# Analytical Approximate Solution of the Fractional Order Biological Population Model by Using Natural Transform

Authors: Nasir Ali; Rashid Nawaz; Laiq Zada; Abir Mouldi; Souhail Mohamed Bouzgarrou; Ndolane Sene
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

In the present work, the natural transform iterative method (NTIM) is implemented to solve the biological population model (BPM) of fractional order. The method is tested for three nonlinear examples. The NTIM is a combination of a new iterative method and natural transform. We see that the solution pattern converges to the exact solution in a few iterations. The method handles an extensive range of differential equations of both fractional and integer order. The fractional order derivative is considered in Caputo’s sense. For mathematical computation, Mathematica 10 is used.


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## Body

## 1. Introduction

The globe and our everyday lives have been revolutionized by modern technologies. Technology is being used in a wide range of engineering applications, including aerodynamics, fluid dynamics, medical sciences, and finance. The essence of technology is influenced and designed by mathematical modeling. The modeling might take the form of mathematical models that can be described using differential equations. These differential equations may have been used to represent the transmission of electromagnetic waves, which is at the root of many present technologies. A variety of applications ranging from wireless communications to radar, medical imaging, and remote sensing have played a great role in our life [1, 2]. Mathematics and biosciences have also numerous practical applications related to real life [3, 4]. Several diseases can be modeled through mathematical calculations and can be controlled by collecting data and making precise analysis [5]. There is a strong and interesting relationship between biology and mathematics utilizing differential equations. The noninteger order differential equations are termed as fractional order differential equations (FDEs) [32–35]. The branch of mathematics dealing with FDEs is known as fractional calculus [6]. Depending on the nature of the problem, differential equations can be linear or nonlinear. Simple analytical methods may be used to analyze linear differential equations, but investigators have developed several ways for solving nonlinear differential equations as their exact solutions are not always feasible. The importance of the FDEs can be discussed in many fields of sciences [7, 23–27]. Many operators for fractional derivatives have been given by several researchers. The most famous is Caputo’s fractional derivative operator [8]. Li et al. introduced the fractional order integral operator for handling differential equations [9]. Recently, many transformations have been used to solve fractional order differential equations. Some of them are the Laplace transform, Sumudu transform, Elzaki transform, etc. [10–12, 28–31]. In this work, we will deal with the natural transform iterative method (NTIM), a combination of the natural transform and the new iterative method (NIM). The proposed techniques have been recently applied by Nawaz et al. for solving noninteger order differential equations [13]. Many other researchers have applied NIM and natural transform for handling the FDEs [14–17]. In this article, we will consider fractional biological population model (FBPM) as [18]
(1)Dtβϕ=ϕ2xx+ϕ2yy+fϕ=0,0<β≤1,t>0,x,y∈R2,fϕ=hϕa1−rhb,ϕx,y,0=gx,y.In Equation (1), ϕ=ϕx,y,t is the population density and f is the supply of population due to births and deaths. The h,a,b,andr are the real numbers and gx,y is the initial condition. The detailed solution of Equation (1) can be found in [19, 20]. FBPM is a mathematical model of biology which will be thoroughly investigated in this paper. FBPM aids in the understanding of the dynamical procedure of population changes in biological population models, as well as providing useful predictions.The remaining paper is structured as follows: Preliminary definitions from fractional calculus are contained in Section2. The notion of NTIM is introduced in Section 3. The NTIM is used to solve three FBPM in Section 4. In Section 5, some results have been discussed. Lastly, a concrete conclusion is given.


## 2. Preliminaries

Definition 1.
Riemann-Liouville (R-L) fractional integral is defined as(2)Jtβft=1Γβ∫0tt−τβ−1fτdτ,β>0,t>0,Jt0ft=ft,where Γ. is the gamma function.Definition 2.
Caputo’s time-fractional derivative operator of orderβ>0is defined as
(3)Dtβϕ℘,t=∂βϕ℘,t∂tβ=1Γn−β∫0tt−τn−β−1∂nϕ℘,τ∂τn,ifn−1<β<n,∂nϕ℘,t∂tn,ifβ=n∈N.Definition 3.
Natural transform ofϕt is defined as [21]
(4)ℕ+θt=Rs,v=1v∫0∞e−stvϕtdt;s,v>0,where s and v are the transform variables.Definition 4.
The inverse of natural transform ofRs,v is defined as
(5)ℕ−Rs,v=ϕt=12πi∫c−i∞c+i∞estvRs,vds,where c∈R and the integral are taken in the complex plane s=a+bi along s=c.Definition 5.
If thenth derivative of ϕt is ϕnt, then its natural transform is given as
(6)ℕ+ϕnt=Rns,v=snvnRs,v−∑k=0n−1sn−k+1vn−kϕn0,n≥1.Theorem 6.
If the natural transform ofht and kt are hs,v and ks,v respectively, defined on set A, then
(7)ℕh∗k=vHs,vKs,v,where ℕh∗k is convolution the functions h and k.


## 3. Natural Transform Iterative Method (NTIM) [13]

Consider FDE of the form(8)Dtβϕ℘,t=f℘,t+Lϕ℘,t+ℵϕ℘,t,0℘,t>0,m−1<α<m,where ℘=x1,x2,⋯,xn and m∈N. The linear operator, nonlinear operator, and the source term are L, ℵ, and f, respectively. The initial condition is given as
(9)ϕ℘,0=g℘.By applying the natural transform to Equation (8), we have
(10)ℕ+Dtβϕ℘,t=ℕ+f℘,t+ℕ+Lϕ℘,t+ℵϕ℘,t.Using the natural transform differentiation property, Equation (10) can be written as
(11)sβvβℕ+ϕ℘,t−sβ−1vβϕ℘,0=ℕ+f℘,t+ℕ+Lϕ℘,t+ℵϕ℘¯,t.By rearranging Equation (11), we have
(12)ℕ+ϕ℘,t=g℘s+vβsβℕ+f℘,t+vβsβℕ+Lϕ℘,t+ℵϕ℘,t.For the NTIM solution, ϕ℘,t is expanded as
(13)u℘,t=∑i=0∞ϕi℘,t,and ℵϕ℘,t, the nonlinear term, is defined as
(14)ℵ∑m=0∞ϕm℘,t=ℵϕ0℘,t+∑m=1∞ℵ∑j=0iϕj℘,t−ℵ∑j=0m−1ϕj℘,t.Using Equation (13) and Equation (14) in Equation (12), we obtain
(15)ℕ+∑i=1∞ϕi=g℘s+vβsβℕ+f℘,t+vβsβℕ+∑m=0∞Lϕm+ℵϕ0+∑m=1∞ℵ∑j=0mϕj−ℵ∑j=0m−1ϕj.Using the recursive relation,
(16)ℕ+ϕ0℘,t=g℘s+vβsβℕ+f℘,t,ℕ+ϕ1℘,t=vβsβℕ+Lϕ0+ℵϕ0,ℕ+ϕ2℘,t=vβsβℕ+Lϕ1x,t+ℵϕ0+ϕ1−ℵϕ0⋮ℕ+ϕi+1℘,t=vβsβℕ+Lϕi+ℵϕ0+ϕ1+⋯+ϕi−ℵϕ0+ϕ1+⋯+ϕi−1.i≥0.Now by taking the inverse natural transform of Equation (16), we have
(17)ϕ0℘,t=ℕ−g℘s+vβsβℕ+f,ϕ1℘,t=ℕ−vβsβℕ+Lϕ0+ℵϕ0,ϕ2℘,t=ℕ−vβsβℕ+Lϕ1+ℵϕ0+ϕ1−ℵϕ0,⋮ϕi+1℘,t=ℕ−vβsβℕ+Lϕi+ℵϕ0+ϕ1+⋯+ϕi−ℵϕ0+ϕ1+⋯+ϕi−1,i≥0.Then by adding the components, the approximate solution of Equations (8) and (9) by NITM is given as
(18)ϕ℘,t=ϕ0℘,t+ϕ1℘,t+⋯+ϕm−1℘,t,m∈N.Convergence of NTIM is as a convergence of NIM and is proved by Bhalekar and Daftardar-Gejji [22].


## 4. Applications of NTIM

In this section, we apply the natural transform iterative method NTIM for handling the three nonlinear cases of FBPM. The method is applied directly to the problems without any discretization by using the given initial conditions. Then, the comparison is made with the help of plots and numerical tables with the existing methods which shows the effectiveness of the proposed method [18].Problem 7.
Consider the population model as [18]
(19)Dtβϕ=ϕ2xx+ϕ2yy+hϕ,t>0,0<β≤1,where ϕ=ϕx,y,t together with initial conditions
(20)ϕx,y,0=xy,and the exact solution is
(21)ϕx,y,z,t=xyeht.
Taking natural transformation of Equation (19), we have
(22)ℕ+Dtβϕ=ℕ+ϕ2xx+ϕ2yy+hϕ.Applying the natural transform differentiation property to Equation (22), we get
(23)sβvβϕx,y,t−vβ−1sβϕx,y,0=ℕ+ϕ2xx+ϕ2yy+hϕ.Taking the inverse natural transform of Equation (23), we have
(24)ϕx,y,t=ϕx,y,0s+ℕ−vβsβℕ+ϕ2xx+ϕ2yy+hϕ.Using the idea of NTIM and the recursive relation of Equation (16), we obtained the solution components as
(25)ϕ0x,y,t=ℕ−ϕx,y,0s,ϕ1x,y,t=ℕ−vαsαℕ+ϕ02xx+ϕ02yy+hϕ0,ϕ2x,y,t=ℕ−vαsαℕ+ϕ0+ϕ12xx+ϕ0+ϕ12yy+hϕ1−ϕ02xx+ϕ02yy,ϕ3x,y,t=ℕ−vαsαℕ+ϕ0+ϕ1+ϕ22xx+ϕ0+ϕ1+ϕ22yy+hϕ2−ϕ0+ϕ12xx+ϕ0+ϕ12yy,⋮By using the software package, the solution components are obtained as
(26)ϕ0x,y,t=xy,ϕ1x,y,t=htβxyΓβ+1,ϕ2x,y,t=h2t2βxyΓ2β+1,ϕ3x,y,t=h3t3βxyΓ3β+1,ϕ4x,y,t=h4t4βxyΓ4β+1,⋮
Combining the components, the 4th order approximate solution is given as
(27)ϕx,y,t=ϕ0+ϕ1+ϕ2+ϕ3+ϕ4+=⋯xy+htβxyΓβ+1+h2t2βxyΓ2β+1+h3t3βxyΓ3β+1+h4t4βxyΓ4β+1+⋯.
Equation (27) can be simplified as
(28)ϕx,y,t=xyhtβΓβ+1+h2t2βΓ2β+1+h3t3βΓ3β+1+h4t4βΓ4β+1+⋯.For β=1, Equation (28) converges as
(29)ϕx,y,t=xy1+ht+ht22!+ht33!+ht44!+⋯,which converges to the exact solution given by Equation (21).Problem 8.
Consider the biological population model as [18]
(30)Dtβϕ=ϕ2xx+ϕ2yy+ϕ,t>0,0<β≤1,where ϕ=ϕx,y,t together with initial conditions
(31)ϕx,y,0=sinxcoshy,and the exact solution is
(32)ϕx,y,t=sinxcoshyet.Taking natural transform of Equation (30), we have
(33)ℕ+Dtβϕ=ℕ+ϕ2xx+ϕ2yy+ϕ.
Using the natural transform differentiation property, we obtain(34)sβvβϕx,y,t−vβ−1sβϕx,y,0=ℕ+ϕ2xx+ϕ2yy+ϕ.Taking the inverse natural transform, we have
(35)ϕx,y,t=ϕx,y,0s+ℕ−vβsβℕ+ϕ2xx+ϕ2yy+ϕ.
Using the recursive relation, the solution components can be obtained as(36)ϕ0x,y,t=ℕ−ϕx,y,0s,ϕ1x,y,t=ℕ−vαsαℕ+ϕ02xx+ϕ02yy+ϕ0,ϕ2x,y,t=ℕ−vαsαℕ+ϕ0+ϕ12xx+ϕ0+ϕ12yy+ϕ1−ϕ02xx+ϕ02yy,ϕ3x,y,t=ℕ−vαsαℕ+ϕ0+ϕ1+ϕ22xx+ϕ0+ϕ1+ϕ22yy+ϕ2−ϕ0+ϕ12xx+ϕ0+ϕ12yy,⋮
By using the software package, the solution components are obtained as(37)ϕ0x,y,t=sinxcoshy,ϕ1x,y,t=tβsinxcoshyΓβ+1,ϕ2x,y,t=t2βsinxcoshyΓ2β+1,ϕ3x,y,t=t3βsinxcoshyΓ3β+1,ϕ4x,y,t=t4βsinxcoshyΓ4β+1,⋮
Combining the components, the 4th order approximate solution is given as
(38)ϕx,y,t=ϕ0+ϕ1+ϕ2+ϕ3+ϕ4+⋯=sinxcoshy+tβsinxcoshyΓβ+1+t2βsinxcoshyΓ2β+1+t3βsinxcoshyΓ3β+1+t4βsinxcoshyΓ4β+1+⋯,
Equation (38) can be simplified as
(39)ϕx,y,t=sinxcoshytβΓβ+1+t2βΓ2β+1+t3βΓ3β+1+t4βΓ4β+1+⋯.
Forβ=1, Equation (39) converges as
(40)ϕx,y,t=sinxcoshy1+t+t22!+t33!+t44!+⋯,which yields the exact solution given by Equation (32).Problem 9.
Consider the biological population model as [18]
(41)Dtβϕ=ϕ2xx+ϕ2yy+hϕ1−rϕ,t>0,0<β≤1,where ϕ=ϕx,y,t subject to the initial condition
(42)ϕx,y,0=ehr/8x+y,and the exact solution is
(43)ϕx,y,z,t=ehr/8x+y+ht.Using the same procedure as for Problems 7 and 8, we obtain the solution as
(44)ϕ0x,y,t=ehr8x+y,ϕ1x,y,t=htβehrx+y/22Γβ+1,ϕ2x,y,t=h2t2βehrx+y/22Γ2β+1,ϕ3x,y,t=h3t3βehrx+y/22Γ3β+1,ϕ4x,y,t=h4t4βehrx+y/22Γ4β+1,⋮
Combining the components, the 3rd order approximate solution is given as
(45)ϕx,y,t=ϕ0+ϕ1+ϕ2+ϕ3+ϕ4+⋯=ehrx+y22+htβehrx+y/22Γβ+1+h2t2βehrx+y/22Γ2β+1+h3t3βehrx+y/22Γ3β+1+h4t4βehrx+y/22Γ4β+1⋮.
Equation (27) can be simplified as
(46)ϕx,y,t=ehrx+y22htβΓβ+1+h2t2βΓ2β+1+h3t3βΓ3β+1+h4t4βΓ4β+1+⋯.For β=1, Equation (29) converges as
(47)ϕx,y,t=ehrx+y/221+ht+ht22!+ht33!+ht44!+⋯,which converges to the exact solution given by Equation (43).


## 5. Results and Discussions

The biological population model of fractional order has been investigated in the present work. We observe that the solution pattern for Examples 1-3 converges very rapidly to the exact solution in a few iterations. The results have been compared through graphs and tables which confirms the convergence of NTIM. Figure1 is the NTIM approximate solution, and Figure 2 is the exact solution for β=1 of Example 1. The 6th order approximate solution for different fractional values of β for Example 1 is depicted in Figure 3. Furthermore, Figures 4 and 5 show the 6th order NTIM solution and the exact solution, respectively, for Example 2 by mean of 3D plots. The comparison for different fractional values of β and exact solution is made in Figure 6 for Example 2. Similarly Figures 7 and 8 show, respectively, the 6th order NTIM solution and exact solution for Example 3. Figure 9 is the comparison of fractional values of β and exact solution for Example 3. In Table 1–3, the approximate solution has been compared in tabular form for β=0.5, β=0.7, β=0.9, and β=1.0, with the exact solution. The value β=1.0 converts the FDE to the classical PDE. The absolute error by the proposed NTIM in Tables 1–3 has been compared with the absolute error obtained by the modified generalized Taylor fractional series method (MGTFSM) for Examples 1-3. It is concluded from the results of Examples 1-3 that as the fractional value of β reaches 1, the NTIM approximate solution meets with the exact solution. We also observe that NTIM yields an excellent approximate solution.Figure 1
NTIM solution of problem 1 ath=1, t=1.5, and β=1.Figure 2
Exact solution of problem 1 ath=1, t=1.5, and β=1.Figure 3
Comparison of an approximate solution by NTIM for different values ofβ at h=1, t=1.5, and y=1 for Problem 7.Figure 4
NTIM Solution of problem 1 att=1.5 and β=1.Figure 5
Exact solution of problem 1 att=1.5 and β=1.Figure 6
Comparison of the approximate solution by NTIM for different values ofβ at t=1.5 and y=1 for Problem 8.Figure 7
NTIM solution of Problem7 at h=1, t=1.5, and β=1.Figure 8
Exact solution of Problem7 at h=1, t=1.5, and β=1.Figure 9
Comparison of the approximate solution by NTIM for different values ofβ at h=1, t=1.5, and y=1 for Problem 9.Table 1
Comparison of the different values ofβ and the absolute error of the 6th order NTIM solution with the 6th order MGTFSM solution for Problem 7.
x,ytβ=0.5β=0.7β=0.9β=1.0ExactNTIM errorError [18]0.10.10.7780640.61350.4928980.1105170.1105172.009212×10−121.4090×10−100.30.7780640.61350.4928980.1349860.1349864.507600×10−91.0576×10−70.50.7780640.61350.4928980.1648720.1648721.652645×10−72.3354×10−60.70.7780640.61350.4928980.2013730.2013751.788941×10−61.8129×10−50.90.7780640.61350.4928980.245950.245961.067487×10−58.4486×10−50.30.12.334191.84051.478690.3315510.3315516.027678×10−124.2269×10−100.32.334191.84051.478690.4049580.4049581.352280×10−83.1727×10−70.52.334191.84051.478690.4946160.4946164.957934×10−77.0062×10−60.72.334191.84051.478690.604120.6041265.366824×10−65.4387×10−50.92.334191.84051.478690.7378490.7378813.202460×10−52.5346×10−40.50.13.890323.06752.464490.5525850.5525851.004596×10−117.0449×10−100.33.890323.06752.464490.6749290.6749292.253800×10−85.2879×10−70.53.890323.06752.464490.824360.8243618.263223×10−71.1677×10−50.73.890323.06752.464491.006871.006888.944707×10−69.0645×10−50.93.890323.06752.464491.229751.22985.337433×10−54.2243×10−4Table 2
Comparison of the different values ofβ and the absolute error of the 6th order NTIM solution with the 6th order MGTFSM solution for Problem 8.
x,ytβ=0.5β=0.7β=0.9β=1.0ExactNTIM errorError [18]0.10.12.464551.943291.561270.3500670.3500676.364298×10−121.4090×10−100.32.464551.943291.561270.4275730.4275731.427800×10−81.0576×10−70.52.464551.943291.561270.5222380.5222395.234815×10−72.3354×10−60.72.464551.943291.561270.6378580.6378645.666541×10−61.8129×10−50.92.464551.943291.561270.7790550.7790893.381305×10−58.4486×10−50.30.14.324513.409862.739550.6142590.6142591.116729×10−114.2268×10−100.34.324513.409862.739550.7502580.7502582.505345×10−83.1726×10−70.54.324513.409862.739550.9163660.9163679.185473×10−77.0059×10−60.74.324513.409862.739551.119241.119259.943018×10−65.4385×10−50.94.324513.409862.739551.3671.367065.933139×10−52.5345×10−40.50.15.720824.510843.62410.8125920.8125921.477307×10−117.0425×10−100.35.720824.510843.62410.9925020.9925023.314275×10−85.2860×10−70.55.720824.510843.62411.212241.212241.21513×10−61.1673×10−50.75.720824.510843.62411.480631.480641.315344×10−59.0614×10−50.95.720824.510843.62411.808381.808467.848841×10−54.2228×10−4Table 3
Comparison of the different values ofβ and the absolute error of the 6th order NTIM solution with the 6th order MGTFSM solution for Example 1.
x,ytβ=0.5β=0.7β=0.9β=1.0ExactNTIM errorError [18]0.10.18.598946.780235.447361.22141.22142.220490×10−111.5572×10−90.38.598946.780235.447361.491821.491824.981669×10−81.1688×10−60.58.598946.780235.447361.822121.822121.826455×10−62.5810×10−50.78.598946.780235.447362.225522.225541.977086×10−52.0036×10−40.98.598946.780235.447362.718162.718281.179755×10−49.3372×10−40.30.110.50288.281396.653421.491821.491822.712142×10−111.9019×10−90.310.50288.281396.653421.822121.822126.084624×10−81.4276×10−60.510.50288.281396.653422.225542.225542.230837×10−63.1525×10−50.710.50288.281396.653422.718262.718282.414818×10−52.4472×10−40.910.50288.281396.653423.319973.320121.440956×10−41.1404×10−30.50.112.828110.11498.126511.822121.822123.312617 × 10-112.3230×10−90.312.828110.11498.126512.225542.225547.431777×10−81.7436×10−60.512.828110.11498.126512.718282.718282.724750×10−63.8504×10−50.712.828110.11498.126513.320093.320122.949466×10−52.9890×10−40.912.828110.11498.126514.055024.05521.759988×10−41.3929×10−3


## 6. Conclusion

Three nonlinear problems of fractional order biological population model have been investigated by the natural transform iterative method in the current study. The method is applied to nonlinear problems without any discretization. We found that NTIM converges very rapidly to the exact solution. The advantage of the method is that it is free of any large or small parameter assumptions or to find any constant at the end of the solution. The obtained results of the FBPM have been compared through 3D and 2D plots, and also, the numerical values have been compared in tabular form for different values ofβ. The comparison between absolute errors of the NTIM approximate solution and modified generalized Taylor fractional series method solution is done with the help of tables. In each case, NTIM reveals an efficient approximate solution as compared with other methods in the literature.



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