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Collocation method using sinc and Rational Legendre functions for solving Volterra’s population model
Commun Nonlinear Sci Numer Simulat 16 (2011) 1811–1819
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Collocation method using sinc and Rational Legendre functions for solving Volterra’s population model K. Parand a,⇑,1, Z. Delafkar a, N. Pakniat a, A. Pirkhedri b, M. Kazemnasab Haji a a b
Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran Department of Computer Sciences, Islamic Azad University of Marivan, Iran
a r t i c l e
i n f o
Article history: Received 11 May 2010 Received in revised form 15 July 2010 Accepted 12 August 2010 Available online 17 August 2010 Keywords: Volterra’s population model Collocation method Sinc functions Rational Legendre functions Semi-infinite interval Nonlinear ODE
a b s t r a c t This paper proposes two approximate methods to solve Volterra’s population model for population growth of a species in a closed system. Volterra’s model is a nonlinear integro-differential equation on a semi-infinite interval, where the integral term represents the effect of toxin. The proposed methods have been established based on collocation approach using Sinc functions and Rational Legendre functions. They are utilized to reduce the computation of this problem to some algebraic equations. These solutions are also compared with some well-known results which show that they are accurate. Ó 2010 Published by Elsevier B.V.
1. Introduction 1.1. spectral methods While spectral approximations for ordinary differential equations (ODEs) in bounded domains have achieved great success and popularity in recent years, spectral approximations for ODEs in unbounded domains have only received limited attention. Several spectral methods for treating unbounded domains have been proposed by different researchers. Direct approaches using Laguerre polynomials were investigated by Maday et al. [1], Funaro [2] and [3,4]. Indirect approaches, e.g. Guo [5–7] have 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 class of spectral methods is based on Rational approximations. For example, Christov [8] and Boyd [9,10] developed some spectral methods on unbounded intervals by using mutually orthogonal systems of Rational functions. Boyd [10] defined a new spectral basis, named Rational Chebyshev functions on the semi-infinite interval, by mapping to the Chebyshev polynomials. Recently, [11] proposed and analyzed a set of Legendre Rational functions which are mutually orthogonal in Lv2(0, 1) with a non-uniform weight function v(x) = (x + 1)2. Another approach consists of replacing the infinite domain with [L, L] and semi-infinite interval with [0, L] by choosing L, sufficiently large. This method is named domain truncation [12].
⇑ Corresponding author. Tel.: +98 21 22431653; fax: +98 21 22431650. E-mail addresses: [email protected] (K. Parand), [email protected] (Z. Delafkar), [email protected] (N. Pakniat), alipirkhedri@gmail. com (A. Pirkhedri), [email protected] (M.K. Haji). . 1 Member of research group of Scientific Computing. 1007-5704/$ - see front matter Ó 2010 Published by Elsevier B.V. doi:10.1016/j.cnsns.2010.08.018
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Ref. [13] applied pseudospectral methods on a semi-infinite interval and compared Rational Chebyshev, Laguerre and mapped Fourier sine. Refs.[14–18] applied Rational Chebyshev and Legendre functions with tau and collocation method were applied to solve nonlinear ordinary differential equations on semi-infinite intervals. Refs. [14–16] obtained the operational matrices of derivative and the product of Rational Chebyshev and Legendre functions to reduce the solution of these problems to the solution of some algebraic equations. Another approach is based on Sinc functions. Using Sinc functions for obtaining approximate solutions of ODE and integral equation is widely discussed in [19–24]. For solving differential equations, two methods based on Sinc approximation were presented by Sinc Collocation and by Sinc-Galerkin. In [25] the Sinc-Galerkin method was applied first. Ref. [22] applied the Sinc-Galerkin method to solve a certain class of singular two-point boundary value problems and expressed the exact solution of the differential equations via the use of Green’s functions as an integral type. A Sinc quadrature rule is also presented for the evaluation of Hadamard finite-part integrals of analytic functions in [24]. They showed that the implementation of the Sinc quadrature did not take any additional work, since the nodes and weights were given by very simple formulas. Sinc-Collocation procedure for the numerical solution of the initial value problem is developed in [26] and approved that Sinc procedure converges to the solution at an exponential converge. In [27] Sinc-Collocation is discussed for solving the planar coulomb Schrodinger Equation. They applied Sinc procedure with ln(sinh(x)) function to map the infinite interval to semi-infinite interval and computed the eigen values and eigen functions of the position-space planar Coulomb equation by this method. In [28] Sinc-Collocation method has been extended to handle system of second-order boundary value problems. They applied this method and reduced the computation to some algebraic equations. Ref. [29] used the Sinc-Galerkin method to solve fourth-order ordinary differential equations. Even at this high order, the consistency of the method in exhibiting an exponential convergence rate could be shown. In [19] it is shown that both Sinc-Collocation and Sinc-Galerkin converge with exponential rate. 1.2. Volterra’s model The Volterra model for population growth of a species within a closed system is given in [30,31] as
Z
dp 2 ¼ ap bp cp d^t
^t
pðxÞdx;
0
pð0Þ ¼ p0 ;
ð1Þ
where a > 0 is the birth rate coefficient, b > 0 is the crowding coefficient, c > 0 is the toxicity coefficient, p0 is the initial population and p ¼ pð^tÞ denotes the population at time ^t. Several time scales and population scales may be employed [32]. Here, we employ time and population by introducing the non-dimensional variables
t¼
c^t ; b=c
u¼
bp ; a
which produce the non-dimensional problem
j
du ¼ u u2 u dt
Z
t
uðxÞdx;
uð0Þ ¼ u0 ;
ð2Þ
0
where j = c/ab is a prescribed non-dimensional parameter. One may show that the only equilibrium solution to Eq. (2) is the trivial solution u(t) = 0. Furthermore, the analytical solution [32]
uðtÞ ¼ u0 exp
Z t Z s 1 1 uðsÞ uðxÞdx ds ;
j
0
0
shows that u(t) > 0 for all t if u0 > 0. The solution to Eq. (2) has been of considerable concern. Although a closed-form solution has been achieved in [31], it was formally shown that the closed-form solution cannot lead to any insight into the behavior of the population evolution [33]. Some approximate and numerical solutions for Volterra’s population model have been reported. In [30], the successive approximations method was offered for the solution to Eq. (2) but was not implemented. In [32], the singular perturbation method for Volterra’s population model is considered. It is shown in [32] that for the case j 1, where populations are weakly sensitive to toxins, a rapid rise occurs along the logistic curve that will reach a peak and be followed by a slow exponential decay. In addition, for large j, where populations are strongly sensitive to toxins, the solution is proportional to sech2(t). In this case the solution u(t) has a smaller amplitude compared with the amplitude of u(t) for the case j 1. In [31], three numerical algorithms, namely, the Euler method, the modified Euler method and the fourth-order Runge–Kutta method for the solution to Eq. (2) are obtained. Moreover, a phase-plane analysis is implemented. In [31], the numerical results are correlated to give insight into the problem and its solution without using perturbation techniques. However, it requires extensive calculations. In [33], the series solution method and the Padé 3 approximations are applied to Eq. (2) and to a related ordinary differential equation. In [34], a comparison of the Adomian decomposition method and Sinc-Galerkin method is given and it is shown that the Adomian decomposition method is more efficient for the solution of Volterra’s population model. Ref. [35] used second derivative multistep method (SDMM) to solve this equation. In [36] Adomian decomposition approach and Padé approximates is applied to Eq. (2). Ref. [37,38] used composite spectral functions approximation
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and hybrid function approximation, respectively to approximate a solution for this equation. In [39] homotopy analysis method (HAM) was applied to this equation. The organization of the paper is as follows: In the following, we describe formulation of Sinc and Rational Legendre functions, our solutions for this problem and the application of these methods for Volterra’s population model is summarized in Section 3 and a comparison with existing methods is also made in the literature. The conclusions are described in the final section. 2. Properties of Sinc and Rational Legendre functions 2.1. Sinc functions properties The Sinc functions is defined on the whole real line I = (1, 1) by
( SincðxÞ ¼
sinðpxÞ px ;
x – 0;
1;
x ¼ 0:
ð3Þ
For each integer k and the mesh size h, the Sinc basis functions are defined on R by [28]
8 < sin ðph ðxkhÞÞ x kh ; x – kh; pðxkhÞ ¼ Sk ðh; xÞ Sinc h : h 1; x ¼ kh:
ð4Þ
The Sinc functions form an interpolatory set of functions, i.e.,
Sk ðh; jhÞ ¼ dkj ¼
1; j ¼ k;
ð5Þ
0; j – k:
If a function f(x) is defined on the real axis, then for h > 0 the series
Cðf ; hÞðxÞ ¼
1 X
f ðkhÞSinc
k¼1
x kh ; h
ð6Þ
is called the Whittaker cardinal expansion of f whenever this series converges. The properties of the Whittaker cardinal expansion have been extensively studied in [40]. These properties are derived in the infinite strip DS of the complex x-plane, where d > 0,
n po DS ¼ x ¼ t þ is : jsj < d 6 : 2
ð7Þ
Approximations can be constructed for infinite, semi-infinite and finite intervals. To construct approximations on the interval (0, +1) which is used in this paper, the eye-shaped domain in the z-plane
n po DE ¼ z ¼ x þ iy : jargðsinhðzÞÞj < d 6 ; 2
ð8Þ
is mapped conformally onto the infinite strip DS via
x ¼ /ðzÞ ¼ lnðsinhðzÞÞ:
ð9Þ
The basis functions on (0, +1) are taken to be the composite translated Sinc functions,
Sk ðxÞ Sðk; hÞ /ðxÞ ¼ Sinc
/ðxÞ kh ; h
ð10Þ
where S(k, h)/(x) is defined by S(k, h)(/(x)). The inverse map of x = /(z) is
z ¼ /1 ðxÞ ¼ lnðex þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2x þ 1Þ:
Thus we may define the inverse images of the real line and of the evenly spaced nodes
ð11Þ fkhgk¼þ1 k¼1
C ¼ f/1 ðtÞ 2 DE : 1 < t < þ1g ¼ ð0; þ1Þ
as
ð12Þ
and
xk ¼ /1 ðkhÞ ¼ lnðekh þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2kh þ 1Þ; k ¼ 0; 1; 2; . . .
ð13Þ
Let w(x) denotes a non-negative, integrable, real-valued function over the interval C. We define
L2w ðCÞ ¼ fv : C ! Rjv
is measurable and kv kw < 1g;
ð14Þ
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where
kv kw ¼
Z
1
12 jv ðxÞj2 wðxÞdx ;
ð15Þ
0
is the norm induced by the inner product of the space L2w ðCÞ,
< u; v >w ¼
Z
1
uðxÞv ðxÞwðxÞdx:
ð16Þ
0
Thus {Sk(x)}k2Z with constant h denotes a system which is mutually orthogonal under (16), i.e.,
< Skn ðxÞ; Skm ðxÞ>wðxÞ ¼ hdnm ;
ð17Þ
where w(x) = coth(x) and dnm is the Kronecker delta function. This system is complete in following expansion holds
f ðxÞ ffi
þN X
L2w ð
CÞ. For any function, f 2
L2w ð
fk Sk ðxÞ;
CÞ the
ð18Þ
k¼N
with
fk ¼
< f ðxÞ; Sk ðxÞ>wðxÞ kSk ðxÞk2wðxÞ
ð19Þ
:
The fk = f(xk) are the discrete expansion coefficients associated with the family {Sk(x)} and the mesh size is given by
rffiffiffiffiffiffiffiffiffi 2p d ; h¼ aN where N is suitably chosen and a depends on the asymptotic behavior of f(x) [41]. Definition 1. Let
Dd ¼ fz ¼ x þ iy : 0 < jyj < dg; the function f is in the space of H2(Dd) [26], if f is analytic in Dd and satisfies
Z
d
ðjf ðx þ iyÞjÞdy ¼ Oðxc Þ;
x ! 1; 0 6 c < 1
d
and on the boundary of Dd, (denoted @Dd), satisfies
Nðf Þ ¼
Z
jf ðzÞdzj < 1:
@D
Theorem 1. Assume that f/0 2 H2(Dd) then for all x 2 C
1 X Nðf /0 Þ Nðf /0 Þ pd=h Eðf ; hÞðxÞ ¼ f ðxÞ f ðkhÞSðk; hÞ /ðxÞ 6 : 62 e 2pd sinhðpd=hÞ pd k¼1 Moreover, if jf(x)j 6 Ceaj/(x)j, x 2 C, for some positive constants C and a, and if the selection h ¼
N X pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi f ðkhÞSðk; hÞ /ðxÞ 6 C 2 N expð pdaN Þ; f ðxÞ k¼N
qffiffiffiffiffi pd aN
2pd 6 lnð2Þ , then
where C2 depends only on f, d and a [42]. Also, the nth derivative of the function f at some points xk can be approximated [41] ð0Þ
dk;j ¼ ½Sðk; hÞ /ðxÞjx¼xj ¼ ð1Þ
dk;j
ð2Þ
dk;j
1; k ¼ j;
0; k – j; ( k ¼ j; d 1 0; ½Sðk; hÞ /ðxÞjx¼xj ¼ ¼ jk d/ h ð1Þ ; k – j; jk 8 2 p 2 < ; k ¼ j; 3 d 1 ¼ 2 ½Sðk; hÞ /ðxÞjx¼xj ¼ 2 2ð1Þjk : d/ ; k – j: h ðjkÞ2
ð20Þ ð21Þ
ð22Þ
K. Parand et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1811–1819
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2.2. Rational Legendre functions properties This section is devoted to introducing Rational Legendre functions and expressing some basic properties of them. Rational Legendre functions denoted by Rn(x) are generated from well known Legendre polynomials by using the algebraic mapping /(x) = (x L)/(x + L), as noted in [17]
Rn ðxÞ ¼ Pn ð/ðxÞÞ;
ð23Þ
where L is a constant parameter and Pn(y) is the Legendre polynomial of degree n. The constant parameter L sets the length scale of the mapping. Boyd [10,12,13] offered guidelines for optimizing the map parameter L where L > 0. Numerical results deponed smoothly on constant parameter L, and therefore are not very sensitive to L, so the error varies very slowly with L around the minimum. A little trial and error is usually sufficient to find a value that is nearly optimum. In general, there is no way to avoid a small amount of trial and error in choosing L when solving problems on an unbounded domain. Experience and the asymptotic approximations of Boyd [12] can help, but some experimentation is always necessary as he explains in his book [12]. Properties of Rational Legendre functions and a complete discussion on approximating functions by Rational Legendre functions are given in [17]. 2.2.1. Rational Legendre functions approximation Let
RN ¼ spanfR0 ; R1 ; . . . ; RN g; We define P N :
L2w ð
PN uðxÞ ¼
ð24Þ
KÞ ! RN by
N X
ak Rk ðxÞ;
ð25Þ
k¼0
To obtain the order of convergence of Rational Legendre approximation, we define the space
Hrw;A ðKÞ ¼ fv : v
is measurable and kv kr;w;A < 1g;
ð26Þ
where the norm is induced by
0
kv kr;w;A
2 112 k r X r þk d A ¼@ v ; ðx þ 1Þ2 k dx k¼0
ð27Þ
w
and A is the Sturm-Liouville operator as follows:
Av ðxÞ ¼ w1 ðxÞ
d d x v ðxÞ : dx dx
ð28Þ
We have the following theorem for the convergence: Theorem 2. For any
v 2 Hrw;A ðKÞ and r P 0,
kPN v v kw 6 cN r kv kr;w;A :
ð29Þ
A complete proof of the theorem and discussion on convergence is given in [45]. To apply a collocation method, we consider the residual Res(x) when the expansion is substituted into the governing equation. It requires that ak’s be selected so that the boundary condition are satisfied, but make the residual zero at as many (suitable chosen) spatial points as possible.
3. Solving Volterra’s population equation In this section we first convert Volterra’s population model Eq. (2) to an equivalent nonlinear ordinary differential equation, and then we apply Sinc and Rational Legendre collocation methods to this ODE. 3.1. Converting Volterra’s population model to a nonlinear ODE equation Let
yðxÞ ¼
Z 0
x
uðtÞdt:
ð30Þ
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This leads to
y0 ðxÞ ¼ uðxÞ;
y00 ðxÞ ¼ u0 ðxÞ:
ð31Þ
Inserting Eqs. (30) and (31) into Eq. (2) yields the nonlinear differential equation
jy00 ðxÞ ¼ y0 ðxÞ y0 ðxÞ2 yðxÞy0 ðxÞ;
ð32Þ
with the initial conditions
y0 ð0Þ ¼ u0 :
yð0Þ ¼ 0;
ð33Þ
3.2. Solving with Sinc functions collocation method For the boundary conditions in Eq. (33), the Sinc basis functions in Eq. (10) do not have a derivative when x tends to zero. Thus, we modify the Sinc basis functions as
xSk ðxÞ;
ð34Þ
Now the derivative of the modified Sinc basis functions are defined as x approaches zero and are equal to zero. In order to approximate the solution of Eq. (32) with boundary conditions, we construct a polynomial p(x) that satisfies Eq. (33). This polynomial is given by
pðxÞ ¼ kx2 þ u0 x:
ð35Þ
The approximate solution for y(x), in Eq. (32) with boundary conditions in Eq. (33) is represented by
yN ðxÞ ¼ uN ðxÞ þ pðxÞ;
ð36Þ
where
uN ðxÞ ¼
N X
ck xSk ðxÞ:
ð37Þ
k¼N
In Eq. (35), k is constant to be determined. It is noted that the approximate solution yN(x), satisfy boundary conditions in Eq. (33), since uN ð0Þ ¼ u0N ð0Þ ¼ 0, furthermore we have p(0) = 0 and p0 (0) = u0. The 2 N+1 coefficients fck gk¼N k¼N and the unknown k determined by substituting yN(x) into Eq. (32) and evaluating the result at the Sinc points:
xj ¼ lnðejh þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2jh þ 1Þ j ¼ N 1; . . . ; N;
ð38Þ
Obviously by using Eqs. (20)–(22) and (36) we have
uN ðxj Þ ¼ cj xj ; u0N ðxj Þ ¼
N X
ð39Þ n o ð1Þ ð0Þ ck xj /0 ðxj Þdk;j þ dk;j ;
ð40Þ
k¼N
u0N ðxj Þ ¼
N X
ð1Þ
ð1Þ
2
ð2Þ
ck f2/0 ðxj Þdk;j þ xj /00 ðxj Þdk;j þ xj /0 ðxj Þdk;j g:
ð41Þ
k¼N
Substituting Eqs. (36), (38)–(41) in Eq. (32) we obtain:
jy00N ðxj Þ ¼ y0N ðxj Þ ðy0N ðxj ÞÞ2 yN ðxj Þy0N ðxj Þ; j ¼ N 1; . . . ; N:
ð42Þ
Eq. (42) gives 2N+2 nonlinear algebraic equations which can be solved for the unknown coefficients ck and k by using the well-known Newton’s method. Consequently, y(x) given in Eq. (32) can be calculated. 3.3. Solving with Rational Legendre collocation method In this section, we apply the collocation method by using Rational Legendre functions to find solutions of Volterra’s population model. At the first step, by [15], let PNu be the approximation of u,
PN u ¼
N X k¼0
ak Rk ðxÞ:
ð43Þ
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Thus, our goal is to find the coefficients ak, 0 6 k 6 N. here we solve Volterra’s population model by collocation method. Let
jd2 PN uðxÞ dPN uðxÞ
2
d x
dx
þ
2 dPN uðxÞ dPN uðxÞ ; þ PN uðxÞ dx dx
ð44Þ
be the residual function of the Volterra’s population model. PNu is a good approximation of function u if it is zero on the whole domain. In other words, we should select coefficients aks so that the residual function approaches zero on the most of the domain. The collocation method for Volterra’s population model is to find P N u 2 RN such that
Resðxj Þ ¼ 0;
j ¼ 0; . . . ; N 2;
ð45Þ
for satisfying boundary conditions,
PN uðx0 Þ ¼ 1;
ð46Þ
dP N uðxÞ jx¼0 ¼ u0 ; dx
ð47Þ
where the xjs are Rational Legendre-Gauss nodes. This generates a set of N + 1 nonlinear equations that can be solved by Newton method for unknown coefficients aks. 3.4. Results Table 1 Shows a comparison of Sinc and Rational Legendre collocation method with the exact values of
umax ¼ 1 þ j ln
j ; 1 þ j u0
evaluated in [31].
Table 1 A comparison of Sinc and Rational Legendre collocation method with the exact values of umax. SCM
RLCM
Exact value
j
N
h
umax
N
L
umax
umax
0.02 0.04 0.10 0.20 0.50
35 30 24 22 20
0.1516 0.2403 0.4712 0.4301 0.4002
0.92342724 0.87372001 0.76974157 0.65905071 0.48519064
8 8 8 8 8
0.033605345 0.134702840 0.277226000 1.004023500 1.681736500
0.92342717 0.87371998 0.76974149 0.65905038 0.48519030
0.92342717 0.87371998 0.76974149 0.65905038 0.48519030
Fig. 1. Volterra’ Population graph obtained by Sinc collocation method (SCM) for j = 0.02, 0.04, 0.1, 0.2, 0.5.
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Fig. 2. Volterra’ Population graph obtained by Rational Legendre collocation method (RLCM) for j = 0.02, 0.04, 0.1, 0.2, 0.5.
Figs. 1 and 2 by use of Sinc and Rational Legendre functions respectively, shows the rapid rise along the logistic curve followed by the slow exponential decay after reaching the maximum point and when j increases, the amplitude of u(t) decreases whereas the exponential decay increases, for j = 0.02, 0.04, 0.1, 0.2, 0.5. 4. Conclusion In this paper, the collocation method with Sinc and Rational Legendre functions are employed to construct an approximation to the solution of the nonlinear Volterra’s Population model in a semi-infinite interval. The comparison are also made between the results of present methods and other methods. It is found that the results of the present works agree well with other methods. The validity of this method is based on the assumption that it converges by increasing the number of collocation points. Our aim was to apply an accurate and well-conditioned method that gives more accurate answers without reformulating the equation to bounded domains. Numerical results indicate the convergence and effectiveness of the present approach. In general, an important concern of spectral methods is the choice of basis functions; the basis functions have three properties: easy computation, rapid convergence, and completeness. In this way the Volterra model population is an integral equation in semi-infinite domain. moreover, there exist the same nonlinear ODE and integro-differential equation problems in biological content which as a further application. Acknowledgments The authors are very grateful to the reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper. The corresponding author would like to thank Shahid Beheshti University for the awarded grant. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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[16] Parand K, Razzaghi M. Rational Legendre approximation for solving some physical problems on semi-infinite intervals. Phys Scr 2004;64:353–7. [17] Parand K, Shahini M, Dehghan M. Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type. J Comput Phys 2009;228(23):8830–40. [18] Parand K, Shahini M. Rational Chebyshev pseudospectral approach for solving ThomasFermi equation. Phys Lett A 2009;373:210–3. [19] Stenger F. Numerical Methods Based on Sinc and Analytic Functions. New York: Springer-Verlag; 1993. [20] Lund J, Bowers K. Sinc Methods for Quadrature and Differential Equations. Philadelphia: SIAM; 1992. [21] Lund J, Vogel C. A fully-Galerkin method for the solution of an inverse problem in a parabolic partial differential equation. Numer Solut Inverse Probl 1990;6:205–17. [22] Al-Khaled K. Theory and computation in singular boundary value problems. Chaos Soliton Fract 2007;33:678–84. [23] Smith R, Bowers K. A Sinc–Galerkin estimation of diffusivity in parabolic problems. Inverse Probl 1993;9:113–35. [24] Bialecki B. A Sinc quadrature rule for Hadamard finite-part integrals. Numerische Mathematik 1990;57:263–9. [25] Stenger F. Sinc–Galerkin method of solution of boundary value problems. Math Comp 1979;33:85–109. [26] Carlson TS, Dockery J, Lund J. A Sinc-collocation method for initial value problems. Math Comput 1997;217:215–35. [27] Koures VG, Harris FE. Sinc collocation in quantum chemistry: solving the planar Coulomb Schrodinger equation. Int J Quant Chem: Quant Chem Symp 1996;30:1311–8. [28] Dehghan M, Saadatmandi A. The numerical solution of a nonlinear system of second-order boundary value problems using the Sinc-collocation method. Math Comp Model 2007;46:1434–41. [29] Smith RC, Bogar GA, Bowers KL, Lund J. Sinc–Galerkin method for fourth-order differential equations. SIAM J Numer Anal 1991;28:760–88. [30] Scudo FM. Volterra and theoretical ecology. Theor Popul Biol 1971;2:1–23. [31] TeBeest KG. Numerical and analytical solutions of Volterras population model. SIAM Review 1997;39:484–93. [32] Small RD. In: Klamkin MS, editor. In Population Growth in a Closed System and Mathematical Modelling. Philadelphia, PA: SIAM; 1989. p. 317–20. [33] Wazwaz AM. Analytical approximations and Padé approximant for Volterras population model. Appl Math Comput 1999;100:13–25. [34] Al-Khaled K. Numerical approximations for population growth models. Appl Math Comput 2005;160:865–73. [35] Parand K, Hojjati G. Solving Volterra’s Population Model using new Second Derivative Multistep Methods. Am J Appl Sci 2008;5:1019–22. [36] Momani S, Qaralleh R. Numerical approximations and Padé approximants for a fractional population growth model. Appl Math Model 2007;31:1907–14. [37] Ra mezani M, Razzaghi M, Dehghan M. Composite spectral functions for solving Volterras population model. Chaos Soliton Fract 2007;34:588–93. [38] Marzban HR, Hoseini S, Razzaghi M. Solution of Volterras population model via block-pulse functions and Lagrange-interpolating polynomials. Math Meth Appl Sci 2009;32:127–34. [39] Xu H. Analytical approximations for a population growth model with fractional order. Commun Nonlinear Sci Numer Simulat 2009;14:1978–83. [40] Lund J, Bowers K. Sinc Methods for Quadrature and Differential Equations. Philadelphia: SIAM; 1992. [41] El-Gamel M, Behiry SH, Hashish H. Numerical method for the solution of special nonlinear fourth-order boundary value problems. Appl Math Comput 2003;145:717–34. [42] Stenger F. Integration formulas via the trapezoidal formula. J Institut Math Appl 1973;12:103–14. [45] Guo BY, Shen J, Wang ZQ. A rational approximation and its applications to differential equations on the half line. J Sci Comput 2000;15(2):117–47.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Collocation method using sinc and Rational Legendre functions for solving Volterra’s population model K. Parand a,⇑,1, Z. Delafkar a, N. Pakniat a, A. Pirkhedri b, M. Kazemnasab Haji a a b
Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran Department of Computer Sciences, Islamic Azad University of Marivan, Iran
a r t i c l e
i n f o
Article history: Received 11 May 2010 Received in revised form 15 July 2010 Accepted 12 August 2010 Available online 17 August 2010 Keywords: Volterra’s population model Collocation method Sinc functions Rational Legendre functions Semi-infinite interval Nonlinear ODE
a b s t r a c t This paper proposes two approximate methods to solve Volterra’s population model for population growth of a species in a closed system. Volterra’s model is a nonlinear integro-differential equation on a semi-infinite interval, where the integral term represents the effect of toxin. The proposed methods have been established based on collocation approach using Sinc functions and Rational Legendre functions. They are utilized to reduce the computation of this problem to some algebraic equations. These solutions are also compared with some well-known results which show that they are accurate. Ó 2010 Published by Elsevier B.V.
1. Introduction 1.1. spectral methods While spectral approximations for ordinary differential equations (ODEs) in bounded domains have achieved great success and popularity in recent years, spectral approximations for ODEs in unbounded domains have only received limited attention. Several spectral methods for treating unbounded domains have been proposed by different researchers. Direct approaches using Laguerre polynomials were investigated by Maday et al. [1], Funaro [2] and [3,4]. Indirect approaches, e.g. Guo [5–7] have 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 class of spectral methods is based on Rational approximations. For example, Christov [8] and Boyd [9,10] developed some spectral methods on unbounded intervals by using mutually orthogonal systems of Rational functions. Boyd [10] defined a new spectral basis, named Rational Chebyshev functions on the semi-infinite interval, by mapping to the Chebyshev polynomials. Recently, [11] proposed and analyzed a set of Legendre Rational functions which are mutually orthogonal in Lv2(0, 1) with a non-uniform weight function v(x) = (x + 1)2. Another approach consists of replacing the infinite domain with [L, L] and semi-infinite interval with [0, L] by choosing L, sufficiently large. This method is named domain truncation [12].
⇑ Corresponding author. Tel.: +98 21 22431653; fax: +98 21 22431650. E-mail addresses: [email protected] (K. Parand), [email protected] (Z. Delafkar), [email protected] (N. Pakniat), alipirkhedri@gmail. com (A. Pirkhedri), [email protected] (M.K. Haji). . 1 Member of research group of Scientific Computing. 1007-5704/$ - see front matter Ó 2010 Published by Elsevier B.V. doi:10.1016/j.cnsns.2010.08.018
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Ref. [13] applied pseudospectral methods on a semi-infinite interval and compared Rational Chebyshev, Laguerre and mapped Fourier sine. Refs.[14–18] applied Rational Chebyshev and Legendre functions with tau and collocation method were applied to solve nonlinear ordinary differential equations on semi-infinite intervals. Refs. [14–16] obtained the operational matrices of derivative and the product of Rational Chebyshev and Legendre functions to reduce the solution of these problems to the solution of some algebraic equations. Another approach is based on Sinc functions. Using Sinc functions for obtaining approximate solutions of ODE and integral equation is widely discussed in [19–24]. For solving differential equations, two methods based on Sinc approximation were presented by Sinc Collocation and by Sinc-Galerkin. In [25] the Sinc-Galerkin method was applied first. Ref. [22] applied the Sinc-Galerkin method to solve a certain class of singular two-point boundary value problems and expressed the exact solution of the differential equations via the use of Green’s functions as an integral type. A Sinc quadrature rule is also presented for the evaluation of Hadamard finite-part integrals of analytic functions in [24]. They showed that the implementation of the Sinc quadrature did not take any additional work, since the nodes and weights were given by very simple formulas. Sinc-Collocation procedure for the numerical solution of the initial value problem is developed in [26] and approved that Sinc procedure converges to the solution at an exponential converge. In [27] Sinc-Collocation is discussed for solving the planar coulomb Schrodinger Equation. They applied Sinc procedure with ln(sinh(x)) function to map the infinite interval to semi-infinite interval and computed the eigen values and eigen functions of the position-space planar Coulomb equation by this method. In [28] Sinc-Collocation method has been extended to handle system of second-order boundary value problems. They applied this method and reduced the computation to some algebraic equations. Ref. [29] used the Sinc-Galerkin method to solve fourth-order ordinary differential equations. Even at this high order, the consistency of the method in exhibiting an exponential convergence rate could be shown. In [19] it is shown that both Sinc-Collocation and Sinc-Galerkin converge with exponential rate. 1.2. Volterra’s model The Volterra model for population growth of a species within a closed system is given in [30,31] as
Z
dp 2 ¼ ap bp cp d^t
^t
pðxÞdx;
0
pð0Þ ¼ p0 ;
ð1Þ
where a > 0 is the birth rate coefficient, b > 0 is the crowding coefficient, c > 0 is the toxicity coefficient, p0 is the initial population and p ¼ pð^tÞ denotes the population at time ^t. Several time scales and population scales may be employed [32]. Here, we employ time and population by introducing the non-dimensional variables
t¼
c^t ; b=c
u¼
bp ; a
which produce the non-dimensional problem
j
du ¼ u u2 u dt
Z
t
uðxÞdx;
uð0Þ ¼ u0 ;
ð2Þ
0
where j = c/ab is a prescribed non-dimensional parameter. One may show that the only equilibrium solution to Eq. (2) is the trivial solution u(t) = 0. Furthermore, the analytical solution [32]
uðtÞ ¼ u0 exp
Z t Z s 1 1 uðsÞ uðxÞdx ds ;
j
0
0
shows that u(t) > 0 for all t if u0 > 0. The solution to Eq. (2) has been of considerable concern. Although a closed-form solution has been achieved in [31], it was formally shown that the closed-form solution cannot lead to any insight into the behavior of the population evolution [33]. Some approximate and numerical solutions for Volterra’s population model have been reported. In [30], the successive approximations method was offered for the solution to Eq. (2) but was not implemented. In [32], the singular perturbation method for Volterra’s population model is considered. It is shown in [32] that for the case j 1, where populations are weakly sensitive to toxins, a rapid rise occurs along the logistic curve that will reach a peak and be followed by a slow exponential decay. In addition, for large j, where populations are strongly sensitive to toxins, the solution is proportional to sech2(t). In this case the solution u(t) has a smaller amplitude compared with the amplitude of u(t) for the case j 1. In [31], three numerical algorithms, namely, the Euler method, the modified Euler method and the fourth-order Runge–Kutta method for the solution to Eq. (2) are obtained. Moreover, a phase-plane analysis is implemented. In [31], the numerical results are correlated to give insight into the problem and its solution without using perturbation techniques. However, it requires extensive calculations. In [33], the series solution method and the Padé 3 approximations are applied to Eq. (2) and to a related ordinary differential equation. In [34], a comparison of the Adomian decomposition method and Sinc-Galerkin method is given and it is shown that the Adomian decomposition method is more efficient for the solution of Volterra’s population model. Ref. [35] used second derivative multistep method (SDMM) to solve this equation. In [36] Adomian decomposition approach and Padé approximates is applied to Eq. (2). Ref. [37,38] used composite spectral functions approximation
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and hybrid function approximation, respectively to approximate a solution for this equation. In [39] homotopy analysis method (HAM) was applied to this equation. The organization of the paper is as follows: In the following, we describe formulation of Sinc and Rational Legendre functions, our solutions for this problem and the application of these methods for Volterra’s population model is summarized in Section 3 and a comparison with existing methods is also made in the literature. The conclusions are described in the final section. 2. Properties of Sinc and Rational Legendre functions 2.1. Sinc functions properties The Sinc functions is defined on the whole real line I = (1, 1) by
( SincðxÞ ¼
sinðpxÞ px ;
x – 0;
1;
x ¼ 0:
ð3Þ
For each integer k and the mesh size h, the Sinc basis functions are defined on R by [28]
8 < sin ðph ðxkhÞÞ x kh ; x – kh; pðxkhÞ ¼ Sk ðh; xÞ Sinc h : h 1; x ¼ kh:
ð4Þ
The Sinc functions form an interpolatory set of functions, i.e.,
Sk ðh; jhÞ ¼ dkj ¼
1; j ¼ k;
ð5Þ
0; j – k:
If a function f(x) is defined on the real axis, then for h > 0 the series
Cðf ; hÞðxÞ ¼
1 X
f ðkhÞSinc
k¼1
x kh ; h
ð6Þ
is called the Whittaker cardinal expansion of f whenever this series converges. The properties of the Whittaker cardinal expansion have been extensively studied in [40]. These properties are derived in the infinite strip DS of the complex x-plane, where d > 0,
n po DS ¼ x ¼ t þ is : jsj < d 6 : 2
ð7Þ
Approximations can be constructed for infinite, semi-infinite and finite intervals. To construct approximations on the interval (0, +1) which is used in this paper, the eye-shaped domain in the z-plane
n po DE ¼ z ¼ x þ iy : jargðsinhðzÞÞj < d 6 ; 2
ð8Þ
is mapped conformally onto the infinite strip DS via
x ¼ /ðzÞ ¼ lnðsinhðzÞÞ:
ð9Þ
The basis functions on (0, +1) are taken to be the composite translated Sinc functions,
Sk ðxÞ Sðk; hÞ /ðxÞ ¼ Sinc
/ðxÞ kh ; h
ð10Þ
where S(k, h)/(x) is defined by S(k, h)(/(x)). The inverse map of x = /(z) is
z ¼ /1 ðxÞ ¼ lnðex þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2x þ 1Þ:
Thus we may define the inverse images of the real line and of the evenly spaced nodes
ð11Þ fkhgk¼þ1 k¼1
C ¼ f/1 ðtÞ 2 DE : 1 < t < þ1g ¼ ð0; þ1Þ
as
ð12Þ
and
xk ¼ /1 ðkhÞ ¼ lnðekh þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2kh þ 1Þ; k ¼ 0; 1; 2; . . .
ð13Þ
Let w(x) denotes a non-negative, integrable, real-valued function over the interval C. We define
L2w ðCÞ ¼ fv : C ! Rjv
is measurable and kv kw < 1g;
ð14Þ
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where
kv kw ¼
Z
1
12 jv ðxÞj2 wðxÞdx ;
ð15Þ
0
is the norm induced by the inner product of the space L2w ðCÞ,
< u; v >w ¼
Z
1
uðxÞv ðxÞwðxÞdx:
ð16Þ
0
Thus {Sk(x)}k2Z with constant h denotes a system which is mutually orthogonal under (16), i.e.,
< Skn ðxÞ; Skm ðxÞ>wðxÞ ¼ hdnm ;
ð17Þ
where w(x) = coth(x) and dnm is the Kronecker delta function. This system is complete in following expansion holds
f ðxÞ ffi
þN X
L2w ð
CÞ. For any function, f 2
L2w ð
fk Sk ðxÞ;
CÞ the
ð18Þ
k¼N
with
fk ¼
< f ðxÞ; Sk ðxÞ>wðxÞ kSk ðxÞk2wðxÞ
ð19Þ
:
The fk = f(xk) are the discrete expansion coefficients associated with the family {Sk(x)} and the mesh size is given by
rffiffiffiffiffiffiffiffiffi 2p d ; h¼ aN where N is suitably chosen and a depends on the asymptotic behavior of f(x) [41]. Definition 1. Let
Dd ¼ fz ¼ x þ iy : 0 < jyj < dg; the function f is in the space of H2(Dd) [26], if f is analytic in Dd and satisfies
Z
d
ðjf ðx þ iyÞjÞdy ¼ Oðxc Þ;
x ! 1; 0 6 c < 1
d
and on the boundary of Dd, (denoted @Dd), satisfies
Nðf Þ ¼
Z
jf ðzÞdzj < 1:
@D
Theorem 1. Assume that f/0 2 H2(Dd) then for all x 2 C
1 X Nðf /0 Þ Nðf /0 Þ pd=h Eðf ; hÞðxÞ ¼ f ðxÞ f ðkhÞSðk; hÞ /ðxÞ 6 : 62 e 2pd sinhðpd=hÞ pd k¼1 Moreover, if jf(x)j 6 Ceaj/(x)j, x 2 C, for some positive constants C and a, and if the selection h ¼
N X pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi f ðkhÞSðk; hÞ /ðxÞ 6 C 2 N expð pdaN Þ; f ðxÞ k¼N
qffiffiffiffiffi pd aN
2pd 6 lnð2Þ , then
where C2 depends only on f, d and a [42]. Also, the nth derivative of the function f at some points xk can be approximated [41] ð0Þ
dk;j ¼ ½Sðk; hÞ /ðxÞjx¼xj ¼ ð1Þ
dk;j
ð2Þ
dk;j
1; k ¼ j;
0; k – j; ( k ¼ j; d 1 0; ½Sðk; hÞ /ðxÞjx¼xj ¼ ¼ jk d/ h ð1Þ ; k – j; jk 8 2 p 2 < ; k ¼ j; 3 d 1 ¼ 2 ½Sðk; hÞ /ðxÞjx¼xj ¼ 2 2ð1Þjk : d/ ; k – j: h ðjkÞ2
ð20Þ ð21Þ
ð22Þ
K. Parand et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1811–1819
1815
2.2. Rational Legendre functions properties This section is devoted to introducing Rational Legendre functions and expressing some basic properties of them. Rational Legendre functions denoted by Rn(x) are generated from well known Legendre polynomials by using the algebraic mapping /(x) = (x L)/(x + L), as noted in [17]
Rn ðxÞ ¼ Pn ð/ðxÞÞ;
ð23Þ
where L is a constant parameter and Pn(y) is the Legendre polynomial of degree n. The constant parameter L sets the length scale of the mapping. Boyd [10,12,13] offered guidelines for optimizing the map parameter L where L > 0. Numerical results deponed smoothly on constant parameter L, and therefore are not very sensitive to L, so the error varies very slowly with L around the minimum. A little trial and error is usually sufficient to find a value that is nearly optimum. In general, there is no way to avoid a small amount of trial and error in choosing L when solving problems on an unbounded domain. Experience and the asymptotic approximations of Boyd [12] can help, but some experimentation is always necessary as he explains in his book [12]. Properties of Rational Legendre functions and a complete discussion on approximating functions by Rational Legendre functions are given in [17]. 2.2.1. Rational Legendre functions approximation Let
RN ¼ spanfR0 ; R1 ; . . . ; RN g; We define P N :
L2w ð
PN uðxÞ ¼
ð24Þ
KÞ ! RN by
N X
ak Rk ðxÞ;
ð25Þ
k¼0
To obtain the order of convergence of Rational Legendre approximation, we define the space
Hrw;A ðKÞ ¼ fv : v
is measurable and kv kr;w;A < 1g;
ð26Þ
where the norm is induced by
0
kv kr;w;A
2 112 k r X r þk d A ¼@ v ; ðx þ 1Þ2 k dx k¼0
ð27Þ
w
and A is the Sturm-Liouville operator as follows:
Av ðxÞ ¼ w1 ðxÞ
d d x v ðxÞ : dx dx
ð28Þ
We have the following theorem for the convergence: Theorem 2. For any
v 2 Hrw;A ðKÞ and r P 0,
kPN v v kw 6 cN r kv kr;w;A :
ð29Þ
A complete proof of the theorem and discussion on convergence is given in [45]. To apply a collocation method, we consider the residual Res(x) when the expansion is substituted into the governing equation. It requires that ak’s be selected so that the boundary condition are satisfied, but make the residual zero at as many (suitable chosen) spatial points as possible.
3. Solving Volterra’s population equation In this section we first convert Volterra’s population model Eq. (2) to an equivalent nonlinear ordinary differential equation, and then we apply Sinc and Rational Legendre collocation methods to this ODE. 3.1. Converting Volterra’s population model to a nonlinear ODE equation Let
yðxÞ ¼
Z 0
x
uðtÞdt:
ð30Þ
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This leads to
y0 ðxÞ ¼ uðxÞ;
y00 ðxÞ ¼ u0 ðxÞ:
ð31Þ
Inserting Eqs. (30) and (31) into Eq. (2) yields the nonlinear differential equation
jy00 ðxÞ ¼ y0 ðxÞ y0 ðxÞ2 yðxÞy0 ðxÞ;
ð32Þ
with the initial conditions
y0 ð0Þ ¼ u0 :
yð0Þ ¼ 0;
ð33Þ
3.2. Solving with Sinc functions collocation method For the boundary conditions in Eq. (33), the Sinc basis functions in Eq. (10) do not have a derivative when x tends to zero. Thus, we modify the Sinc basis functions as
xSk ðxÞ;
ð34Þ
Now the derivative of the modified Sinc basis functions are defined as x approaches zero and are equal to zero. In order to approximate the solution of Eq. (32) with boundary conditions, we construct a polynomial p(x) that satisfies Eq. (33). This polynomial is given by
pðxÞ ¼ kx2 þ u0 x:
ð35Þ
The approximate solution for y(x), in Eq. (32) with boundary conditions in Eq. (33) is represented by
yN ðxÞ ¼ uN ðxÞ þ pðxÞ;
ð36Þ
where
uN ðxÞ ¼
N X
ck xSk ðxÞ:
ð37Þ
k¼N
In Eq. (35), k is constant to be determined. It is noted that the approximate solution yN(x), satisfy boundary conditions in Eq. (33), since uN ð0Þ ¼ u0N ð0Þ ¼ 0, furthermore we have p(0) = 0 and p0 (0) = u0. The 2 N+1 coefficients fck gk¼N k¼N and the unknown k determined by substituting yN(x) into Eq. (32) and evaluating the result at the Sinc points:
xj ¼ lnðejh þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2jh þ 1Þ j ¼ N 1; . . . ; N;
ð38Þ
Obviously by using Eqs. (20)–(22) and (36) we have
uN ðxj Þ ¼ cj xj ; u0N ðxj Þ ¼
N X
ð39Þ n o ð1Þ ð0Þ ck xj /0 ðxj Þdk;j þ dk;j ;
ð40Þ
k¼N
u0N ðxj Þ ¼
N X
ð1Þ
ð1Þ
2
ð2Þ
ck f2/0 ðxj Þdk;j þ xj /00 ðxj Þdk;j þ xj /0 ðxj Þdk;j g:
ð41Þ
k¼N
Substituting Eqs. (36), (38)–(41) in Eq. (32) we obtain:
jy00N ðxj Þ ¼ y0N ðxj Þ ðy0N ðxj ÞÞ2 yN ðxj Þy0N ðxj Þ; j ¼ N 1; . . . ; N:
ð42Þ
Eq. (42) gives 2N+2 nonlinear algebraic equations which can be solved for the unknown coefficients ck and k by using the well-known Newton’s method. Consequently, y(x) given in Eq. (32) can be calculated. 3.3. Solving with Rational Legendre collocation method In this section, we apply the collocation method by using Rational Legendre functions to find solutions of Volterra’s population model. At the first step, by [15], let PNu be the approximation of u,
PN u ¼
N X k¼0
ak Rk ðxÞ:
ð43Þ
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Thus, our goal is to find the coefficients ak, 0 6 k 6 N. here we solve Volterra’s population model by collocation method. Let
jd2 PN uðxÞ dPN uðxÞ
2
d x
dx
þ
2 dPN uðxÞ dPN uðxÞ ; þ PN uðxÞ dx dx
ð44Þ
be the residual function of the Volterra’s population model. PNu is a good approximation of function u if it is zero on the whole domain. In other words, we should select coefficients aks so that the residual function approaches zero on the most of the domain. The collocation method for Volterra’s population model is to find P N u 2 RN such that
Resðxj Þ ¼ 0;
j ¼ 0; . . . ; N 2;
ð45Þ
for satisfying boundary conditions,
PN uðx0 Þ ¼ 1;
ð46Þ
dP N uðxÞ jx¼0 ¼ u0 ; dx
ð47Þ
where the xjs are Rational Legendre-Gauss nodes. This generates a set of N + 1 nonlinear equations that can be solved by Newton method for unknown coefficients aks. 3.4. Results Table 1 Shows a comparison of Sinc and Rational Legendre collocation method with the exact values of
umax ¼ 1 þ j ln
j ; 1 þ j u0
evaluated in [31].
Table 1 A comparison of Sinc and Rational Legendre collocation method with the exact values of umax. SCM
RLCM
Exact value
j
N
h
umax
N
L
umax
umax
0.02 0.04 0.10 0.20 0.50
35 30 24 22 20
0.1516 0.2403 0.4712 0.4301 0.4002
0.92342724 0.87372001 0.76974157 0.65905071 0.48519064
8 8 8 8 8
0.033605345 0.134702840 0.277226000 1.004023500 1.681736500
0.92342717 0.87371998 0.76974149 0.65905038 0.48519030
0.92342717 0.87371998 0.76974149 0.65905038 0.48519030
Fig. 1. Volterra’ Population graph obtained by Sinc collocation method (SCM) for j = 0.02, 0.04, 0.1, 0.2, 0.5.
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Fig. 2. Volterra’ Population graph obtained by Rational Legendre collocation method (RLCM) for j = 0.02, 0.04, 0.1, 0.2, 0.5.
Figs. 1 and 2 by use of Sinc and Rational Legendre functions respectively, shows the rapid rise along the logistic curve followed by the slow exponential decay after reaching the maximum point and when j increases, the amplitude of u(t) decreases whereas the exponential decay increases, for j = 0.02, 0.04, 0.1, 0.2, 0.5. 4. Conclusion In this paper, the collocation method with Sinc and Rational Legendre functions are employed to construct an approximation to the solution of the nonlinear Volterra’s Population model in a semi-infinite interval. The comparison are also made between the results of present methods and other methods. It is found that the results of the present works agree well with other methods. The validity of this method is based on the assumption that it converges by increasing the number of collocation points. Our aim was to apply an accurate and well-conditioned method that gives more accurate answers without reformulating the equation to bounded domains. Numerical results indicate the convergence and effectiveness of the present approach. In general, an important concern of spectral methods is the choice of basis functions; the basis functions have three properties: easy computation, rapid convergence, and completeness. In this way the Volterra model population is an integral equation in semi-infinite domain. moreover, there exist the same nonlinear ODE and integro-differential equation problems in biological content which as a further application. Acknowledgments The authors are very grateful to the reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper. The corresponding author would like to thank Shahid Beheshti University for the awarded grant. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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