 Research
 Open Access
A method for solving nonlinear Volterra’s population growth model of noninteger order
 D Baleanu^{1, 2}Email author,
 B Agheli^{3},
 M Adabitabar Firozja^{3} and
 M Mohamed Al Qurashi^{4}
https://doi.org/10.1186/s136620171421x
© The Author(s) 2017
 Received: 3 August 2017
 Accepted: 7 November 2017
 Published: 25 November 2017
Abstract
Many numerical methods have been developed for nonlinear fractional integrodifferential Volterra’s population model (FVPG). In these methods, to approximate a function on a particular interval, only a restricted number of points have been employed. In this research, we show that it is possible to use the fuzzy transform method (Ftransform) to tackle with FVPG. It makes the Ftransform preferable to other methods since it can make full use of all points on this interval. We also make a comparison showing that this method is less computational and is more convenient to be utilized for coping with nonlinear integrodifferential equation (IDEs), fractional nonlinear integrodifferential equation (FIDEs), and fractional ordinary differential equations (FODEs).
Keywords
 population growth
 fuzzy transform
 Caputo derivative
 integrodifferential equation
MSC
 45J05
 26A33
 26A33
1 Introduction
Fractional arithmetic and fractional differential equations appear in many sciences, including medicine [1], economics [2], dynamical problems [3, 4], chemistry [5], mathematical physics [6], traffic model [7], fluid flow [8], and so on. Scholars and researchers are invited to check books that have been written to take advantage of fractional arithmetic [9–11].

\(D^{\beta}\) denotes the fractional differential operator of order β defined by$$ D^{\beta}P(t)=\frac{1}{\Gamma(k\beta)} \int_{0}^{t}(ts)^{\beta1}P^{(k)}(s) \,ds, \quad k1< \beta\leq k, k\in \mathbb{N}, $$(1.2)

the positive parameter A stands for the coefficient of birth rate,

the positive parameter B is the coefficient of crowding,

the positive parameter C is the coefficient of toxicity, which demonstrates the important treatment of the population evolution as long as its level falls to zero in the long run,

\(P_{0} \) is the primary population,

\(P=P( \widetilde{t} ) \) denotes the population at time t̃, and

the integral \(I_{0}^{\widetilde{t}} P(s)= \int_{0}^{\widetilde{s}} P(s) \,ds \) indicates the effect of toxin gathering.

\(u(t) \) at time t is the scaled population of similar individuals,

\(\lambda=\frac{C}{A B} \) is a prescribed nondimensional parameter.
Various approximate methods for approximating the VPG and FVPG have been investigated by scholars: Hicdurmaz and Can [13], the pseudospectral method of the Legendre functions of noninteger order; Parand and Delkhosh [14], the generalized Chebyshev orthogonal functions of noninteger order of the first kind and the collocation method; Maleki and Maleki [15], the multidomain LegendreGauss pseudospectral approach; Suat Erturk et al. [16], the Padé approximations and differential transform method; Khan et al. [17], a new homotopy perturbation method; Krishnaveni et al. [18], the shifted Legendre polynomial method, Dehghan and Shahini [19], the rational pseudospectral approximation, Ghasemi [20], a new homotopy analysis method; Yildirim and Gulkanat [21], the homotopyPadé technique; Fathizadeh [22], the hybrid rational Haar wavelets; and so on.
The Ftransform has been implemented for dealing with ordinary differential equations as compared with many other classical procedures [23]. In approximate methods, for the purpose of approximating a function on particular interval, only a restricted number of points are used. The Ftransform is preferable to other methods because it uses all points in this interval.
The FTM has recently been utilized in [24–26] to find an approximate solution of the firstorder fuzzy differential equations and twopoint boundary value problems.
Along the same line of research, Chen and Shen [27] have established an algorithm to gain the numerical solutions of secondorder primary amount problems.
2 Fuzzy partition and fuzzy transform
In this section, we outline the main definitions of the Ftransform to be utilized in the subsequent sections of numerical implementations.
Definition 2.1
([23])
 (1)
the functions \(B_{k}\) on \([a,b ] \) taking values in \([0,1 ] \) are continuous, \(\sum_{k=1}^{n} B_{k}(t)=1\) for \(t\in [a,b ] \), and \(B_{k}(t_{k})=1 \),
 (2)
\(B_{k}(t)=0 \) if \(t\notin (t_{k1},t_{k+1} ) \) with \(t_{0}=a \) and \(t_{n+1}=b\),
 (3)
\(B_{k}(t) \) increases on \([t_{k1},t_{k}]\) for \(k=2,\dots,n \) and decreases on \([t_{k},t_{k+1} ] \), \(k=1,\dots,n1\).
The membership functions \(B_{1},B_{2},\dots,B_{n} \) are called basic functions (BFs).
The following formulas give standard triangular membership functions:The following formulas for \(k=2,\dots, n1 \) give standard sinusoidal membership functions:$$\begin{aligned} &B_{1}(t)= \textstyle\begin{cases} 1\frac{tt_{1}}{h_{1}}, & t_{1}\leq t\leq t_{2},\\ 0 & \textit{otherwise}, \end{cases}\displaystyle \\ &B_{k}(t)= \textstyle\begin{cases} \frac{tt_{k1}}{h_{k1}}, & t_{k1} \leq t \leq t_{k},\\ 1\frac{tt_{k}}{h_{k}}, & t_{k} \leq t \leq t_{k+1},\\ 0 & \textit{otherwise}, \end{cases}\displaystyle \\ &B_{n}(t)= \textstyle\begin{cases} \frac{tt_{n1}}{h_{n1}}, & t_{n1} \leq t\leq t_{n},\\ 0 & \textit{otherwise}. \end{cases}\displaystyle \end{aligned}$$(2.1)where \(h_{k}=t_{k+1}t_{k}\), \(k=1,\dots, n1 \). A fuzzy partition of \([a,b] \) for \(k=1,2,\dots,n1 \) is uniform if \(t_{k+1} t_{k}=h=\frac{ba}{n1}\) and two additional properties are satisfied:$$\begin{aligned} &B_{1}(t)= \textstyle\begin{cases} 0.5 (1+\cos\frac{\pi}{h}(tt_{1}) ), & t_{1}\leq t\leq t_{2},\\ 0 & \textit{otherwise}, \end{cases}\displaystyle \\ & B_{k}(t)=\textstyle\begin{cases} 0.5 (1+\cos\frac{\pi}{h}(tt_{k}) ), & t_{k1}\leq t\leq t_{k+1},\\ 0 & \textit{otherwise}, \end{cases}\displaystyle \\ &B_{n}(t)= \textstyle\begin{cases} 0.5 (1+\cos\frac{\pi}{h}(tt_{n}) ), & t_{n1} \leq t\leq t_{n},\\ 0 & \textit{otherwise}, \end{cases}\displaystyle \end{aligned}$$(2.2)  (4)
\(B_{k}(t_{k}t)=B_{k}(t_{k}+t) \) for all \(t\in [0,h] \), \(k=2,\dots,n1 \),
 (5)
\(B_{k}(t)=B_{k1}(th) \) and \(B_{k+1}(t)=B_{k}(th) \) for all \(k=2,\dots,n1 \) and \(t\in[t_{k},t_{k+1}] \).
Lemma 2.2
([23])
Lemma 2.3
Proof
Regarding Definition 2.1, if \(t\notin (t_{i1},t_{i+1} ) \), then \(B_{i}(t)=0 \). Otherwise, if \(k\leq i1 \), then \(t\leq t_{k} \), so \(B_{i}(t)=0\), and we have \(\int_{0}^{t_{k}}B_{i}(t) \,dt=0\).
Definition 2.4
([23])
Definition 2.5
([23])
Theorem 2.6
([23])
Lemma 2.7
([23] (Convergence))
Theorem 2.8
3 Description of the new approach
Remark 3.1
Using the boundary condition \(u_{0}=u(0) \), we can calculate \(u_{1},u_{2}, \ldots, u_{n} \) and then get the approximate solution \(u(t)\approx u_{\mathrm{BFT}}(t) \) for (1.3).
4 Numerical results
A comparison for \(\pmb{\beta=1 }\) of the FTM and approximate methods
λ  NHPM  HPM  HFC  RCC  SDMM  \(\boldsymbol{u_{\max}}\)  Fuzzy transform method  

h = 0.01, n = 500  h = 0.001, n = 5000  
0.02  0.922942037  0.90383805  0.92342704  0.92342715  0.92342714  0.92342717  0.910786  0.923059 
0.04  0.873725344  0.86124017  0.87371998  0.87371998  0.87371998  0.87371998  0.863645  0.873381 
0.10  0.765113089  0.76511308  0.76974149  0.76974149  0.76974140  0.76974149  0.760485  0.769751 
0.20  0.659050432  0.65791230  0.65905038  0.65905038  0.65905037  0.65905038  0.651225  0.659057 
0.50  0.485190290  0.48528234  0.48519030  0.48519030  0.48519029  0.48519030  0.480624  0.485189 
In the last two columns of Table 1, we can see that by increasing the amount n a more accurate answer can be achieved.
5 Conclusion
In the present paper, we have applied the sinusoidal BFs for an approximate solution of FVPG. The advantage of this method can be its simple way with approximate accuracy indicated through a given example. As it has been illustrated in the previous section, the approximate solution resulted in this paper is consistent with either exact solution or with other counterparts. Eventually, it is recommended to utilize the proposed approach to solve differential equations, partial integrodifferential equations, and equation systems of arbitrary order.
Declarations
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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