ′ψ=−ℏ0−g−p⊙Lℏψ where ℏ is (ii)-differentiable [37].Proof.
For arbitrary fixedσ∈0,1, we have
(15)p⊙Lℏψ−gℏ0=pℓ−ℏ¯ψ,σ−ℏ0,σpℓℏ¯ψ,σ,since ℓℏ′¯ψ,σ=pℓℏ¯ψ,σ−ℏ¯0,σ and ℓℏ′¯ψ,σ=pℓℏψ,σ−ℏ0,σ then
(16)p⊙Lℏψ−gℏ0=ℓℏ′ψ,σ,ℓℏ′¯ψ,σ,by linearity of L,
(17)p⊙Lℏψ−gℏ0=ℓℏ′ψ,σ,ℏ′¯ψ,σ.
Sinceℏ is (i)-differentiable, it follows that
(18)p⊙Lℏψ−gℏ0=Lℏ′ψ.
Now, we assume thatℏ is the (ii)-differentiable; for arbitrary fixed σ∈0,1, we have
(19)−ℏ0g−p⊙Lℏφ=−ℏ¯0,σ+pℓℏ¯ψ,σ,−ℏ0,σ+pℓℏ¯ψ,σ.This is equivalent to the following:
(20)pℓℏ¯ψ,σ−ℏ¯0,σ,pℓℏψ,σ−ℏ¯0,σ,since ℓℏ′¯ψ,σ=pℓℏ¯ψ,σ−ℏ¯0,σ and ℓℏ′ψ,σ=pℓℏψ,σ−ℏ0,σ then
(21)−ℏ0−ℏ−peLℏφ=ℓℏ′¯ψ,σ,ℓℏ′ψ,σ,−ℏ0−g−peLℏφ=ℓℏ′¯ψ,σ,ℏ′ψ,σs,since ℏ is (ii)-differentiable then it follows that −ℏ0−g−peLℏφ=Lℏ′ψ.Theorem 9.
Letℏψ,gψ be continuous fuzzy-valued functions suppose that c1,c2 are constant, then Lc1⊙ℏψ⊕c2⊙gψ=c1⊙Lℏψ⊕c2⊙Lgψ [37].Proof.
(22)Lc1⊙ℏψ⊕c2⊙gψ=∫0∞c1⊙ℏψ⊕c2⊙gψ⊙e−pxdxLc1⊙ℏψ⊕c2⊙gψ=∫0∞c1⊙ℏψ⊕c2⊙gψ⊙e−pψdψ=∫0∞c1⊙ℏψ⊙e−pψdψ⊕∫0∞c2⊙gψ⊙e−pψdψ=c1⊙∫0∞ℏψ⊙e−pψdψ⊕c2⊙∫0∞gψ⊙e−pψdψ=c1⊙Lℏψ⊕c2⊙Lgψ.
Hence,Lc1⊙ℏψ⊕c2⊙gψ=c1⊙Lℏψs⊕c2⊙Lssgψ.Remark 10.
Letℏψ be continuous fuzzy-value function on 0,∞ and λ≥0; then, Lλ⊙ℏψ=λ⊙Lℏψ.Proof.
Fuzzy Laplace transformλ⊙ℏψ is denoted as Lλ⊙ℏψ=∫0∞λ⊙ℏψ⊙e−pψdψ (p>0 and integer), and also, we have
(23)∫0∞λ⊙ℏψ⊙e−pψdψ=λ⊙∫0∞ℏψ⊙e−pψdψ,then Lλ⊙ℏψ=λ⊙Lℏψ.Remark 11.
Letℏψ be continuous fuzzy-value function and gψ≥0. Suppose that ℏψ⊙gψ⊙e−pψ is improper fuzzy Rimann-integrable on 0,∞, then
(24)∫0∞ℏψ⊙gψ⊙e−pψdψ=∫0∞gψℏψ,σe−pψdψ,∫0∞gψℏ¯ψ,σe−pψdψ.Theorem 12.
Letℏ is continuous fuzzy-value function and Lℏψ=Gp, then [37]
(25)Leaψ⊙ℏψ=Gp−a,where eaψ is real value function and p−a>0.Proof.
(26)Leaψ⊙ℏψ=∫0∞eaψ−pψ⊙ℏψ=∫0∞eaψ−pψℏψ,σdψ,∫0∞eaψ−pψℏ¯ψ,σdψ=∫0∞e−p−aψ⊙ℏψ=Gp−a.


## 3. Road Map of the Current Method

Consider the fuzzy partial differential equation is given as(27)Dφϱυ~ϖ,ψ,φ=Dϖ2υ~ϖ,ψ,φ+υ~ϖ,ψ,φ+κ~σ,0<ϱ≤1,with the initial fuzzy conditions
(28)υ~ϖ,ψ,0=g~ϖ.In this situation, we apply the Laplace transformation to (27) as follows:
(29)LDϖϱυ~ϖ,ψ,φ=LDϖ2υ~ϖ,ψ,φ+υ~ϖ,ψ,φ+κ~,using the initial condition
(30)sϱLυ~ϖ,ψ,φ=sϱ−1g~ϖ+LDϖ2υ~ϖ,ψ,φ+υ~ϖ,ψ,φ+κ~,Lυ~ϖ,ψ,φ=g~ϖs+1sϱLDϖ2υ~ϖ,ψ,φ+υ~ϖ,ψ,φ+κ~.Suppose that the result asυ~ϖ,ψ,φ=∑n=0∞Unϖ,ψ,φ, then (30) defines
(31)L∑n=0∞υ~nϖ,ψ,φ=g~ϖs+1sρLDϖ2∑n=0∞υ~nϖ,ψ,φ+∑n=0∞υ~nϖ,ψ,φ+κ~,comparisons on both sides, we have
(32)Lυ~0ϖ,ψ,φ=g~ϖs+1sϱLκ~,Lυ~1ϖ,ψ,φ=1sϱLDϖ2υ~0ϖ,ψ,φ+υ~0ϖ,ψ,φ,Lυ~2ϖ,ψ,φ=1sϱLDϖ2υ~1ϖ,ψ,φ+υ~1ϖ,ψ,φ,⋮Lυ~n+1ϖ,ψ,φ=1sϱLDϖ2υ~nϖ,ψ,φ+υ~nϖ,ψ,φ,n≥0.Applying Laplace inverse transform, we get(33)υ~0ϖ,ψ,φ=g~ϖ+L−11sϱLκ~,υ~1ϖ,ψ,φ=L−11sϱLDϖ2υ~0ϖ,ψ,φ+υ~0ϖ,ψ,φ,⋮υ~n+1ϖ,ψ,φ=L−11sϱLDϖ2υ~nϖ,ψ,φ+υ~nϖ,ψ,φ,n≥0.The series type solution is obtained as(34)υ~ϖ,ψ,φ=υ~0ϖ,ψ,φ+υ~1ϖ,ψ,φ+υ~2ϖ,ψ,φ+⋯.


### 3.1. Numerical Results

Example 1.
Consider the fractional fuzzy biological population model(35)∂ϱυ~∂φϱ=∂2∂ϖ2υ~2+∂2∂ψ2υ~2+hυ~−11−rυ~,0<ρ≤1,ϖ,ψ∈R,φ>0,with the fuzzy initial condition
(36)υ~ϖ,ψ,0=κ~σhr4ϖ2+hr4ψ2+ψ+5,where κ~σ=κ¯σ,κ¯σ=σ−1,1−σ,0≤σ≤1. Applying the abovementioned methodology as defined in (33), we achieved the following solutions.
(37)υ¯0ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5,υ¯0ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5,υ¯1ϖ,ψ,φ=κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φρΣρ+1,υ¯1ϖ,ψ,φ=κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φρΣρ+1,υ¯2ϖ,ψ,φ=−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ρΣ2ρ+1,υ¯2ϖ,ψ,φ=−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ρΣ2ρ+1.
We can write the series form solution(38)υ~ϖ,ψ,φ=υ~0ϖ,ψ,φ+υ~1ϖ,ψ,φ+υ~2ϖ,ψ,φ+⋯,such that
(39)υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯,υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯.
In general, we can write as follows:(40)υ¯ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5+κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φϱΣϱ+1−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ϱΣ2ϱ+1+⋯,υ¯ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5+κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φϱΣϱ+1−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ϱΣ2ϱ+1+⋯.
The exact result is(41)υ~ϖ,ψ,φ=κ~σhr4ϖ2+hr4ψ2+ψ+2hφ+5.We have given simulation of problem 1 at different noninteger order0<ρ≤1 for lower and upper portions of fuzzy solutions given in Figure 1 in two-dimensional form, respectively. The two similar color legends show the lower and upper branches of fuzzy solutions, respectively.Example 2.
Consider the fractional fuzzy biological population model(42)∂ϱυ~∂φϱ=∂2∂ϖ2υ~2+∂2∂ψ2υ~2+hυ~,with the fuzzy initial condition
(43)υ~ϖ,ψ,0=κ~σϖψ,where κ~σ=κ¯σ,κ¯σ=σ−1,1−σ,0≤σ≤1. Applying the abovementioned methodology as expressed (33), we achieved the following solutions.
(44)υ¯0ϖ,ψ,φ=κ¯σϖψ,υ¯0ϖ,ψ,φ=κ¯σϖψ,υ¯1ϖ,ψ,φ=κ¯σhϖψφϱΣϱ+1,υ¯1ϖ,ψ,φ=κ¯σhϖψφϱΣϱ+1,υ¯2ϖ,ψ,φ=κ¯σh2ϖψφ2ϱΣ2ϱ+1,υ¯2ϖ,ψ,φ=κ¯σh2ϖψφ2ϱΣ2ϱ+1.
We can write the series form solution(45)υ~ϖ,ψ,φ=υ~0ϖ,ψ,φ+υ~1ϖ,ψ,φ+υ~2ϖ,ψ,φ+⋯,such that
(46)υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯,υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯.
In general, we can write as follows:(47)υ¯ϖ,ψ,φ=κ¯σϖψ+κ¯σhϖψφϱΣϱ+1+κ¯σh2ϖψφ2ϱΣ2ϱ+1+⋯,υ¯ϖ,ψ,φ=κ¯σϖψ+κ¯σhϖψφϱΣϱ+1+κ¯σh2ϖψφ2ϱΣ2ϱ+1+⋯.
The exact result is(48)υ~ϖ,ψ,φ=κ~σϖψehφ.Figure 1
(a) A two-dimensional fuzzy upper and lower branch graph of an analytical series solution. (b) Different fractions ofρ.We have given simulation of problem 2 at different noninteger order0<ρ≤1 for lower and upper portions of fuzzy solutions given in Figure 2 in two-dimensional form, respectively. The two similar color legends show the lower and upper branches of fuzzy solutions, respectively.Example 3.
Consider the fractional fuzzy biological population model(49)∂ϱυ~∂φϱ=∂2∂ϖ2υ~2+∂2∂ψ2υ~2+υ~,with the fuzzy initial condition
(50)υ~ϖ,ψ,0=κ~σsinϖsinhψ,where κ~σ=κ¯σ,κ¯σ=σ−1,1−σ,0≤σ≤1. Applying the above-mentioned procedure as expressed in (33), we achieved the following solutions.
(51)jυ¯0ϖ,ψ,φ=κ¯σsinϖsinhψ,υ¯0ϖ,ψ,φ=κ¯σsinϖsinhψ,υ¯1ϖ,ψ,φ=κ¯σsinϖsinhψφϱΣϱ+1,υ¯1ϖ,ψ,φ=κ¯σsinϖsinhψφϱΣϱ+1,υ¯2ϖ,ψ,φ=κ¯σsinϖsinhψφ2ϱΣ2ϱ+1,υ¯2ϖ,ψ,φ=−κ¯σsinϖsinhψφ2ϱΣ2ϱ+1.
We can write the series form solution(52)υ~ϖ,ψ,φ=υ~0ϖ,ψ,φ+υ~1ϖ,ψ,φ+υ~2ϖ,ψ,φ+⋯,such that
(53)υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯,υ¯ϖ,ψ,φ=υ0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯.
In general, we can write as follows:(54)υ¯ϖ,ψ,φ=κ¯σsinϖsinhψ+κ¯σsinϖsinhψφϱΣρ+1+κ¯σsinϖsinhψφ2ϱΣ2ϱ+1+⋯,υ¯ϖ,ψ,φ=κ¯σsinϖsinhψ+κ¯σsinϖsinhψφϱΣϱ+1+κ¯σsinϖsinhψφ2ρΣ2ρ+1+⋯.
The exact result is(55)υ~ϖ,ψ,φ=κ~σsinϖsinhψeφ.Figure 2
(a) A two-dimensional fuzzy upper and lower branch graph of an analytical series solution. (b) Different fractions ofϱ.We have given simulation of problem 3 at different noninteger order0<ϱ≤1 for lower and upper portions of fuzzy solutions given in Figure 3 in two-dimensional form, respectively. The two similar color legends show the lower and upper branches of fuzzy solutions, respectively.Figure 3
The upper and lower fuzzy two-dimensional different fractional-order solution graph ofϱ.


## 3.1. Numerical Results

Example 1.
Consider the fractional fuzzy biological population model(35)∂ϱυ~∂φϱ=∂2∂ϖ2υ~2+∂2∂ψ2υ~2+hυ~−11−rυ~,0<ρ≤1,ϖ,ψ∈R,φ>0,with the fuzzy initial condition
(36)υ~ϖ,ψ,0=κ~σhr4ϖ2+hr4ψ2+ψ+5,where κ~σ=κ¯σ,κ¯σ=σ−1,1−σ,0≤σ≤1. Applying the abovementioned methodology as defined in (33), we achieved the following solutions.
(37)υ¯0ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5,υ¯0ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5,υ¯1ϖ,ψ,φ=κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φρΣρ+1,υ¯1ϖ,ψ,φ=κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φρΣρ+1,υ¯2ϖ,ψ,φ=−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ρΣ2ρ+1,υ¯2ϖ,ψ,φ=−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ρΣ2ρ+1.
We can write the series form solution(38)υ~ϖ,ψ,φ=υ~0ϖ,ψ,φ+υ~1ϖ,ψ,φ+υ~2ϖ,ψ,φ+⋯,such that
(39)υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯,υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯.
In general, we can write as follows:(40)υ¯ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5+κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φϱΣϱ+1−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ϱΣ2ϱ+1+⋯,υ¯ϖ,ψ,φ=κ¯σhr4ϖ2+hr4ψ2+ψ+5+κ¯σhhr4ϖ2+hr4ψ2+ψ+5−1/2φϱΣϱ+1−κ¯σ2h2hr4ζ2+hr4ψ2+ψ+5−3/2φ2ϱΣ2ϱ+1+⋯.
The exact result is(41)υ~ϖ,ψ,φ=κ~σhr4ϖ2+hr4ψ2+ψ+2hφ+5.We have given simulation of problem 1 at different noninteger order0<ρ≤1 for lower and upper portions of fuzzy solutions given in Figure 1 in two-dimensional form, respectively. The two similar color legends show the lower and upper branches of fuzzy solutions, respectively.Example 2.
Consider the fractional fuzzy biological population model(42)∂ϱυ~∂φϱ=∂2∂ϖ2υ~2+∂2∂ψ2υ~2+hυ~,with the fuzzy initial condition
(43)υ~ϖ,ψ,0=κ~σϖψ,where κ~σ=κ¯σ,κ¯σ=σ−1,1−σ,0≤σ≤1. Applying the abovementioned methodology as expressed (33), we achieved the following solutions.
(44)υ¯0ϖ,ψ,φ=κ¯σϖψ,υ¯0ϖ,ψ,φ=κ¯σϖψ,υ¯1ϖ,ψ,φ=κ¯σhϖψφϱΣϱ+1,υ¯1ϖ,ψ,φ=κ¯σhϖψφϱΣϱ+1,υ¯2ϖ,ψ,φ=κ¯σh2ϖψφ2ϱΣ2ϱ+1,υ¯2ϖ,ψ,φ=κ¯σh2ϖψφ2ϱΣ2ϱ+1.
We can write the series form solution(45)υ~ϖ,ψ,φ=υ~0ϖ,ψ,φ+υ~1ϖ,ψ,φ+υ~2ϖ,ψ,φ+⋯,such that
(46)υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯,υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯.
In general, we can write as follows:(47)υ¯ϖ,ψ,φ=κ¯σϖψ+κ¯σhϖψφϱΣϱ+1+κ¯σh2ϖψφ2ϱΣ2ϱ+1+⋯,υ¯ϖ,ψ,φ=κ¯σϖψ+κ¯σhϖψφϱΣϱ+1+κ¯σh2ϖψφ2ϱΣ2ϱ+1+⋯.
The exact result is(48)υ~ϖ,ψ,φ=κ~σϖψehφ.Figure 1
(a) A two-dimensional fuzzy upper and lower branch graph of an analytical series solution. (b) Different fractions ofρ.We have given simulation of problem 2 at different noninteger order0<ρ≤1 for lower and upper portions of fuzzy solutions given in Figure 2 in two-dimensional form, respectively. The two similar color legends show the lower and upper branches of fuzzy solutions, respectively.Example 3.
Consider the fractional fuzzy biological population model(49)∂ϱυ~∂φϱ=∂2∂ϖ2υ~2+∂2∂ψ2υ~2+υ~,with the fuzzy initial condition
(50)υ~ϖ,ψ,0=κ~σsinϖsinhψ,where κ~σ=κ¯σ,κ¯σ=σ−1,1−σ,0≤σ≤1. Applying the above-mentioned procedure as expressed in (33), we achieved the following solutions.
(51)jυ¯0ϖ,ψ,φ=κ¯σsinϖsinhψ,υ¯0ϖ,ψ,φ=κ¯σsinϖsinhψ,υ¯1ϖ,ψ,φ=κ¯σsinϖsinhψφϱΣϱ+1,υ¯1ϖ,ψ,φ=κ¯σsinϖsinhψφϱΣϱ+1,υ¯2ϖ,ψ,φ=κ¯σsinϖsinhψφ2ϱΣ2ϱ+1,υ¯2ϖ,ψ,φ=−κ¯σsinϖsinhψφ2ϱΣ2ϱ+1.
We can write the series form solution(52)υ~ϖ,ψ,φ=υ~0ϖ,ψ,φ+υ~1ϖ,ψ,φ+υ~2ϖ,ψ,φ+⋯,such that
(53)υ¯ϖ,ψ,φ=υ¯0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯,υ¯ϖ,ψ,φ=υ0ϖ,ψ,φ+υ¯1ϖ,ψ,φ+υ¯2ϖ,ψ,φ+⋯.
In general, we can write as follows:(54)υ¯ϖ,ψ,φ=κ¯σsinϖsinhψ+κ¯σsinϖsinhψφϱΣρ+1+κ¯σsinϖsinhψφ2ϱΣ2ϱ+1+⋯,υ¯ϖ,ψ,φ=κ¯σsinϖsinhψ+κ¯σsinϖsinhψφϱΣϱ+1+κ¯σsinϖsinhψφ2ρΣ2ρ+1+⋯.
The exact result is(55)υ~ϖ,ψ,φ=κ~σsinϖsinhψeφ.Figure 2
(a) A two-dimensional fuzzy upper and lower branch graph of an analytical series solution. (b) Different fractions ofϱ.We have given simulation of problem 3 at different noninteger order0<ϱ≤1 for lower and upper portions of fuzzy solutions given in Figure 3 in two-dimensional form, respectively. The two similar color legends show the lower and upper branches of fuzzy solutions, respectively.Figure 3
The upper and lower fuzzy two-dimensional different fractional-order solution graph ofϱ.


## 4. Conclusion

In this paper, a concept of fuzzy Caputo fractional derivative was employed and introduced to analysis solutions of fuzzy biological population equations. An important example validated the conclusion reached. Additionally, we supplied graphs of the numerical solution in a variety of fractional order. Moreover, a technique was suggested to analysis results of fuzzy biological population equations in sense of Caputo operator. As a result, some concrete applications are shown to validate the theoretical framework based on the fuzzy Caputo calculus. Using this notion, we confirm that the proposed study can be used effectively as an extended planner in dealing with many sorts of uncertain situations in engineering and applied mathematics. To summarize, obtaining analytical solutions for many forms of fuzzy fractional differential equations is difficult. As a result, future studies must focus on analyzing and solving fractional fuzzy integro differential equations and fractional fuzzy dynamical systems based on the different derivative of fractional orderρ.



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