A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra's Population Growth Model

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 783069 6 pageshttpdxdoiorg1011552013783069

Research ArticleA Multiple-Step Legendre-Gauss Collocation Method forSolving Volterrarsquos Population Growth Model

Majid Tavassoli Kajani1 Mohammad Maleki2 and Adem KJlJccedilman3

1 Department of Mathematics Khorasgan Branch Islamic Azad University Isfahan Iran2Department of Mathematics Mobarakeh Branch Islamic Azad University Isfahan Iran3Department of Mathematics and Institute of Mathematical Research Universiti Putra Malaysia (UPM)43400 Serdang Selangor Malaysia

Correspondence should be addressed to Adem Kılıcman kilicmanyahoocom

Received 6 October 2013 Accepted 15 November 2013

Academic Editor Necdet Bildik

Copyright copy 2013 Majid Tavassoli Kajani et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new shifted Legendre-Gauss collocation method is proposed for the solution of Volterrarsquos model for population growth of aspecies in a closed system Volterrarsquos model is a nonlinear integrodifferential equation on a semi-infinite domain where the integralterm represents the effects of toxin In this method by choosing a step size the original problem is replaced with a sequence ofinitial value problems in subintervals The obtained initial value problems are then step by step reduced to systems of algebraicequations using collocation The initial conditions for each step are obtained from the approximated solution at its previous stepIt is shown that the accuracy can be improved by either increasing the collocation points or decreasing the step size The methodseems easy to implement and computationally attractive Numerical findings demonstrate the applicability and high accuracy ofthe proposed method

1 Introduction

Many science and engineering problems arise in unboundeddomains During the last few years different spectral methodshave been proposed for solving problems on unboundeddomains One of the methods is through the use of orthog-onal polynomials over unbounded domains such as theHermite spectral and the Laguerre spectral methods [1ndash5]However all of these algorithms need certain quadratureson unbounded domains which introduce errors and soweaken the merit of spectral approximations Another directapproach for solving such problems is based on rationalapproximations Christov [6] and Boyd [7 8] developedsome spectral methods on unbounded intervals by usingmutually orthogonal systems of rational functions Boyd [8]defined a new spectral basis named rational Chebyshevfunctions on the semi-infinite interval by mapping it to theChebyshev polynomials Guo et al [9] introduced a new set of

rational Legendre functions which are mutually orthogonalin 1198712(0 +infin) They applied a spectral scheme using therational Legendre functions for solving the Korteweg-deVries equation on the half line Boyd et al [10] applied pseu-dospectral methods on a semi-infinite interval and comparedthe rational Chebyshev Laguerre and the mapped Fouriersine methods Parand et al [11] compared two commoncollocation approaches based on radial basis functions forthe case of heat transfer equations arising in porous mediumThe use of a suitable mapping to transfer infinite domains tothe finite domains and then applying the standard spectralmethods for the transformed problems in finite domains areconsidered another approach that is frequently used see [12ndash16] Another approach is replacing the infinite domain with[minus119879 119879] and the semi-infinite interval with [0 119879] by choosing119879 sufficiently large This method is named as the domaintruncation [17 18]

2 Mathematical Problems in Engineering

In [19 20] the Volterra model for population growth ofa species within a closed system is given by

119889119901

119889= 119886119901 minus 119887119901

2minus 119888119901int

0

119901 (119909) 119889119909 119901 (0) = 1199010 (1)

where 119886 gt 0 is the birth rate coefficient 119887 gt 0 is the crowdingcoefficient 119888 gt 0 is the toxicity coefficient 119901

0is the initial

population and 119901 = 119901() denotes the population at time Also the coefficient 119888 indicates the essential behavior of thepopulation evolution before its level falls to zero in the longterm

This model is an integroordinary differential equationwhere the term 119888119901 int

0119901(119909)119889119909 represents the effect of toxin

accumulation on the species Although several time scalesand population scales may be employed [20] here we willscale time and population by introducing the nondimen-sional variables

119905 =

119887119888 119906 =

119901

119886119887 (2)

which produce the nondimensional problem

120581119889119906

119889119905= 119906 minus 119906

2minus 119906int

119905

0

119906 (119909) 119889119909 119906 (0) = 1199060 (3)

where 119906(119905) is the scaled population of identical individualsat time 119905 and 120581 = 119888(119886119887) is a prescribed nondimensionalparameter One may show that the only equilibrium solutionof (3) is the trivial solution119906(119905) = 0 In addition the analyticalsolution [20]

119906 (119905) = 1199060exp(1

120581int

119905

0

[1 minus 119906 (120591) minus int

120591

0

119906 (119909) 119889119909] 119889120591) (4)

shows that 119906(119905) gt 0 for all 119905 if 1199060gt 0

During the recent years the solution of (3) has been ofconsiderable concern In [19] the successive approximationsmethod was suggested for the solution of (3) but was notimplemented In this case the solution 119906(119905) has a smalleramplitude compared with the amplitude of 119906(119905) for the case120581 ≪ 1 Similarly in [20] the singular perturbation methodfor solving Volterrarsquos population model is considered Theauthor scaled out the parameters of (3) as much as possibleby using four different ways and considered two cases 120581 =119888(119886119887) small and 120581 = 119888(119886119887) large Thus it is shown in [20]that for the case 120581 ≪ 1 where populations are weakly sensi-tive to toxins a rapid rise occurs along the logistic curve thatwill reach a peak and then is followed by a slow exponentialdecay In the case of large 120581 the populations are strongly sen-sitive to toxins and the solutions are proportional to sech2(119905)

In [21] four numerical methods namely the Eulermethod the modified Euler method the classical fourth-order Runge-Kutta method and the Runge-Kutta-Fehlbergmethod for the solution of (3) are proposed Moreover aphase-plane analysis is implemented In [22] a comparisonof the Adomian decomposition method and Sinc-Galerkinmethod is given and it is shown that the Adomiandecomposition method is more efficient for the solution

of Volterrarsquos population model In [23] the series solutionmethod and the decomposition method are implementedindependently to (3) and to a related nonlinear ordinarydifferential equation Furthermore the Pade approximationsare used in the analysis to capture the essential behavior of thepopulation 119906(119905) of identical individuals and approximationof 119906max and the exact value of 119906max for different 120581 werecompared The authors of [24ndash26] applied spectral methodto solve Volterrarsquos population on a semi-infinite intervalbased on a rational Tau method

In [27] the approach is based upon domain truncationand composite spectral functions approximations They firstconsidered an interval [0 119871] where 119871 is any positive integerand divided this interval into subintervals with step size ℎ =1119873 where 119873 is a positive integer They then transformedeach subinterval into [0 1) and utilized the properties ofcomposite spectral functions consisting of few terms oforthogonal functions to reduce the solution of Volterrarsquosmodel to the solution of a system of algebraic equations

In [28] a numerical method based on domain truncationand hybrid functions was proposed to solve Volterrarsquos popu-lation model They considered an interval [0 119905

119891) and then

by utilizing the properties of hybrid functions that consistof block-pulse and Lagrange-interpolating polynomials theyreduced the solution of Volterrarsquos model to the solution of asystem of algebraic equations

In [29] the authors compared the application of rationalChebyshev collocation and Hermite functions collocationmethods for solving Volterrarsquos population model In [30] anew homotopy perturbation method is proposed for directlysolving the Volterrarsquos population model as a nonlinear inte-grodifferential equation

In this paper we introduce a new collocation methodfor solving (3) Volterrarsquos population model in (3) is firstconverted to an equivalent nonlinear initial value problem(IVP) This method solves the problem step by step and isvalid for large domains We first consider a step size and thenreplace the original IVP in the interval [0infin)with a sequenceof IVPs in subintervals with length equal to the consideredstep sizeThen the sequence of IVPs is consecutively reducedto sets of algebraic equations using collocation based onshifted Legendre-Gauss (ShLG) pointsThe initial conditionsof the 119896th step (except for the first step where the initialconditions are available) are obtained from the approximatedsolution obtained earlier at the (119896 minus 1)th step

The paper is organized as follows In Section 2 some basicproperties of Legendre and shifted Legendre polynomialsrequired for our subsequent development are givenThen theapplication of this method to Volterrarsquos population model issummarized In Section 3 we report our numerical findingsand demonstrate the efficiency and accuracy of the proposedscheme

2 Step by Step Spectral Collocation Methodfor Volterrarsquos Population Model

In this section we derive the step by step ShLG spectral collo-cation method for solving Volterrarsquos population model in (3)

Mathematical Problems in Engineering 3

21 Review of Legendre and Shifted Legendre PolynomialsThe Legendre polynomials 119875

119899(119909) 119899 = 0 1 are the

eigenfunctions of the singular Sturm-Liouville problem

((1 minus 1199092) 1198751015840

119899(119909))1015840

+ 119899 (119899 + 1) 119875119899(119909) = 0 (5)

Also they are orthogonal with respect to 1198712 inner product onthe interval [minus1 1] with the weight function 119908(119909) = 1 that is

int

1

minus1

119875119899(119909) 119875119898(119909) 119889119909 =

2

2119899 + 1120575119899119898 (6)

where 120575119899119898

is the Kronecker delta The Legendre polynomialssatisfy the recursion relation

119875119899+1(119909) =

2119899 + 1

119899 + 1119909119875119899(119909) minus

119899

119899 + 1119875119899minus1(119909) (7)

where 1198750(119909) = 1 and 119875

1(119909) = 119909 If 119875

119899(119909) is normalized so that

119875119899(1) = 1 then for any 119899 the Legendre polynomials in terms

of power of 119909 are

119875119899(119909) =

1

2119899

[1198992]

sum

119898=0

(minus1)119898(119899

119898)(2119899 minus 2119898

119899)119909119899minus2119898 (8)

where [1198992] denotes the integer part of 1198992The Legendre-Gauss (LG) collocation points minus1 lt 119909

1lt

1199092lt sdot sdot sdot lt 119909

119873minus1lt 1 are the roots of 119875

119873minus1(119909) Explicit

formulas for the LGpoints are not knownTheLGpoints havethe property that

int

1

minus1

119901 (119909) 119889119909 =

119873minus1

sum

119894=1

119908119894119901 (119909119894) (9)

is exact for polynomials of degree at most 2119873 minus 3 where 119908119894

1 ⩽ 119894 ⩽ 119873 minus 1 are LG quadrature weights For more detailsabout Legendre polynomials see [31]

The shifted Legendre polynomials on the interval 119905 isin[119886 119887] are defined by

119899(119905) = 119875

119899(1

119887 minus 119886(2119905 minus 119886 minus 119887)) 119899 = 0 1 (10)

which are obtained by an affine transformation from theLegendre polynomialsThe set of shifted Legendre polynomi-als is a complete 1198712[119886 119887]-orthogonal system with the weightfunction 119908(119905) = 1 Thus any function 119891 isin 1198712[119886 119887] can beexpanded in terms of shifted Legendre polynomials

The ShLG collocation points 119886 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119873minus1lt 119887

on the interval [119886 119887] are obtained by shifting the LG points119909119894 using the transformation

119905119894=1

2((119887 minus 119886) 119909

119894+ 119886 + 119887) 119894 = 1 2 119873 minus 1 (11)

By using the property of standard LG quadrature itfollows that for any polynomial 119901 of degree at most 2119873 minus 3on (119886 119887)

int

119887

119886

119901 (119905) 119889119905 =119887 minus 119886

2int

1

minus1

119901(1

2((119887 minus 119886) 119909 + 119886 + 119887)) 119889119909

=119887 minus 119886

2

119873minus1

sum

119894=1

119908119894119901(1

2((119887 minus 119886) 119909

119894+ 119886 + 119887))

=

119873minus1

sum

119894=1

119908119894119901 (119905119894)

(12)

where 119908119894= ((119887 minus 119886)2)119908

119894 1 ⩽ 119894 ⩽ 119873 minus 1 are

ShLG quadrature weights The results stated above are alsosatisfied for Legendre-Gauss-Lobatto and Legendre-Gauss-Radau quadrature rules

22 Solution of Volterrarsquos Population Model In this subsec-tion we first convert Volterrarsquos population model (3) to anequivalent nonlinear IVP Let

119910 (119905) = int

119905

0

119906 (119909) 119889119909 (13)

which leads to

1199101015840(119905) = 119906 (119905) 119910

10158401015840(119905) = 119906

1015840(119905) (14)

With substituting (13) and (14) into (3) the following nonlin-ear IVP is obtained

12058111991010158401015840(119905) = 119910

1015840(119905) minus (119910

1015840(119905))2

minus 119910 (119905) 1199101015840(119905) 0 le 119905 lt infin

119910 (0) = 0 1199101015840(0) = 119906

0

(15)

Then to drive a step by step ShLG collocation method forsolving (15) we first choose a step size ℎ where ℎ can be anypositive real number Now let 119910

119896(119905) be the solution of (15)

in subinterval 119868119896= [(119896 minus 1)ℎ 119896ℎ] 119896 = 1 2 The IVP in

(15) on the interval [0infin) can be replaced with the followingsequence of IVPs on subintervals 119868

119896 119896 = 1 2

12058111991010158401015840

119896(119905) = 119910

1015840

119896(119905) minus (119910

1015840

119896(119905))2

minus 119910119896(119905) 1199101015840

119896(119905) 119905 isin 119868

119896

119910119896((119896 minus 1) ℎ) = 119910

119896minus1((119896 minus 1) ℎ)

1199101015840

119896((119896 minus 1) ℎ) = 119910

1015840

119896minus1((119896 minus 1) ℎ)

(16)

where the initial conditions for the 119896th IVP (119896 ⩾ 2) areconsidered using the solution obtained earlier for the (119896minus1)thIVP Note that for the first IVP the initial conditions areavailable from (15) In addition it is important to note thatthe initial conditions in (16) also maintained the continuityand the differentiability at the interface of subintervals Thecalculations begin at the first step with solving the followingIVP on 119868

1= [0 ℎ]

12058111991010158401015840

1(119905) = 119910

1015840

1(119905) minus (119910

1015840

1(119905))2

minus 1199101(119905) 1199101015840

1(119905) 119905 isin 119868

1

1199101(0) = 0 119910

1015840

1(0) = 119906

0

(17)


This then allows the approximation of1199102(119905) on the subinterval

1198682to be obtained at the second step from the IVP in (16) and

so onConsider now the ShLG collocation points (119896 minus 1)ℎ lt


119896119873minus1lt 119896ℎ on the 119896th subinterval 119868

119896 119896 = 1 2

obtained using (11) Obviously

119905119896119894=ℎ

2(119909119894+ 2119896 minus 1) 119894 = 1 2 119873 minus 1 (18)

Also consider two additional noncollocated points 1199051198960= (119896 minus

1)ℎ and 119905119896119873= 119896ℎ We approximate the function 119910

119896(119905) isin

1198712(119868119896) within each subinterval 119868

119896by a polynomial of degree

at most119873 as

119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0

119910119896119895119871119896119895(119905) 119905 isin 119868

119896 (19)

where 119910119896119895= 119910119896(119905119896119895) and

119871119896119895(119905) =

119873

prod

119897=0119897 = 119895

119905 minus 119905119896119897

119905119896119895minus 119905119896119897

119895 = 0 1 119873 (20)

is a basis of119873th-degree Lagrange polynomials on the subin-terval 119868

119896that satisfy 119871

119896119895(119905119896119894) = 120575119894119895 Here it can be easily seen

that for 119895 = 0 1 119873 and 119896 = 1 2 we have

119871119896119895(119905) = 119871

1119895(119905 minus (119896 minus 1) ℎ) 119905 isin 119868

119896 (21)

Thus by utilizing (21) for (19) the approximation of 119910119896(119905)

within each subinterval 119868119896can be restated as

119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0


119896

(22)

It is important to observe that the series (22) includesthe Lagrange polynomials associated with the noncollocatedpoints 119905

1198960= (119896 minus 1)ℎ and 119905

119896119873= 119896ℎ Differentiating the series

of (22) twice and evaluating at the ShLG collocation points119905119896119894 119894 = 1 119873 minus 1 give

119910(119898)

119896(119905119896119894) asymp

119873

sum

119895=0

119910119896119895119871(119898)

1119895(119905119896119894minus (119896 minus 1) ℎ)

=

119873

sum

119895=0

119910119896119895119871(119898)

1119895(1199051119894) =

119873

sum

119895=0

119863(119898)

119894119895119910119896119895 119898 = 1 2

(23)

where 119863(119898)119894119895= 119871(119898)

1119895(1199051119894) The (119873 minus 1) times (119873 + 1) nonsquare

matrices D(119898) = [119863(119898)119894119895] 119898 = 1 2 are the first- and second-

order Gauss pseudospectral differentiation matrices in thesubinterval 119868

1= [0 ℎ] where we note that the extra columns

of D(1) and D(2) are due to the Lagrange polynomials 11987110(119905)

and 1198711119873(119905) associated with the noncollocated points 119905

10= 0

and 1199051119873= ℎ

Further it is seen from (21)ndash(23) that in the present stepby step collocation scheme we only need to produce the basis

of Lagrange polynomials1198711119895(119905) and theGauss pseudospectral

differentiation matrices D(1) and D(2) in the first subintervalThis reduces the number of arithmetic calculations andalso the computational time specially when the numberof subintervals (number of steps) andor the number ofcollocation points are large

Then we define the residual function for the 119896th IVP onthe subinterval 119868

119896in (16) as follows

Res (119905) = 120581(119868119873(119910119896))10158401015840

(119905) minus (119868119873(119910119896))1015840

(119905) + ((119868119873(119910119896))1015840

(119905))2

+ 119868119873(119910119896) (119905) (119868

119873(119910119896))1015840

(119905)

(24)

At step 119896 the algebraic equations for obtaining the coefficients119910119896119895come from equalizing Res(119905) to zero at ShLG points plus

two boundary conditions on the 119896th subinterval by utilizing(22)-(23)

Res (119905119896119894) = 0 119894 = 1 119873 minus 1

1199101198960= 119910119896minus1119873

119873

sum

119895=0

119863(1)

1119895119910119896119895=

119873

sum

119895=0

119863(1)

1119895119910119896minus1119895

(25)

By using (25) we obtain a set of119873+ 1 algebraic equations forunknowns 119910

1198960 119910

119896119873which can be solved using Newtonrsquos

iterative method Again we note that in (25) the valuesof 119910119896minus1119895

119895 = 0 119873 are obtained earlier at step 119896 minus 1Consequently at step 119896 using (25) the approximation of119910119896(119905) in the 119896th subinterval 119868

119896is obtained with substituting

the obtained values of 119910119896119895

into (22) which is indeed theapproximate solution of Volterrarsquos model on the subinterval[(119896 minus 1)ℎ 119896ℎ)

3 Numerical Results

We apply the method presented in this paper to examine themathematical structure of 119906(119905) In particular we seek to studythe rapid growth along the logistic curve that will reach apeak followed by the slow exponential decay where 119906(119905) rarr0 as 119905 rarr infin The mathematical behavior so defined wasintroduced by Scudo [19] and justified by Small [20] basedon singular perturbation methods Further these propertieswere also confirmed byTeBeest [21] upon using a phase-planeanalysis Wazwaz [23] by applying Adomian decompositionmethod (ADM) Ramezani et al [27] by using compositespectral functions (CSF) Marzban et al [28] by using hybridof block-pulse and Lagrange polynomials (HBL) and Parandet al [29] by using rational Chebyshev collocation (RCC) andHermite functions collocation methods (HFC)

We applied themethod presented in this paper and solved(3) for 119906

0= 01 and 120581 = 002 004 01 02 and 05 and then

evaluated 119906max which are also evaluated in [23 27ndash29] InTable 1 the resulting values using the present method with


Table 1 A comparison of methods in [23 27ndash29] and the present method with the exact values for 119906max

120581 ADM [23] CSF [27] RCC [29] HBL [28] ℎ 119873 Present Exact 119906max

002 09038380533 09234262 092342715 09234271721 01 28 092342717207027 092342717207022004 0861240177 08737192 087371998 08737199832 015 22 087371998315393 08737199831539901 07651130834 07697409 076974149 07697414907 025 19 076974149070061 07697414907006002 06579123080 06590497 065905038 06590503815 05 18 065905038155231 06590503815523105 04852823482 04851898 048519030 04851902914 10 20 048519029140942 048519029140942

1 2 3 4 5 6 7 8 9 10

1

09

08

07

06

05

04

03

02

01

u(t)

t

Figure 1The results of the present method calculation for 120581 = 002004 01 02 and 05 in the order of height

different step sizes together with the results given in [23 27ndash29] and exact values

119906max = 1 + 120581 ln(120581

1 + 120581 minus 1199060

) (26)

reported in [21] are presented Compared with other meth-ods our method provides more accurate numerical resultsNote that the step sizes considered in Table 1 are based onthe position of 119906max To this end for all values of 120581 we firstsolved the problem with the step size ℎ = 1 and 119873 = 15 tofind the approximate position of 119906max Then for each value of120581 we selected an appropriate step size

Figure 1 shows the results of the present step by stepcollocation method for 120581 = 002 004 01 02 and 05 Thisfigure shows the rapid rise along the logistic curve followedby the slow exponential decay after reaching the maximumpoint and when 120581 increases the amplitude of 119906(119905) decreaseswhereas the exponential decay increases Also this figureshows the stability of the present method in large number ofsteps calculations

4 Conclusion

A new efficient step by step collocation method based onshifted Legendre-Gauss points has been proposed for solvingVolterra model for population growth of a species in a closedsystem We considered a step size and converted the originalIVP raised from Volterrarsquos population model to a sequenceof IVPs in subintervals and solved them step by step usingcollocationThis approach is easy to implement and possessesthe spectral accuracy Furthermore this method is availablefor large domain calculations Numerical example shows

the excellent agreement between the approximate and exactvalues for 119906max

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to the referees for their valuablesuggestions and comments that improved the paper AdemKılıcman gratefully acknowledges that this research waspartially supported by the University Putra Malaysia underthe ERGS Grant Scheme having project number 5527068

References

[1] O Coulaud D Funaro and O Kavian ldquoLaguerre spectralapproximation of elliptic problems in exterior domainsrdquo Com-puterMethods inAppliedMechanics and Engineering vol 80 no1ndash3 pp 451ndash458 1990

[2] D Funaro ldquoComputational aspects of pseudospectral Laguerreapproximationsrdquo Applied Numerical Mathematics vol 6 no 6pp 447ndash457 1990

[3] J Shen ldquoStable and efficient spectral methods in unboundeddomains using Laguerre functionsrdquo SIAM Journal onNumericalAnalysis vol 38 no 4 pp 1113ndash1133 2000

[4] D Funaro and O Kavian ldquoApproximation of some diffusionevolution equations in unbounded domains by Hermite func-tionsrdquoMathematics of Computation vol 57 no 196 pp 597ndash6191991

[5] B-Y Guo ldquoError estimation of Hermite spectral methodfor nonlinear partial differential equationsrdquo Mathematics ofComputation vol 68 no 227 pp 1067ndash1078 1999

[6] C I Christov ldquoA complete orthonormal system of functions in1198712(infin minusinfin) spacerdquo SIAM Journal on Applied Mathematics vol

42 no 6 pp 1337ndash1344 1982[7] J P Boyd ldquoSpectral methods using rational basis functions on

an infinite intervalrdquo Journal of Computational Physics vol 69no 1 pp 112ndash142 1987

[8] J P Boyd ldquoOrthogonal rational functions on a semi-infiniteintervalrdquo Journal of Computational Physics vol 70 no 1 pp 63ndash88 1987

[9] B-Y Guo J Shen and Z-Q Wang ldquoA rational approximationand its applications to differential equations on the half linerdquoJournal of Scientific Computing vol 15 no 2 pp 117ndash147 2000

[10] J P Boyd C Rangan and P H Bucksbaum ldquoPseudospectralmethods on a semi-infinite interval with application to the


hydrogen atom a comparison of the mapped Fourier-sinemethod with Laguerre series and rational Chebyshev expan-sionsrdquo Journal of Computational Physics vol 188 no 1 pp 56ndash74 2003

[11] K Parand S Abbasbandy S Kazem and A R Rezaei ldquoCom-parison between two common collocation approaches based onradial basis functions for the case of heat transfer equationsarising in porous mediumrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 3 pp 1396ndash14072011

[12] C E Grosch and S A Orszag ldquoNumerical solution of problemsin unbounded regions coordinate transformsrdquo Journal of Com-putational Physics vol 25 no 3 pp 273ndash295 1977

[13] J P Boyd ldquoThe optimization of convergence for Chebyshevpolynomial methods in an unbounded domainrdquo Journal ofComputational Physics vol 45 no 1 pp 43ndash79 1982

[14] B-Y Guo ldquoJacobi spectral approximations to differential equa-tions on the half linerdquo Journal of Computational Mathematicsvol 18 no 1 pp 95ndash112 2000

[15] B-y Guo ldquoJacobi approximations in certain Hilbert spaces andtheir applications to singular differential equationsrdquo Journal ofMathematical Analysis and Applications vol 243 no 2 pp 373ndash408 2000

[16] M Maleki I Hashim and S Abbasbandy ldquoAnalysis of IVPsand BVPs on semi-infinite domains via collocation methodsrdquoJournal of Applied Mathematics vol 2012 Article ID 696574 21pages 2012

[17] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications New York NY USA 2nd edition 2000

[18] M Maleki and M Tavassoli Kajani ldquoA nonclassical collocationmethod for solving two-point boundary value problems overinfinite intervalsrdquo Australian Journal of Basic and AppliedSciences vol 5 no 9 pp 1045ndash1050 2011

[19] F M Scudo ldquoVito Volterra and theoretical ecologyrdquoTheoreticalPopulation Biology vol 2 pp 1ndash23 1971

[20] R D Small Population Growth in a Closed System and mathe-matical Modelling SIAM Philadelphia Pa USA 1989

[21] K G TeBeest ldquoNumerical and analytical solutions of Volterrarsquospopulation modelrdquo SIAM Review vol 39 no 3 pp 484ndash4931997

[22] K Al-Khaled ldquoNumerical approximations for populationgrowth modelsrdquo Applied Mathematics and Computation vol160 no 3 pp 865ndash873 2005

[23] A-M Wazwaz ldquoAnalytical approximations and Pade approx-imants for Volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999

[24] K Parand and M Razzaghi ldquoRational Chebyshev tau methodfor solving Volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 149 no 3 pp 893ndash900 2004

[25] K Parand and M Razzaghi ldquoRational Chebyshev tau methodfor solving higher-order ordinary differential equationsrdquo Inter-national Journal of Computer Mathematics vol 81 no 1 pp 73ndash80 2004

[26] K Parand and M Razzaghi ldquoRational legendre approximationfor solving some physical problems on semi-infinite intervalsrdquoPhysica Scripta vol 69 no 5 pp 353ndash357 2004

[27] M Ramezani M Razzaghi and M Dehghan ldquoCompositespectral functions for solving Volterrarsquos population modelrdquoChaos Solitons and Fractals vol 34 no 2 pp 588ndash593 2007

[28] H R Marzban S M Hoseini and M Razzaghi ldquoSolutionof Volterrarsquos population model via block-pulse functions and

Lagrange-interpolating polynomialsrdquoMathematical Methods inthe Applied Sciences vol 32 no 2 pp 127ndash134 2009

[29] K Parand A R Rezaei and A Taghavi ldquoNumerical approx-imations for population growth model by rational Chebyshevand Hermite functions collocation approach a comparisonrdquoMathematical Methods in the Applied Sciences vol 33 no 17 pp2076ndash2086 2010

[30] N A Khan A Ara and M Jamil ldquoApproximations of thenonlinear Volterrarsquos populationmodel by an efficient numericalmethodrdquoMathematical Methods in the Applied Sciences vol 34no 14 pp 1733ndash1738 2011

[31] M Abramowitz and I Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover Publications New York NY USA 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In [19 20] the Volterra model for population growth ofa species within a closed system is given by

119889119901

119889= 119886119901 minus 119887119901

2minus 119888119901int

0

119901 (119909) 119889119909 119901 (0) = 1199010 (1)

where 119886 gt 0 is the birth rate coefficient 119887 gt 0 is the crowdingcoefficient 119888 gt 0 is the toxicity coefficient 119901

0is the initial

population and 119901 = 119901() denotes the population at time Also the coefficient 119888 indicates the essential behavior of thepopulation evolution before its level falls to zero in the longterm

This model is an integroordinary differential equationwhere the term 119888119901 int

0119901(119909)119889119909 represents the effect of toxin

accumulation on the species Although several time scalesand population scales may be employed [20] here we willscale time and population by introducing the nondimen-sional variables

119905 =

119887119888 119906 =

119901

119886119887 (2)

which produce the nondimensional problem

120581119889119906

119889119905= 119906 minus 119906

2minus 119906int

119905

0

119906 (119909) 119889119909 119906 (0) = 1199060 (3)

where 119906(119905) is the scaled population of identical individualsat time 119905 and 120581 = 119888(119886119887) is a prescribed nondimensionalparameter One may show that the only equilibrium solutionof (3) is the trivial solution119906(119905) = 0 In addition the analyticalsolution [20]

119906 (119905) = 1199060exp(1

120581int

119905

0

[1 minus 119906 (120591) minus int

120591

0

119906 (119909) 119889119909] 119889120591) (4)

shows that 119906(119905) gt 0 for all 119905 if 1199060gt 0

During the recent years the solution of (3) has been ofconsiderable concern In [19] the successive approximationsmethod was suggested for the solution of (3) but was notimplemented In this case the solution 119906(119905) has a smalleramplitude compared with the amplitude of 119906(119905) for the case120581 ≪ 1 Similarly in [20] the singular perturbation methodfor solving Volterrarsquos population model is considered Theauthor scaled out the parameters of (3) as much as possibleby using four different ways and considered two cases 120581 =119888(119886119887) small and 120581 = 119888(119886119887) large Thus it is shown in [20]that for the case 120581 ≪ 1 where populations are weakly sensi-tive to toxins a rapid rise occurs along the logistic curve thatwill reach a peak and then is followed by a slow exponentialdecay In the case of large 120581 the populations are strongly sen-sitive to toxins and the solutions are proportional to sech2(119905)

In [21] four numerical methods namely the Eulermethod the modified Euler method the classical fourth-order Runge-Kutta method and the Runge-Kutta-Fehlbergmethod for the solution of (3) are proposed Moreover aphase-plane analysis is implemented In [22] a comparisonof the Adomian decomposition method and Sinc-Galerkinmethod is given and it is shown that the Adomiandecomposition method is more efficient for the solution

of Volterrarsquos population model In [23] the series solutionmethod and the decomposition method are implementedindependently to (3) and to a related nonlinear ordinarydifferential equation Furthermore the Pade approximationsare used in the analysis to capture the essential behavior of thepopulation 119906(119905) of identical individuals and approximationof 119906max and the exact value of 119906max for different 120581 werecompared The authors of [24ndash26] applied spectral methodto solve Volterrarsquos population on a semi-infinite intervalbased on a rational Tau method

In [27] the approach is based upon domain truncationand composite spectral functions approximations They firstconsidered an interval [0 119871] where 119871 is any positive integerand divided this interval into subintervals with step size ℎ =1119873 where 119873 is a positive integer They then transformedeach subinterval into [0 1) and utilized the properties ofcomposite spectral functions consisting of few terms oforthogonal functions to reduce the solution of Volterrarsquosmodel to the solution of a system of algebraic equations

In [28] a numerical method based on domain truncationand hybrid functions was proposed to solve Volterrarsquos popu-lation model They considered an interval [0 119905

119891) and then

by utilizing the properties of hybrid functions that consistof block-pulse and Lagrange-interpolating polynomials theyreduced the solution of Volterrarsquos model to the solution of asystem of algebraic equations

In [29] the authors compared the application of rationalChebyshev collocation and Hermite functions collocationmethods for solving Volterrarsquos population model In [30] anew homotopy perturbation method is proposed for directlysolving the Volterrarsquos population model as a nonlinear inte-grodifferential equation

In this paper we introduce a new collocation methodfor solving (3) Volterrarsquos population model in (3) is firstconverted to an equivalent nonlinear initial value problem(IVP) This method solves the problem step by step and isvalid for large domains We first consider a step size and thenreplace the original IVP in the interval [0infin)with a sequenceof IVPs in subintervals with length equal to the consideredstep sizeThen the sequence of IVPs is consecutively reducedto sets of algebraic equations using collocation based onshifted Legendre-Gauss (ShLG) pointsThe initial conditionsof the 119896th step (except for the first step where the initialconditions are available) are obtained from the approximatedsolution obtained earlier at the (119896 minus 1)th step

The paper is organized as follows In Section 2 some basicproperties of Legendre and shifted Legendre polynomialsrequired for our subsequent development are givenThen theapplication of this method to Volterrarsquos population model issummarized In Section 3 we report our numerical findingsand demonstrate the efficiency and accuracy of the proposedscheme

2 Step by Step Spectral Collocation Methodfor Volterrarsquos Population Model

In this section we derive the step by step ShLG spectral collo-cation method for solving Volterrarsquos population model in (3)



119899(119909) 119899 = 0 1 are the


((1 minus 1199092) 1198751015840

119899(119909))1015840

+ 119899 (119899 + 1) 119875119899(119909) = 0 (5)


int

1

minus1

119875119899(119909) 119875119898(119909) 119889119909 =

2

2119899 + 1120575119899119898 (6)

where 120575119899119898


119875119899+1(119909) =

2119899 + 1

119899 + 1119909119875119899(119909) minus

119899

119899 + 1119875119899minus1(119909) (7)

where 1198750(119909) = 1 and 119875

1(119909) = 119909 If 119875




119875119899(119909) =

1

2119899

[1198992]

sum

119898=0

(minus1)119898(119899

119898)(2119899 minus 2119898

119899)119909119899minus2119898 (8)


1lt





int

1

minus1

119901 (119909) 119889119909 =

119873minus1

sum

119894=1

119908119894119901 (119909119894) (9)




119899(119905) = 119875

119899(1




119873minus1lt 119887


119905119894=1

2((119887 minus 119886) 119909

119894+ 119886 + 119887) 119894 = 1 2 119873 minus 1 (11)


int

119887

119886

119901 (119905) 119889119905 =119887 minus 119886

2int

1

minus1

119901(1

2((119887 minus 119886) 119909 + 119886 + 119887)) 119889119909

=119887 minus 119886

2

119873minus1

sum

119894=1

119908119894119901(1

2((119887 minus 119886) 119909

119894+ 119886 + 119887))

=

119873minus1

sum

119894=1

119908119894119901 (119905119894)

(12)

where 119908119894= ((119887 minus 119886)2)119908

119894 1 ⩽ 119894 ⩽ 119873 minus 1 are



119910 (119905) = int

119905

0

119906 (119909) 119889119909 (13)

which leads to

1199101015840(119905) = 119906 (119905) 119910

10158401015840(119905) = 119906

1015840(119905) (14)


12058111991010158401015840(119905) = 119910

1015840(119905) minus (119910

1015840(119905))2

minus 119910 (119905) 1199101015840(119905) 0 le 119905 lt infin

119910 (0) = 0 1199101015840(0) = 119906

0

(15)





119896 119896 = 1 2

12058111991010158401015840

119896(119905) = 119910

1015840

119896(119905) minus (119910

1015840

119896(119905))2

minus 119910119896(119905) 1199101015840

119896(119905) 119905 isin 119868

119896

119910119896((119896 minus 1) ℎ) = 119910

119896minus1((119896 minus 1) ℎ)

1199101015840

119896((119896 minus 1) ℎ) = 119910

1015840

119896minus1((119896 minus 1) ℎ)

(16)


1= [0 ℎ]

12058111991010158401015840

1(119905) = 119910

1015840

1(119905) minus (119910

1015840

1(119905))2

minus 1199101(119905) 1199101015840

1(119905) 119905 isin 119868

1

1199101(0) = 0 119910

1015840

1(0) = 119906

0

(17)







119896 119896 = 1 2


119905119896119894=ℎ

2(119909119894+ 2119896 minus 1) 119894 = 1 2 119873 minus 1 (18)



119896(119905) isin



at most119873 as

119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0

119910119896119895119871119896119895(119905) 119905 isin 119868

119896 (19)

where 119910119896119895= 119910119896(119905119896119895) and

119871119896119895(119905) =

119873

prod

119897=0119897 = 119895

119905 minus 119905119896119897

119905119896119895minus 119905119896119897

119895 = 0 1 119873 (20)





119871119896119895(119905) = 119871


119896 (21)



119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0


119896

(22)


1198960= (119896 minus 1)ℎ and 119905



119910(119898)

119896(119905119896119894) asymp

119873

sum

119895=0

119910119896119895119871(119898)

1119895(119905119896119894minus (119896 minus 1) ℎ)

=

119873

sum

119895=0

119910119896119895119871(119898)

1119895(1199051119894) =

119873

sum

119895=0

119863(119898)

119894119895119910119896119895 119898 = 1 2

(23)

where 119863(119898)119894119895= 119871(119898)







10= 0

and 1199051119873= ℎ






Res (119905) = 120581(119868119873(119910119896))10158401015840

(119905) minus (119868119873(119910119896))1015840

(119905) + ((119868119873(119910119896))1015840

(119905))2

+ 119868119873(119910119896) (119905) (119868

119873(119910119896))1015840

(119905)

(24)



Res (119905119896119894) = 0 119894 = 1 119873 minus 1

1199101198960= 119910119896minus1119873

119873

sum

119895=0

119863(1)

1119895119910119896119895=

119873

sum

119895=0

119863(1)

1119895119910119896minus1119895

(25)


1198960 119910







3 Numerical Results



0= 01 and 120581 = 002 004 01 02 and 05 and then





002 09038380533 09234262 092342715 09234271721 01 28 092342717207027 092342717207022004 0861240177 08737192 087371998 08737199832 015 22 087371998315393 08737199831539901 07651130834 07697409 076974149 07697414907 025 19 076974149070061 07697414907006002 06579123080 06590497 065905038 06590503815 05 18 065905038155231 06590503815523105 04852823482 04851898 048519030 04851902914 10 20 048519029140942 048519029140942

1 2 3 4 5 6 7 8 9 10

1

09

08

07

06

05

04

03

02

01

u(t)

t



119906max = 1 + 120581 ln(120581

1 + 120581 minus 1199060

) (26)



4 Conclusion





Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899(119909) 119899 = 0 1 are the


((1 minus 1199092) 1198751015840

119899(119909))1015840

+ 119899 (119899 + 1) 119875119899(119909) = 0 (5)


int

1

minus1

119875119899(119909) 119875119898(119909) 119889119909 =

2

2119899 + 1120575119899119898 (6)

where 120575119899119898


119875119899+1(119909) =

2119899 + 1

119899 + 1119909119875119899(119909) minus

119899

119899 + 1119875119899minus1(119909) (7)

where 1198750(119909) = 1 and 119875

1(119909) = 119909 If 119875




119875119899(119909) =

1

2119899

[1198992]

sum

119898=0

(minus1)119898(119899

119898)(2119899 minus 2119898

119899)119909119899minus2119898 (8)


1lt





int

1

minus1

119901 (119909) 119889119909 =

119873minus1

sum

119894=1

119908119894119901 (119909119894) (9)




119899(119905) = 119875

119899(1




119873minus1lt 119887


119905119894=1

2((119887 minus 119886) 119909

119894+ 119886 + 119887) 119894 = 1 2 119873 minus 1 (11)


int

119887

119886

119901 (119905) 119889119905 =119887 minus 119886

2int

1

minus1

119901(1

2((119887 minus 119886) 119909 + 119886 + 119887)) 119889119909

=119887 minus 119886

2

119873minus1

sum

119894=1

119908119894119901(1

2((119887 minus 119886) 119909

119894+ 119886 + 119887))

=

119873minus1

sum

119894=1

119908119894119901 (119905119894)

(12)

where 119908119894= ((119887 minus 119886)2)119908

119894 1 ⩽ 119894 ⩽ 119873 minus 1 are



119910 (119905) = int

119905

0

119906 (119909) 119889119909 (13)

which leads to

1199101015840(119905) = 119906 (119905) 119910

10158401015840(119905) = 119906

1015840(119905) (14)


12058111991010158401015840(119905) = 119910

1015840(119905) minus (119910

1015840(119905))2

minus 119910 (119905) 1199101015840(119905) 0 le 119905 lt infin

119910 (0) = 0 1199101015840(0) = 119906

0

(15)





119896 119896 = 1 2

12058111991010158401015840

119896(119905) = 119910

1015840

119896(119905) minus (119910

1015840

119896(119905))2

minus 119910119896(119905) 1199101015840

119896(119905) 119905 isin 119868

119896

119910119896((119896 minus 1) ℎ) = 119910

119896minus1((119896 minus 1) ℎ)

1199101015840

119896((119896 minus 1) ℎ) = 119910

1015840

119896minus1((119896 minus 1) ℎ)

(16)


1= [0 ℎ]

12058111991010158401015840

1(119905) = 119910

1015840

1(119905) minus (119910

1015840

1(119905))2

minus 1199101(119905) 1199101015840

1(119905) 119905 isin 119868

1

1199101(0) = 0 119910

1015840

1(0) = 119906

0

(17)







119896 119896 = 1 2


119905119896119894=ℎ

2(119909119894+ 2119896 minus 1) 119894 = 1 2 119873 minus 1 (18)



119896(119905) isin



at most119873 as

119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0

119910119896119895119871119896119895(119905) 119905 isin 119868

119896 (19)

where 119910119896119895= 119910119896(119905119896119895) and

119871119896119895(119905) =

119873

prod

119897=0119897 = 119895

119905 minus 119905119896119897

119905119896119895minus 119905119896119897

119895 = 0 1 119873 (20)





119871119896119895(119905) = 119871


119896 (21)



119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0


119896

(22)


1198960= (119896 minus 1)ℎ and 119905



119910(119898)

119896(119905119896119894) asymp

119873

sum

119895=0

119910119896119895119871(119898)

1119895(119905119896119894minus (119896 minus 1) ℎ)

=

119873

sum

119895=0

119910119896119895119871(119898)

1119895(1199051119894) =

119873

sum

119895=0

119863(119898)

119894119895119910119896119895 119898 = 1 2

(23)

where 119863(119898)119894119895= 119871(119898)







10= 0

and 1199051119873= ℎ






Res (119905) = 120581(119868119873(119910119896))10158401015840

(119905) minus (119868119873(119910119896))1015840

(119905) + ((119868119873(119910119896))1015840

(119905))2

+ 119868119873(119910119896) (119905) (119868

119873(119910119896))1015840

(119905)

(24)



Res (119905119896119894) = 0 119894 = 1 119873 minus 1

1199101198960= 119910119896minus1119873

119873

sum

119895=0

119863(1)

1119895119910119896119895=

119873

sum

119895=0

119863(1)

1119895119910119896minus1119895

(25)


1198960 119910







3 Numerical Results



0= 01 and 120581 = 002 004 01 02 and 05 and then





002 09038380533 09234262 092342715 09234271721 01 28 092342717207027 092342717207022004 0861240177 08737192 087371998 08737199832 015 22 087371998315393 08737199831539901 07651130834 07697409 076974149 07697414907 025 19 076974149070061 07697414907006002 06579123080 06590497 065905038 06590503815 05 18 065905038155231 06590503815523105 04852823482 04851898 048519030 04851902914 10 20 048519029140942 048519029140942

1 2 3 4 5 6 7 8 9 10

1

09

08

07

06

05

04

03

02

01

u(t)

t



119906max = 1 + 120581 ln(120581

1 + 120581 minus 1199060

) (26)



4 Conclusion





Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119896 119896 = 1 2


119905119896119894=ℎ

2(119909119894+ 2119896 minus 1) 119894 = 1 2 119873 minus 1 (18)



119896(119905) isin



at most119873 as

119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0

119910119896119895119871119896119895(119905) 119905 isin 119868

119896 (19)

where 119910119896119895= 119910119896(119905119896119895) and

119871119896119895(119905) =

119873

prod

119897=0119897 = 119895

119905 minus 119905119896119897

119905119896119895minus 119905119896119897

119895 = 0 1 119873 (20)





119871119896119895(119905) = 119871


119896 (21)



119910119896(119905) asymp 119868

119873(119910119896) (119905) =

119873

sum

119895=0


119896

(22)


1198960= (119896 minus 1)ℎ and 119905



119910(119898)

119896(119905119896119894) asymp

119873

sum

119895=0

119910119896119895119871(119898)

1119895(119905119896119894minus (119896 minus 1) ℎ)

=

119873

sum

119895=0

119910119896119895119871(119898)

1119895(1199051119894) =

119873

sum

119895=0

119863(119898)

119894119895119910119896119895 119898 = 1 2

(23)

where 119863(119898)119894119895= 119871(119898)







10= 0

and 1199051119873= ℎ






Res (119905) = 120581(119868119873(119910119896))10158401015840

(119905) minus (119868119873(119910119896))1015840

(119905) + ((119868119873(119910119896))1015840

(119905))2

+ 119868119873(119910119896) (119905) (119868

119873(119910119896))1015840

(119905)

(24)



Res (119905119896119894) = 0 119894 = 1 119873 minus 1

1199101198960= 119910119896minus1119873

119873

sum

119895=0

119863(1)

1119895119910119896119895=

119873

sum

119895=0

119863(1)

1119895119910119896minus1119895

(25)


1198960 119910







3 Numerical Results



0= 01 and 120581 = 002 004 01 02 and 05 and then





002 09038380533 09234262 092342715 09234271721 01 28 092342717207027 092342717207022004 0861240177 08737192 087371998 08737199832 015 22 087371998315393 08737199831539901 07651130834 07697409 076974149 07697414907 025 19 076974149070061 07697414907006002 06579123080 06590497 065905038 06590503815 05 18 065905038155231 06590503815523105 04852823482 04851898 048519030 04851902914 10 20 048519029140942 048519029140942

1 2 3 4 5 6 7 8 9 10

1

09

08

07

06

05

04

03

02

01

u(t)

t



119906max = 1 + 120581 ln(120581

1 + 120581 minus 1199060

) (26)



4 Conclusion





Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












002 09038380533 09234262 092342715 09234271721 01 28 092342717207027 092342717207022004 0861240177 08737192 087371998 08737199832 015 22 087371998315393 08737199831539901 07651130834 07697409 076974149 07697414907 025 19 076974149070061 07697414907006002 06579123080 06590497 065905038 06590503815 05 18 065905038155231 06590503815523105 04852823482 04851898 048519030 04851902914 10 20 048519029140942 048519029140942

1 2 3 4 5 6 7 8 9 10

1

09

08

07

06

05

04

03

02

01

u(t)

t



119906max = 1 + 120581 ln(120581

1 + 120581 minus 1199060

) (26)



4 Conclusion





Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra's Population Growth Model

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Transcript of A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra's Population Growth Model