# Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions

Authors: Saleh Alshammari; Mohammed Al-Smadi; Mohammad Al Shammari; Ishak Hashim; Mohd Almie Alias
Journal: International Journal of Differential Equations
 (2019)

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## Abstract

In this article, an analytical reliable treatment based on the concept of residual error functions is employed to address the series solution of the differential logistic system in the fractional sense. The proposed technique is a combination of the generalized Taylor series and minimizing the residual error function. The solution methodology depends on the generation of a fractional expansion in an effective convergence formula, as well as on the optimization of truncated errors,Resqjt, through the use of repeated Caputo derivatives without any restrictive assumptions of system nature. To achieve this, some logistic patterns are tested to demonstrate the reliability and applicability of the suggested approach. Numerical comparison depicts that the proposed technique has high accuracy and less computational effect and is more efficient.


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## Body

## 1. Introduction

Scientists were interested in studying differential equations in the fractional sense as flexible mathematical frameworks for modeling, measuring, and describing genetic structures, memory and material transfer, and multiple processes that have recently become an increasingly stimulating area of engineering and science applications, including but not limited to viscosity, fluid mechanics, optimal control, oscillation, signal processing, anomalous diffusion, electromagnetic, and fractal geometry [1–6]. When comparing with the integer order issues, the memory and hereditary properties of several substances are well and fully described by noninteger order issues. Anyhow, numerous attempts to provide numerical solutions to such equations exist in the literature, often due to the difficulty in finding analytical solutions accurately. Therefore, different numerical and approximation methods were introduced to handle those fractional systems [7–10].In this research direction, the application of fractional residual power series method is employed based on residual error concept to find the analytic-numeric solutions of the fractional logistic model:(1)D0αqt=λqt1−qt,t≥0,0<α≤1,along with the initial condition(2)D0αq0=p0,p0>0,where λ>0, D0α indicates Caputo fractional derivatives, while qt indicates smooth solution to be obtained. In this regard, suppose that the logistic models (1) and (2) have a unique analytical solution for t≥0. However, when α=1, the differential equation (1) will be called the standard logistic model in the following form dqt/dt=λqt1−qt, which has the exact solution qt=p0eλt/1−p0+p0eλt. Further, P.-F. Verhulst was the first who presented the standard logistic model. Indeed, there are two basic kinds of the logistic models: continuous and discrete models. The continuous cases are explained by ordinary differential equations of the first order that are called “standard logistic,” while the discrete case representing a simple recurrence formula detects the properties of chaotic in specific domains. Furthermore, logistic solutions provide a consistent classification of the rates of population growth and their rapid and startling development, which does not include reducing food supplies, basic needs, or disease outbreaks. For solution behavior, the logistic system curve increases exponentially starting from the factor of multiplication up to the limit of saturation, that is, the maximum carrying capacity. Consequently, dQ/dt=λQ1−Q/K, where Q represents the size of population growth in terms of t, λ indicates the maximum population growth rate, while K indicates carrying capacity. On the other hand, the continuous logistic system has a steady population growth rate in the form of Qt=p0eλt with initial population data p0. In addition, the theory of existence and uniqueness of the continuous logistic system in a fractional sense has been presented in the literature; for more details, see [11–15].In this research work, a reliable numerical treatment, the fractional residual power series method (FRPSM), is suggested for solving a sort of fractional logistic system. The FRPSM is an easy and reliable tool to obtain the values of unknown coefficients of desired fractional series solution for different types of linear and nonlinear FDEs without discretization, perturbation, and linearization by solving sequence of algebraic system [16–23]. The FRPS technique is primarily applied using the residual error concept and the repetition of Caputo derivatives to obtain the appropriate series solution by choosing a fit initial data, whereas the gained series solution and all fractional derivatives are valid for all mesh points of the domain of interest. To view the characteristics and advantages of numerical methods developed in dealing with various physical and engineering phenomena in the fractional sense, we refer to [24–31].The rest of the current study is outlined in five sections. Characterization and primary results of the theory of fractional calculus are given as well as representation of fractional series solution is also provided in Section2. In Section 3, the main procedures of the proposed algorithm are discussed to construct the required series solution. In Section 4, several applications are considered to confirm the performance and reliability of the present FRPSM. The conclusion is briefly presented finally.


## 2. Primary Mathematical Concepts

This section is devoted to concepts and results concerning the Caputo fractional derivatives and generalized power series representations. Throughout this research, the orderα of fractional derivatives is a nonnegative real constant.Definition 1.
[2]. The Riemann–Liouville fractional integral operator J0α with order α≥0 is given by(3)J0αqt=1Γα∫0tqτt−τ1−αdτ,0≤ε<t,α>0,qt,α=0.Definition 2.
[2]. The operator of fractional derivative D0α with order α≥0 for qt is given by(4)D0αqt=1Γn−α∫0tqnτt−τα−n+1dτ,0<n−1<α<n,n∈ℕ,dndtnqt,α=n,which is called the Caputo fractional operator.
The operatorD0α satisfies the following properties:(i)
D0αc=0, c∈ℝ(ii)
D0αtγ=Γγ+1/Γγ+1−αtγ−α, n−1<α<n, γ>n−1, and is equal to zero otherwise(iii)
D0αAqt+Bpt=AD0αqt+BD0αpt for A,B∈ℝ, that is, D0α is linear operator
Moreover,J0αD0αqt=qt−∑k=0n−1qkτ+t−τkτ/Γk+1 for 0≤τ<t, q∈Cna,b, and n−1<α≤n, n∈ℕ, as well as for α≥0, we have D0αJ0αqt=qt.Remark 1.
[19]. For an arbitrary function qt,0≤τ<t, the Caputo fractional derivative can be computed by the following formula:(5)D0αqt=∑n=0∞qnτΓn+1−αt−τnα.
It is worth mentioning that the fractional calculus has a nonlocal property, so solving fractional differential equations is a challenge, especially for numerical calculations. This property indeed is the main reason why fractional calculus is more popular and good tool for modeling reality. However, Taylor expansion in the fractional sense does not give an approximation of the function at a point because of nonlocality. Anyhow, the local fractional Taylor formula has been successfully generalized and applied in science and engineering problems based on the theory of fractal geometry; for more details, we refer to [32–34].Definition 3.
[20]. The fractional power series “FPS” about t=t0 for n−1<α≤n is given by(6)∑i=0∞ait−t0iα=a0+a1t−t0α+a1t−t02α+….Theorem 1.
[20]. According to the FPS ∑i=0∞ait−t0iα, there are the following possibilities:(1)
The series converges only fort=t0 whenever R is equal to zero(2)
The series converges ∀t≥t0 whenever R is equal to infinity(3)
The series converges att∈t0,t0+Rfor R>0 and diverges for t>t0+RwhereR indicates the radius of convergence of the power series.Lemma 1.
Assume thatqt∈Ct0,t0+R,R>0,D0iαqt∈Ct0,t0+R, and 0<α≤1. Then,(7)J0iαD0iαqt−J0i+1αD0i+1αqt=Dt0iαqt0Γiα+1t−t0iα.Proof.
From the properties of Caputo fractional operator, one can deduce that(8)J0iαD0iαqt−J0i+1αD0i+1αqt=J0iαD0iαqt−J0αJ0iαD0iαD0αqt=J0iαD0iαqt−J0αJ0iαD0iαD0αqt=J0iαD0iαqt−J0iαD0iαD0αqt.
On the other hand, forJ0iαD0iαD0αqt, we infer that(9)J0iαD0iαqt−J0i+1αD0i+1αqt=J0iαD0iαqt−D0iαqt+D0iαqt0=J0iαD0iαqt0=D0iαqt0Γiα+1t−t0iα,withai=D0iαqt0.Theorem 2.
[19]. Assuming that qt has the following power series expansion about t=t0:(10)qt=∑i=0∞ait−t0iα.
Ifqt∈Ct0,t0+R and Diαqt∈Ct0,t0+R, i=0,1,2,…, then coefficient ai is given by ai=D0iαqt0/Γiα+1,where Diα=Dα⋅Dα…Dα (i-times).


## 3. The FRPS Approach

This section is dedicated to presenting the procedures needed to implement the FRPS algorithm to solve the continuous logistic equation in the fractional sense by expanding FPS and utilizing repeated fractional differentiation of the truncated residual functions. To perform this, suppose that the fractional logistic equations (1) and (2) have the solution form about t=0:(11)qt=∑n=0∞antnαΓnα+1.This analysis aims at extending the application of fractional Taylor series framework to get an accurate analytic series solution of fractional systems (1) and (2). Thus, if we use the initial data given by (2), D0αq0=p0, as initial truncated series of qt, so the FPS solution of equation (1) can be written by(12)qt=p0+∑n=1∞antnαΓnα+1.Therefore, thej‐th truncated series solution of qt is given by(13)qjt=p0+∑n=1jantnαΓnα+1.According the FRPS approach, we define thej‐th residual function, Resqjt, for the proposed logistic model as follows:(14)Resqjt=D0αqjt−λqjt1−qjt,j=1,2,3,…,whereas the residual function, Resqt, can be defined by(15)Resqt=limj⟶∞Resqjt=D0αqt−λqt1−qt,0≤t<R.In this point, we noted thatResqt=0 for all t≥0, which leads to D0kαResq0=D0kαResqj0=0, for all k=1,2,…,j. Consequently, the following fractional relations assist us to determine the unknown coefficients, an, n=1,2,…,j, of equation (13):(16)D0j−1αResqj0=0,j=1,2,3,….To show the iteration concept of the FRPS technique to find outa1, put q1t=p0+q1tα/Γα+1 in j‐th residual function of equation (14) at j=1 to get that(17)Resq1t=D0αq1t−λq1t1−q1t=a1−λp0+a1tαΓα+11−p0−a1tαΓα+1=λp0−1p0+a11+λ2p0−1Γα+1tα+λa12Γ2α+1t2α.Thus, by using the fact thatResq10=0, it yields(18)a1=λp01−p0.Therefore, the first FRPS approximation for equations (1) and (2) will be(19)q1t=p01+λ1−p0Γα+1tα.In the same way, to find outa2, put the second truncated series q2t=p01+λ1−p0/Γα+1tα+a2t2α/Γ2α+1 in Resq2t such that(20)Resq2t=D0αq2t−λq2t1−q2t=λp01−p0+a2tαΓα+1−λp01+λ1−p0Γα+1tα+a2t2αΓ2α+11−p01+λ1−p0Γα+1tα−a2t2αΓ2α+1=λp01−p01+1+λ2p0−1Γα+1tα+a22λΓα+1Γ2α+1t3α+λ2p021−p02Γ2α+1t2α+a21Γα+1tα+λ2p0−1Γ2α+1t2α+a22λΓ22α+1t4α.According to the fact on equation (16) at j=2, by applying the operator D0α on both sides of equation (20), it follows that(21)D0αResq2t=D0αλp01−p01+1+λ2p0−1Γα+1tα+a22λΓα+1Γ2α+1t3α+λ2p021−p02Γ2α+1t2α+a21Γα+1tα+λ2p0−1Γ2α+1t2α+a22λΓ22α+1t4α=λ3Γ2α+1p021−p02Γ3α+1tα+λ2p01−p02p0−1+a22Γ3α+1Γα+1Γ22α+1t2α+a21+λ2p0−1Γα+1tα+λΓ4α+1a2Γ22α+1Γ3α+1t3α.Consequently, by applying the factD0αResq20=0 in equation (21), one can get a2=−λ2p01−p02p0−1. The second FRPS approximation is(22)q2t=p01−λp0−1Γα+1tα+λ2p0−12p0−1Γ2α+1t2α.Similarly, substitutingq3t=p0+∑n=13antnα/Γnα+1 into the residual function Resq3t such that Resq3t=D0αq3t−λq3t1−q3t, calculating the fractional derivative D02α of Resq3t, and finally solving the following obtained result by D02αResq3tt=0=0, the third coefficient a3 is determined such that(23)a3=λ3p01−p0Γ2α+1−p01−p04Γ2α+1+Γ2α+1Γ2α+1.So, the third FRPS approximation is given by(24)q3t=p01−λp0−1Γα+1tα+λ2p0−12p0−1Γ2α+1t2α−λ3p0−1Γ2α+1+p0p0−14Γ2α+1+Γ2α+1Γ2α+1Γ3α+1t3α.Further, when the same routine is repeated as above up to the arbitrary orderk, the coefficients an,n=4,5,6,…,k, can be obtained.Theorem 3.
Assuming thatqt satisfies the conditions of (11) with R>0 such that qt∈Ct0,t0+R,D0iαqt∈Ct0,t0+R,i=0,1,2,…,N+1. Then,(25)qt=qNt+RNζ,where qNt=∑i=0ND0iαqt0/Γiα+1t−t0iα and RNζ=D0N+1αqζ/ΓN+1α+1t−t0N+1α, for some ζ∈t0,t. Here, qNtis the N-term truncation of qt and RNζ is the remainder error function.Proof.
Clearly,(26)qt−J0N+1αD0N+1αqt=∑i=0NJ0iαD0iαqt−J0i+1αD0i+1αqt.
Using Lemma 1, it is obvious that(27)qt−J0N+1αD0N+1αqt=∑i=0ND0iqt0Γiα+1t−t0iα.
This, in turn, implies that(28)qt=∑i=0ND0iαqt0Γiα+1t−t0iα+J0N+1αD0N+1αqt.
On the other hand, we get that(29)J0N+1αD0N+1αqt=J0N+1αD0N+1αqt=1ΓN+1α∫t0tD0N+1αqτt−τN+1α−1dτ=D0N+1αqζΓN+1α∫t0tt−τN+1α−1dτby mean−value theorem for integrals=D0N+1αqζΓN+1αt−t0N+1αN+1α=D0N+1αqζΓN+1α+1t−t0N+1α.
Hence, one can deduce the stated result.Remark 2.
IfD0N+1αuζ<M on t0,t0+R,then the upper bound of RNζ can be obtained by(30)RNζ≤Supt∈t0,t0+RMt−t0N+1αΓN+1α+1.


## 4. Illustrative Examples

In this section, numerical applications of the fractional logistic differential equations are presented and quantified at some mesh points. Numerical outcomes highlight the globality of the proposed algorithm in obtaining string solutions consistently and also show that the approximate values are highly acceptable in terms of stability and accuracy. In the calculation, all symbolic and digital calculations are performed using Mathematica software package.Example 1.
Consider the fractional logistic differential equation(31)D0αqt=12qt1−qt,t>0,with the initial condition(32)D0αq0=12,0<α≤1.
The exact solution of IVP (31) and (32) at α=1 is given by(33)qt=e1/2t1+e1/2t.
According the FRPS algorithm, the FPS solution to (31) has the form(34)qt=12+∑n=1∞antnαΓnα+1.
Now, define the residual error for equation (31) by(35)Resqt=D0αqt−12qt1−qt,and the j‐th residual function Resqjt by(36)Resqjt=D0αqjt−12qjt1−qjt,j=1,2,3,….
Following the FRPS algorithm to find out the coefficientsan,n=1,2,3,…,j, of equation (34). Let the first truncated PS approximation has the form(37)q1t=12+a1Γ1+αtα.
From equation (36) at j=1, we have(38)Resq1t=D0α12+a1Γ1+αtα−1212+a1Γ1+αtα1−12+a1Γ1+αtα=a1−1214−a12Γ21+αt2α,and depending on the fact on equation (16), Resq10=0, we have a1=1/8. So, the first approximation is(39)q1t=184+1Γ1+αtα.
Forj=2, the second truncated PS approximation has the form(40)q2t=184+1Γ1+αtα+a2Γ1+2αt2α,and the second residual function is(41)Resq2t=D0α184+1Γ1+αtα+a2Γ1+2αt2α−12184+1Γ1+αtα+a2Γ1+2αt2α1−184+1Γ1+αtα+a2Γ1+2αt2α=a2Γ1+αtα+12182Γ21+αt2α+a28Γ1+αΓ1+2αt3α+a222Γ21+2αt4α.
Now, applyingD0α on both sides of equation (41) such that(42)D0αResq2t=D0α18+a21Γ1+αtα−12D0α184+1Γ1+αtα+a2Γ1+2αt2α12−18Γ1+αtα−a2Γ1+2αt2α=a2+αΓ2α82Γ31+αtα+a2Γ1+3α8Γ1+αΓ21+2αt2α+a222αΓ4αΓ21+2αΓ1+3αt3α.
Using the results of equation (16) at j=2, Resq20=0, we have a2=0. So, the second approximation is(43)q2t=184+1Γ1+αtα.
Forj=3, the third truncated PS approximation has the form(44)q3t=184+1Γ1+αtα+a3Γ1+3αt3α,and the third residual function is(45)Resq3t=D0α184+1Γ1+αtα+a3Γ1+3αt3α−12184+1Γ1+αtα+a3Γ1+3αt3α1−184+1Γ1+αtα+a3Γ1+3αt3α.
Now, applyingD02α on both sides of (46) such that(46)D02αResq3t=D02αD0α184+1Γ1+αtα+a3Γ1+3αt3α−12184+1Γ1+αtα+a3Γ1+3αt3α1−184+1Γ1+αtα+a3Γ1+3αt3α=αΓ2α64Γ21+α+a3+a3Γ1+αΓ1+4α8Γ21+αΓ1+2αΓ1+3αt2α+a323αΓ6αΓ21+3αΓ1+4αt4α.
UsingD02αResq30=0, and then continuing in this process, one can get that(47)a3=−αΓ2α64Γ21+α,a4=0,a5=αΓ2αΓ4α128ΓαΓ21+αΓ1+3α,and so on.
Consequently, few terms of RPS solution are(48)q3t=16432+8Γ1+αtα−αΓ2αΓ21+αΓ1+3αt3α,q4t=16432+8Γ1+αtα−αΓ2αΓ21+αΓ1+3αt3α,q5t=112864+16Γ1+αtα−2αΓ2αΓ21+αΓ1+3αt3α+αΓ2αΓ4αΓαΓ21+αΓ1+3αΓ1+5αt5α.
The numerical results of the 5th FRP solution are given in Table 1 with α=1 in the interval 0,1 with step size h=0.1. Figure 1 shows a comparison between the behavior of the exact solution and the approximate solution at α=1, while in Table 2, numerical comparison is given between the proposed method and the optimal homotopy asymptotic method (OHAM) [15] at α=1. In Figure 2, the behavior of the 5th FPRS approximation is presented with different values of α, where α∈1.0,0.9,0.75,0.5,0.25 and with step size of 0.2, whereas in Table 3, we review the numerical comparison between the FRPS solutions and the OHAM [15] when α=1 with step size of 0.3. Anyhow, Table 4 shows the representation of the 6th approximate solution with different values of α such that α∈1.0,0.75,0.5,0.25. From these results, it can be observed that the behavior of the approximate solutions for different values of α is in good agreement with each other that depends on the fractional order α.Table 1
Numerical results of 5th FRPS at α=1 for Example 1.
t
Exact solution
Approximations
Absolute error
Relative error
0.1
0.5124973965
0.5124973965
0
0
0.2
0.5249791875
0.5249791875
0
0
0.3
0.5374298453
0.5374298457
3.59375
×
10
−
10
6.68692
×
10
−
10
0.4
0.5498339973
0.5498340000
2.68752
×
10
−
9
4.88787
×
10
−
9
0.5
0.5621765009
0.5621765137
1.27861
×
10
−
8
2.27438
×
10
−
8
0.6
0.5744425168
0.5744425625
4.56883
×
10
−
8
7.95351
×
10
−
8
0.7
0.5866175789
0.5866177129
1.33973
×
10
−
7
2.28382
×
10
−
7
0.8
0.5986876601
0.5986880000
3.39887
×
10
−
7
5.67721
×
10
−
7
0.9
0.6106392339
0.6106400059
7.71910
×
10
−
7
1.26410
×
10
−
6Figure 1
The behavior of FRPS solution atα=1 for Example 1.Table 2
Comparison between the FRPS solutions and the OHAM [15] at α=1 for Example 1.
t
Exact
Approximations
Absolute error
FRPSM
OHAM
FRPSM
OHAM
0.0
0.5
0.5
0.5
0.00
0.00
0.3
0.5374298453
0.5374298457
0.5374288935
3.59375
×
10
−
10
9.52
×
10
−
7
0.6
0.5744425168
0.5744425625
0.5744461429
4.56883
×
10
−
8
3.63
×
10
−
6
0.9
0.6106392339
0.6106400059
0.6106416979
7.71910
×
10
−
7
2.46
×
10
−
6Figure 2
FRPS solution plots of Example 1 for different values ofα.Table 3
The 5th FRPS solution of Example 1 for several values t and α.
t
α
=
1
α
=
0.9
α
=
0.75
α
=
0.5
0.0
0.5
0.5
0.5
0.5
0.2
0.5249791875
0.5304889353505724
0.5405474790276749
0.5624203934410724
0.4
0.5498340000
0.5566921519428415
0.5678024178207643
0.5873774649503688
0.6
0.5744425625
0.5812250955770806
0.5912278649203895
0.6059542706735661
0.8
0.5986880000
0.6266870034872078
0.6122323913149167
0.6211659591174139
1.0
0.6224609375
0.6266870034872078
0.6314223950369852
0.6341997586473209Table 4
Representation of approximate solutions for different values ofα.
α
i
FPS approximate solutionq6t
1
/
4
q
6
t
=
1
/
2
+
1
/
960
π
Γ
3
5
/
4
2
t
+
10
Γ
2
5
/
4
12
π
−
2
t
1
/
2
t
1
/
4
1
/
2
q
6
t
=
1
/
2
+
1
/
32
π
8
π
−
16
/
45
π
3
/
2
15
π
/
4
−
t
t
t
1
/
2
3
/
4
q
6
t
=
1
/
2
+
1
/
1800
Γ
2
7
/
4
225
Γ
7
/
4
+
15
t
3
−
Γ
3
/
4
Γ
19
/
4
t
3
/
2
/
2
π
Γ
19
/
4
t
3
/
4
1
q
6
t
=
1
/
2
+
1
/
8
t
+
1
/
384
t
3
+
1
/
15360
t
5Example 2.
Consider the fractional logistic differential equation(49)D0αqt=14qt1−qt,t>0,0<α≤1,with the initial condition(50)D0αq0=13.
The exact solution of IVP (49) and (50) at α=1 is given by(51)qt=e1/4t2+e1/4t.
According to the FRPS algorithm, the FPS solution of (49) has the form(52)qt=13+∑n=1∞antnαΓnα+1.
Now, the residual error of equation (49) can be defined by(53)Resqt=D0αqt−14qt1−qt,and the j‐th residual function Resqjt by(54)Resqjt=D0αqjt−14qjt1−qjt,j=1,2,3,….
In view of the FRPS algorithm, few termsan,n=1,2,3,4, of equation (52) are given by(55)a1=118,a2=1216,a3=12592−Γ1+2α1296Γ21+α,a4=131104−Γ21+2α+2Γ1+αΓ1+3α15552Γ21+αΓ1+2α.
In Table5, the numerical error of the 5th FRPS solution is presented for α=1 on 0,1 with step size of 0.1. In Table 6, the numerical results of the 5th FRPS solution are given for different values of α on 0,1 with step size of 0.2. The exact and approximate solutions for α=1, t∈0,1 and n=5 are shown in Figure 3, while the behaviors of FRPS solutions are plotted for different values of α in Figure 4. From these results, it can be observed that the behavior of the approximate solutions for different values of α is in good agreement with each other that depends on the fractional order α.Table 5
Numerical results of 5th FRPS at α=1 for Example 2.
t
Exact solution
Approximations
Absolute error
Relative error
0.1
0.3389118421
0.3389118421
0
0
0.2
0.3445354618
0.3445354618
1.00
×
10
−
11
3.00
×
10
−
11
0.3
0.3502029635
0.3502029634
1.00
×
10
−
10
2.00
×
10
−
10
0.4
0.3559130712
0.3559130706
5.60
×
10
−
9
1.50
×
10
−
9
0.5
0.3616644631
0.3616644609
2.15
×
10
−
8
5.90
×
10
−
9
0.6
0.3674557720
0.3674557656
6.40
×
10
−
8
1.74
×
10
−
8
0.7
0.3732855868
0.3732855706
1.61
×
10
−
7
4.33
×
10
−
8
0.8
0.3791524531
0.3791524170
3.60
×
10
−
7
9.51
×
10
−
8
0.9
0.3850548747
0.3850548016
7.31
×
10
−
7
1.90
×
10
−
7Table 6
The 5th FRPS solution of Example 2 for different values of t and α.
t
α
=
1
α
=
0.9
α
=
0.75
α
=
0.5
0.0
0.3333333333
0.33333333333
0.33333333333
0.33333333333
0.2
0.3445354618
0.34705266128
0.35171387863
0.36224905383
0.4
0.3559130706
0.35916526544
0.36457322181
0.37469934920
0.6
0.3674557656
0.37084329986
0.37604627351
0.38441631947
0.8
0.3791524170
0.38228876744
0.38673297588
0.39270737781
1.0
0.3909911774
0.39358561694
0.39688357406
0.40007972096Figure 3
The behavior of FRPS solution atα=1 for Example 2.Figure 4
RPS solution plots of Example 2 for different values ofα.


## 5. Conclusions

This study targets to implement the FRPS method for investigating the approximate solution of fractional logistic system subject to fit initial condition. The target has been accomplished successfully through extending the FRPS algorithm to handle such type of FDEs. The present method provides a solution in rapidly convergent FPS without linearization, or any limitations. Some numerical applications are performed to show the efficiency and reliability of the presented algorithm using Mathematica software package. The obtained results showed that the FRPSM is an easy, efficient, and systematic algorithm to provide analytic series solutions to various emerging systems of science.



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*Source: 7609879-2019-09-25.xml*