Numerical approximations for population growth model by Rational Chebyshev and Hermite Functions collocation approach: A comparison

# Numerical approximations for population growth model by Rational Chebyshev and Hermite Functions collocation approach: A comparison

## Abstract

This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve the Volterra’s model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations.

C

a]K. Parand\correfcor \cortext[cor]Corresponding author. Tel:+98 21 22431653; Fax:+98 21 22431650. \fntext[a]Member of research group of Scientific Computing.

ollocation method; Spectral method; Volterra’s population model; Nonlinear integro-differential equation; Hermite functions; Rational Chebyshev; Numerical approximations; Population growth model

\PACS

02.60.Lj, 02.70.Hm, 87.23.Cc

## 1 Introduction

Many science and engineering problems arise in unbounded domains. Different spectral methods have been proposed for solving problems in unbounded domains. The most common method is through the use of polynomials that are orthogonal over unbounded domains, such as the Hermite spectral method and the Laguerre spectral method Coulaud (); Funaro.Kavian (); Funaro.Appl. Numer. Math.1990 (); Guo.Math. Comp.1999 (); Guo.num2000 (); Maday (); Shen (); Siyyam (); Iranzo ().

Guo Guo.J. Math. Anal. Appl.1998 (); Guo.com2000 (); Guo.J. Math. Anal. Appl.2000 () proposed a method that proceeds by mapping the original problem in an unbounded domain to a problem in a bounded domain, and then using suitable Jacobi polynomials to approximate the resulting problems.

Another approach is replacing infinite domain with and semi-infinite interval with by choosing , sufficiently large. This method is named domain truncation BoydBook ().

Another effective direct approach for solving such problems is based on rational approximations. Christov Christov.SIAM J. Appl. Math.1982 () and Boyd Boyd.J. Comput. Phys.1987(69) (); Boyd1987 () developed some spectral methods on unbounded intervals by using mutually orthogonal systems of rational functions. Boyd Boyd1987 () defined a new spectral basis, named rational Chebyshev functions on the semi-infinite interval, by mapping to the Chebyshev polynomials. Guo et al. Guo.sci2000 () introduced a new set of rational Legendre functions which are mutually orthogonal in . They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half line. Boyd et al. Boyd2003 () applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine.

Parand et al. Parand.Appl. Math. Comput.2004 (); Parand.Int. J. Comput. Math.2004 (); Parand.Phys. Scripta2004 (); Parand.Shahini.Phys.let.A (); Parand.CAM (); Parand.JCP (); Parand.Dehghan.Rezaei.CPC () applied spectral method to solve nonlinear ordinary differential equations on semi-infinite intervals. Their approach was based on rational Tau and collocation method.

Among these, an approach consists of using the collocation method or the pseudospectral method based on the nodes of Gauss formulas related to unbounded intervals Iranzo ().

Collocation method has become increasingly popular for solving differential equations also they are very useful in providing highly accurate solutions to differential equations.

We aim to compare rational Chebyshev collocation (RCC) approach and Hermite functions collocation (HF) collocation approach to solve a population growth of a species within a closed system.

This paper is arranged as follows: in subsection 1.1, the Volterra’s population model is taken into consideration and some of the traditional methods that solved it are discussed. In sections 2 and 3, we describe the properties of rational Chebyshev and Hermite functions. In section 4, the proposed methods are applied to solve Volterra’s population model. This equation is first converted to an equivalent nonlinear ordinary differential equation and then our methods are applied to solve this new equation, and then a comparison is made with existing methods that were reported in the literature. The numerical results and advantages of the methods are discussed in the final section.

### 1.1 Volterra’s Population Model

The Volterra’s model for population growth of a species within a closed system is given in Scudo (); Small () as

 dpd~t=ap−bp2−cp∫~t0p(x)dx,    p(0)=p0, (1)

where is the birth rate coefficient, is the crowding coefficient, and is the toxicity coefficient. The coefficient indicates the essential behavior of the population evolution before its level falls to zero in the long term. is the initial population, and denotes the population at time .
This model is a first-order integro-ordinary differential equation where the term represents the effect of toxin accumulation on the species. We apply scale time and population by introducing the nondimensional variables

 t=~tcbandu=pba,

to obtain the nondimensional problem

 κdudt=u−u2−u∫t0u(x)dx,    u(0)=u0, (2)

where is the scaled population of identical individuals at time , and is a prescribed nondimensional parameter. The only equilibrium solution of Eq. (2) is the trivial solution and the analytical solution Tebeest ()

 u(t)=u0exp(1κ∫t0[1−u(τ)−∫τ0u(x)dx]dτ),

shows that for all if .

The solution of Eq. (1) has been of considerable concern. Although a closed form solution has been achieved in Scudo (); Small (), it was formally shown that the closed form solution cannot lead to any insight into the behavior of the population evolution Scudo (). In the literature, several numerical solutions for Volterra’s population model have been reported. In Scudo (), the successive approximations method was suggested for the solution of Eq. (2), but was not implemented. In this case, the solution has a smaller amplitude compared to the amplitude of for the case .

In Small (), the singular perturbation method for solving Volterra’s population model is considered. The author scaled out the parameters of Eq. (1) as much as possible and considered four different ways to do this. He considered two cases small and large.

It is shown in Small () that for the case , where populations are weakly sensitive to toxins, a rapid rise occurs along the logistic curve that will reach a peak and then is followed by a slow exponential decay. And, for large , where populations are strongly sensitive to toxins, the solutions are proportional to

In Tebeest (), several numerical algorithms namely Euler method, modified Euler method, classical fourth-order Runge-Kutta method and Runge-Kutta-Fehlberg method for the solution of Eq. (2) are obtained. Moreover, a phase-plane analysis is implemented. In Tebeest (), the numerical results are correlated to give insight on the problem and its solution without using perturbation techniques. However, the performance of the traditional numerical techniques is well known in that it provides grid points only, and in addition, it requires large amounts of calculations.

In Al-Khaled () Adomian decomposition method and Sinc-Galerkin method were compared for the solution of some mathematical population growth models.

In Wazwaz (), the series solution method and the decomposition method are implemented independently to Eq. (2) and to a related nonlinear ordinary differential equation. Furthermore, the Padé approximations are used in the analysis to capture the essential behavior of the populations of identical individuals and approximation of and the exact value of for different were compared.

The authors of Parand.Appl. Math. Comput.2004 (); Parand.Int. J. Comput. Math.2004 (); Parand.Phys. Scripta2004 () applied spectral method to solve Volterra’s population on a semi-infinite interval. This approach is based on a rational Tau method. They obtained the operational matrices of derivative and the product of rational Chebyshev and Legendre functions and then applied these matrices together with the Tau method to reduce the solution of this problem to the solution of a system of algebraic equations.

In Parand.Hojati () second derivative multistep methods (denoted SDMM) are used to solve Volterra’s model. They first converted the model to a nonlinear ordinary differential equation and then the new SDMM applied to solve this equation.

In Ramezani.Razzaghi () the approach is based upon composite spectral functions approximations. The properties of composite spectral functions consisting of few terms of orthogonal functions are utilized to reduce the solution of the Volterra’s model to the solution of a system of algebraic equations.

In Marzban.Hoseini () a numerical method based on hybrid function approximations was proposed to solve Volterra’s population model. These hybrid functions consist of block-pulse and Lagrange-interpolating polynomials.

Momani et al. Momani.Qaralleh () and Xu Xu () used a numerical and Analytical algorithm for approximate solutions of a fractional population growth model respectively. The first algorithm is based on Adomian decomposition method (ADM) with Padé approximants and the second algorithm is based on homotopy analysis method (HAM).

In total, in recent years, numerous works have been focusing on the development of more advanced and efficient methods for initial value problems especially for stiff systems.

## 2 Rational Chebyshev Functions

This section is devoted to introducing rational Chebyshev functions (which we denote (RC)) and expressing some basic properties of them that will be used to construct the RC collocation (RCC) method. rational Chebyshev functions denoted by are generated from well known Chebyshev polynomials by using the algebraic mapping BoydBook (); Boyd1987 (); Parand.Shahini.Phys.let.A (); Guo.Shen.Wang2002 ()

 Rn(x)=Tn(ϕ(x)), (3)

where is a constant parameter and is the Chebyshev polynomial of degree . The constant parameter sets the length scale of the mapping. Boyd BoydBook (); Boyd1982 () offered guidelines for optimizing the map parameter where . Using properties of Chebyshev polynomials and RC, we have

 Rn(x) =⌊n2⌋∑i=0(−1)i2n−2i(n−ii)(x−Lx+L)n−2i =⌊n−12⌋∑i=0(−1)i2n−2i−1(n−i−1i)(x−Lx+L)n−2i (4)

Other properties of RC and a complete discussion on approximating functions by RC are given in Parand.Shahini.Phys.let.A (); Guo.Shen.Wang2002 ().

### 2.1 Rational Chebyshev functions approximation

Let

 RN=span{R0,R1,...,RN}. (5)

We define by

 PNu(x)=N∑k=0akRk(x) (6)

To obtain the order of convergence of rational Chebyshev approximation, we define the space

 Hrw,A(λ)={v:v is measurable and ∥v∥r,x,A <∞}, (7)

where the norm is induced by

 ∥v∥r,x,A=(r∑k=0∥∥∥(x+1)r2+kdkdxkv∥∥∥2w)12, (8)

and A is the Sturm-Liouville operator as follows:

 Av(x)=−w−1(x)ddx(w−1(x)ddxv(x)). (9)

is the weight function and . We have the following theorem for the convergence:

###### Theorem 1

For any and ,

 ∥PNv−v∥w≤cN−r∥v∥r,w,A. (10)
###### Proof 1

A complete proof is given by Guo et al.Guo.Shen.Wang2002 ().

This theorem shows that the rational Chebyshev approximation has exponential convergence.

## 3 Properties of Hermite Functions

In this section, we detail the properties of the Hermite functions (HF) that will be used to construct the Hermite functions collocation (HFC) method. First we note that the Hermite polynomials are generally not suitable in practice due to their wild asymptotic behavior at infinities ShenWang2008 ().
Hermite polynomials with large can be written in direct formula as follow:

 Hn(x) ∼Γ(n+1)Γ(n/2+1)ex2/2cos(√2n+1x−nπ2) ∼nn/2ex2/2cos(√2n+1x−nπ2).

Hence, we shall consider the so called Hermite functions. The normalized Hermite functions of degree is defined by

 ˜Hn(x)=1√2nn!e−x2/2Hn(x),n≥0,x∈R.

Clearly, {} is an orthogonal system in ,i.e.,

 ∫+∞−∞˜Hn(x)˜Hm(x)dx=√πδmn.

where is the Kronecker delta function.
In contrast to the Hermite polynomials, the Hermite functions are well behaved with the decay property:

 |˜Hn(x)|⟶0,as|x|⟶∞,

and the asymptotic formula with large is

 ˜Hn(x)∼n−14cos(√2n+1x−nπ2)

The three-term recurrence relation of Hermite polynomials implies

 ˜Hn+1(x)=x√2n+1˜Hn(x)−√nn+1˜Hn−1(x),n≥1, ˜H0(x)=e−x2/2,˜H1(x)=√2xe−x2/2.

Using recurrence relation of Hermite polynomials and the above formula leads to

 ˜H′n(x) =√2n˜Hn−1(x)−x˜Hn(x) =√n2˜Hn−1(x)−√n+12˜Hn+1(x).

and this implies

 ∫R˜H′n(x)˜H′m(x)dx=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−√n(n−1)π2,m=n−2,√π(n+12),m=n,−√(n+1)(n+2)π2,m=n+2,0,otherwise.

Let us define

 ˜PN:={u:u=e−x2/2v,∀v∈PN}.

where is the set of all Hermite polynomials of degree at most .
We now introduce the Gauss quadrature associated with the Hermite functions approach.
Let be the Hermite-Gauss nodes and define the weights

 ˜wj=√π(N+1)˜H2N(xj),0≤j≤N.

Then we have

 ∫Rp(x)dx=N∑j=0p(xj)˜wj,∀p∈˜P2N+1.

For a more detailed discussion of these early developments, see the ShenTangHighOrder (); ShenTangWang ().

### 3.1 Approximations by Hermite Functions

Let us define and

 HN=span{˜H0(x),˜H1(x),...,˜HN(x)}

The -orthogonal projection is a mapping in a way that for any ,

 <~ξNv−v,ϕ>=0,∀ϕ∈HN

or equivalently,

 ~ξNv(x)=N∑l=0~vl˜Hl(x).

To obtain the convergence rate of Hermite functions we define the space defined by

 HrA(Λ)={v|v is measurable on Λ% and ∥v∥r,A<∞},

and equipped with the norm . For any , the space and its norm are defined by space interpolation. By induction, for any non-negative integer ,

 Arv(x)=r∑k=0(x2+1)(r−k)/2pk(x)∂kxv(x),

where are certain rational functions which are bounded uniformly on . Thus,

 ∥v∥r,A≤c(r∑k=0∥(x2+1)(r−k)/2pk(x)∂kxv∥)1/2.
###### Theorem 2

For any , and ,

 ∥~ξNv−v∥μ≤cN1/3+(μ−1)/2∥v∥r,A. (11)

Proof. A complete proof is given by Guo et al. GuoShenXu2003 (). Also same theorem has been proved by Shen et al. ShenWang2008 ().

## 4 Solving Volterra’s Population Model

In this section, we study an algorithm for solving Volterra’s population model by using the collocation method based on rational Chebyshev and Hermite functions. We first convert Volterra’s population model in Eq. (2) to an equivalent nonlinear ordinary differential equation. Let

 y(t)=∫t0u(x)dx. (12)

 y′(t)=u(t),y′′(t)=u′(t). (13)

Inserting Eq. (12) and Eq. (13) into Eq. (2) yields the nonlinear differential equation

 κy′′(t)=y′(t)−(y′(t))2−y(t)y′(t), (14)

with the initial conditions

 y(0)= 0, (15) y′(0)= u0,

that were obtained by using Eq. (12) and Eq. (13) respectively.
We are going to solve this model for and various and .

### 4.1 Solving Volterra’s Population Model by Rational Chebyshev Functions

In the first step of our analysis, we apply operator on the function as follows:

 PNy(t)=N∑k=0akRk(t) (16)

Then, we construct the residual function by substituting by in the volterra’s population in Eq. (14):

 Res(t)= κd2dt2PNy(t)−ddtPNy(t)+(ddtPNy(t))2 +(PNy(t))(ddtPNy(t)), (17)

The equations for obtaining the coefficients s come from equalizing to zero at rational Chebyshev-Gauss-Radau points plus two boundary conditions:

 ⎧⎪ ⎪⎨⎪ ⎪⎩Res(xj)=0,j=1,2,...,N−1,PNy(0)=0,ddxPNy(t)∣∣t=0=u0. (18)

Solving the set of equations we have the approximating function .

### 4.2 Solving Volterra’s Population Model by Hermite functions

As mentioned before, Volterra’s Population Model is defined on the interval ; but we know the properties of Hermite functions are derived in the infinite domain .
Also we know approximations can be constructed for infinite, semi-infinite and finite intervals. One of the approaches to construct approximations on the interval which is used in this paper, is the use of mapping, that is a change of variable of the form

 ω=ϕ(z)=ln(sinh(kz)).

where is a constant.
The basis functions on are taken to be the transformed Hermite functions,

 ˆHn(x)≡˜Hn(x)∘ϕ(x)=˜Hn(ϕ(x)).

where is defined by . The inverse map of is

 z=ϕ−1(ω)=1kln(eω+√e2ω+1). (19)

Thus we may define the inverse images of the spaced nodes as

 Γ={ϕ−1(t):−∞

and

 ~xj=ϕ−1(xj)=ekxj,j=0,1,2,...

Let denotes a non-negative, integrable, real-valued function over the interval . We define

 L2w(Γ)={v:Γ→R∣v % is measurable and∥v∥w<∞}

where

 ∥v∥w=(∫∞0∣v(x)∣2w(x)dx)12,

is the norm induced by the inner product of the space ,

 w=∫∞0u(x)v(x)w(x)dx. (20)

Thus denotes a system which is mutually orthogonal under (20), i.e.,

 <ˆHn(x),ˆHm(x)>w(x)=√πδnm,

where and is the Kronecker delta function. This system is complete in . For any function the following expansion holds

 f(x)≅+N∑k=−NfkˆHk(x),

with

 fk=w(x)∥ˆHk(x)∥2w(x).

Now we can define an orthogonal projection based on transformed Hermite functions as below:
Let

 ˆHN=span{ˆH0(x),ˆH1(x),...,ˆHn(x)}

The -orthogonal projection is a mapping in a way that for any ,

 <^ξNy−y,ϕ>=0,∀ϕ∈ˆHN

or equivalently,

 ^ξNy(x)=N∑i=0^aiˆHi(x). (21)

Then to apply the HFC method to approximate the Volterra’s Population Eq.(14) with initial conditions (15), we use the operator introduced in Eq.(21) basically.

We note that the Hermite functions are not differentiable at the point , therefore to approximate the solution of Eq. (14) with the initial conditions Eq. (15) we construct a polynomial that satisfy in Eq. (15) and also multiply operator Eq. (21) by to make it differentiable at point in Eq. (15). This polynomial is given by

 p(t)=λt2+0.1t, (22)

where is constant to be determined.

Therefore, the approximate solution of , in Eq. (14) with initial conditions Eq. (15) is represented by

 ˆξNy(t)=p(t)+t^ξNy(t), (23)

Now we construct the residual function by substituting by in the Volterra’s population Eq. (14):

 Resl(t)= κd2dx2ˆξNy(t/l)−ddxˆξNy(t/l)+(ddxˆξNy(t/l))2 +(ˆξNy(t/l))(ddxˆξNy(t/l)), (24)

where is a constant that is called domain scaling.

It has already been mentioned in Liu.Liu.Tang () that when using a spectral approach on the whole real line one can possibly increase the accuracy of the computation by a suitable scaling of the underlying time variable . For example, if denotes a solution of the ordinary differential equation, then the rescaled function is , where is constant. Domain scaling is used in several of the applications presented in the next section. For more detail we refer the reader to Tang.1993 ().

The equations for obtaining the coefficients s come from equalizing to zero at transformed Hermite-Gauss points:

 Res(xj)=0,j=0,…,N+1, (25)

where the s are transformed Hermite-Gauss nodes by Eq. (19). This generates a set of nonlinear equations that can be solved by Newton method for unknown coefficients s and .

Table 1 Shows a comparison of methods in Parand.Hojati (); Ramezani.Razzaghi (), and the present methods with the exact values

 umax=1+κln(κ1+κ−u0)

evaluated in Tebeest ().

Figures 1 and 2 show the results of rational Chebyshev and Hermite functions collocation methods for = , , , , . These figures show the rapid rise along the logistic curve followed by the slow exponential decay after reaching the maximum point and when increases, the amplitude of decreases whereas the exponential decay increases.

Figure 3 illustrates a comparison between the two presented methods for .

Logarithmic graphs of absolute coefficients of rational Chebyshev and Hermite functions in the approximate solutions for are shown in Figures 4 and 5, respectively. These graphs illustrate that both of methods have an appropriate convergence rate.

## 5 Conclusions

The aim of the this study is to develop an efficient and accurate numerical method based on orthogonal functions for solving the Volterra model for the population growth of a species in a closed system. The methods were used in a direct way on a semi-infinite domain without using linearization, perturbation or restrictive assumptions. In this paper, we have applied both the rational Chebyshev and the Hermite functions in solving nonlinear integro-differential equations and compared the results obtained by the two methods and others reported in Parand.Hojati (); Ramezani.Razzaghi (). The study showed that rational Chebyshev collocation method is simple and easy to use. It also minimizes the computational results. In total an important concern of spectral methods is the choice of basis functions; the basis functions have three properties: easy to computation, rapid convergence and completeness, which means that any solution can be represented. The stability and convergence of rational Chebyshev and Hermite functions approximations make this approach very attractive and contributing to the good agreement between the approximate and exact values for in the numerical example.

## Acknowledgments

The first author’s research (K. Parand) was supported by a grant from Shahid Beheshti University.

### References

1. Coulaud O, Funaro D, Kavian O. Laguerre spectral approximation of elliptic problems in exterior domains. Computer Methods in Applied Mechanics and Engineering. 1990; 80 (1-3):451–458.DOI: 10.1016/0045-7825(90)90050-V.
2. Funaro D, Kavian O. Approximation of some diffusion evolution equations in unbounded domains by Hermite functions. Mathematics of Computation. 1991; 57:597–619.
3. Funaro D. Computational aspects of pseudospectral Laguerre approximations. Applied Numerical Mathematics. 1990; 6(6):447–457. DOI: 10.1016/0168-9274(90)90003-X.
4. Guo BY. Error estimation of Hermite spectral method for nonlinear partial differential equations. Mathematics of Computation. 1999; 68(227):1067–1078. DOI: 10.1090/S0025-5718-99-01059-5.
5. Guo BY, Shen J. Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numerische Mathematik. 2000; 86(4):635–654.DOI: 10.1007/s002110000168.
6. Maday Y, Pernaud-Thomas B, Vandeven H. Reappraisal of Laguerre type spectral methods. La Recherche Aerospatiale. 1985; 6:13–35.
7. Shen J. Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM Journal on Numerical Analysis. 2000; 38(4):1113–1133.DOI: 10.1137/S0036142999362936.
8. Siyyam HI. Laguerre Tau methods for solving higher-order ordinary differential equations. Journal of Computational Analysis and Applications. 2001; 3(2):173–182.DOI: 10.1023/A:1010141309991.
9. Iranzo V, Falques A. Some spectral approximations for differential equations in unbounded domains. Computer Methods in Applied Mechanics and Engineering, 1992; 98(1):105–126.DOI: 10.1016/0045-7825(92)90171-F.
10. Guo BY. Gegenbauer approximation and its applications to differential equations on the whole line. Journal of Mathematical Analysis and Applications. 1998; 226(1):180–206.
11. Guo BY. Jacobi spectral approximation and its applications to differential equations on the half line. Journal of Computational Mathematics. 2000; 18:95–112.
12. Guo BY. Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. Journal of Mathematical Analysis and Applications. 2000; 243(2):373–408.DOI: 10.1006/jmaa.1999.6677.
13. Boyd JP. Chebyshev and Fourier Spectrals Method. Dover Press: New York; 2001.
14. Christov CI. A complete orthogonal system of functions in space. SIAM Journal on Applied Mathematics. 1982; 42:1337–1344.
15. Boyd JP. Spectral methods using rational basis functions on an infinite interval. Journal of Computational Physics. 1987; 69(1):112–142.DOI: 10.1016/0021-9991(87)90158-6.
16. Boyd JP. Orthogonal rational functions on a semi-infinite interval. Journal of Computational Physics. 1987; 70(1):63–88.DOI: 10.1016/0021-9991(87)90002-7.
17. Guo BY, Shen J, Wang ZQ. A rational approximation and its applications to differential equations on the half line. Journal of Scientific Computing. 2000; 15(2):117–147.DOI: 10.1023/A:1007698525506.
18. Boyd JP, Rangan C, Bucksbaum PH. Pseudospectral methods on a semi-infinite interval with application to the Hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions. Journal of Computational Physics. 2003; 188(1):56–74.DOI: 10.1016/S0021-9991(03)00127-X.
19. Parand K, Razzaghi M. Rational Chebyshev Tau method for solving Volterra’s population model. Applied Mathematics and Computation. 2004; 149(3):893–900.DOI: 10.1016/j.amc.2003.09.006.
20. Parand K, Razzaghi M. Rational Chebyshev Tau method for solving higher-order ordinary differential equations. International Journal of Computer Mathematics. 2004; 81(1):73–80.DOI: 10.1080/00207160310001614981.
21. Parand K, Razzaghi M. Rational Legendre approximation for solving some physical problems on semi-infinite intervals. Physica Scripta. 2004; 69:353–357.DOI: 10.1238/Physica.Regular.069a00353.
22. Parand K, Shahini M. Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation. Physics Letters A. 2009; 373:210–213.DOI: 10.1016/j.physleta.2008.10.044.
23. Parand K, Taghavi A. Rational scaled generalized Laguerre function collocation method for solving the Blasius equation. Journal of Computational and Applied Mathematics. 2009; 233(4):980–989. DOI: 10.1016/j.cam.2009.08.106.
24. Parand K, Shahini M, Dehghan M. Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type. Journal of Computational Physics. 2009; 228(23):8830–8840.DOI: 10.1016/j.jcp.2009.08.029.
25. Parand K, Dehghan M, Rezaei AR, Ghaderi SM.An approximational algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method Computer Physics Communications. 2010; DOI:10.1016/j.cpc.2010.02.018.
26. Scudo FM. Vito Volterra and theoretical ecology. Theoretical Population Biology. 1971; 2(1):1–23.DOI: 10.1016/0040-5809(71)90002-5.
27. Small RD. Population growth in a closed system. SIAM review. 1983; 25(1):93–95.
28. TeBeest KG. Numerical and analytical solutions of Volterra’s population model. SIAM review. 1997; 39(3):484–493.
29. Al-Khaled K. Numerical approximations for population growth models. Applied Mathematics and Computation. 2005; 160(3):865–873.DOI: 10.1016/j.amc.2003.12.005.
30. Wazwaz AM. Analytical approximations and Padé approximants for Volterra’s population model. Applied Mathematics and Computation. 1999; 100(1):13–25.DOI: 10.1016/S0096-3003(98)00018-6.
31. Parand K, Hojjati G. Solving Volterra’s Population Model using new Second Derivative Multistep Methods. American Journal of Applied Sciences. 2008; 5(8) 1019–1022.DOI: 10.3844/ajassp.2008.1019.1022.
32. Ramezani M, Razzaghi M, Dehghan M. Composite spectral functions for solving Volterras population model. Chaos, Solitons & Fractals. 2007; 34(2):588-593.DOI: 10.1016/j.chaos.2006.03.067.
33. Marzban HR, Hoseini SM, Razzaghi M. Solution of Volterras population model via block-pulse functions and Lagrange-interpolating polynomials. Mathematical Methods in the Applied Sciences. 2009; 32:127–134. DOI: 10.1002/mma.1028.
34. Momani S, Qaralleh R. Numerical approximations and Padé approximants for a fractional population growth model. Applied Mathematical Modelling. 2007; 31:1907–1914.DOI: 10.1016/j.apm.2006.06.015.
35. Xu H. Analytical approximations for a population growth model with fractional order. Communications in Nonlinear Science and Numerical Simulation. 2009; 14:1978–1983.DOI: 10.1016/j.cnsns.2008.07.006.
36. Guo BY, Shen J, Wang ZQ. Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. International journal for numerical methods in engineering. 2002; 53(1):65–84.DOI: 10.1002/nme.392.
37. Boyd JP. The Optimzation of Convergence for Chebyshev Polynomial Methods in an Unbounded Domain. Journal of Computational Physics. 1982; 45:43–79.DOI: 10.1016/0021-9991(82)90102-4.
38. Shen J, Wang L-L. Some Recent Advances on Spectral Methods for Unbounded Domains. Communications in computational physics. 2009; 5 (2-4):195–241.
39. Shen J, Tang T. High Order Numerical Methods and Algorithms. Chinese Science Press, Beijing; 2005.
40. Shen J, Tang T, Wang L-L. Spectral Methods Algorithms, Analyses and Applications. Springer, First edition, 2010.
41. Guo BY, Shen J, Xu C-l. Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation. Advances in Computational Mathematics. 2003; 19(1-3):35–55. DOI: 10.1023/A:1022892132249.
42. Liu Y, Liu L, Tang T. The numerical computation of connecting orbits in dynamical systems: a rational spectral approach. Journal of Computational Physics. 1994; 111(2):373–380.DOI: 10.1006/jcph.1994.1070.
43. Tang T. The Hermite spectral method for Gaussian-type functions. SIAM Journal on Scientific Computing.DOI: 10.1137/0914038. 1993; 14(3):594–606.
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters