numerical solutions of integro-differential equations and application of a population model with an...

Applied Mathematical Modelling 37 (2013) 2086–2101

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Numerical solutions of integro-differential equations and applicationof a population model with an improved Legendre method

S�uayip Yüzbas�ı ⇑, Mehmet Sezer, Bayram KemancıDepartment of Mathematics, Faculty of Science, Mugla University, Mugla, Turkey

a r t i c l e i n f o

Article history:Received 7 December 2011Received in revised form 1 May 2012Accepted 8 May 2012Available online 17 May 2012

Keywords:Population modelIntegro-differential equationsImproved Legendre collocation methodLegendre polynomialsNumerical solutions

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.05.012

⇑ Corresponding author. Tel.: +90 252 211 15 81;E-mail addresses: [email protected] (S�. Yüzbas�ı)

a b s t r a c t

In this paper, an improved Legendre collocation method is presented for a class of integro-differential equations which involves a population model. This improvement is made byusing the residual function of the operator equation. The error differential equation, gainedby residual function, has been solved by the Legendre collocation method (LCM). By sum-ming the approximate solution of the error differential equation with the approximatesolution of the problem, a better approximate solution is obtained. We give the illustrativeexamples to demonstrate the efficiency of the method. Also we compare our results withthe results of the known some methods. In addition, an application of the population modelis made.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Integro-differential equations are encountered as model in many fields of science and engineering such as populationgrowth, one dimensional viscoelasticity and reactor dynamics [1–4]. In this study, we consider a class of integro-differentialequations included some problems such as a stable population model in continuous time [1,2]. In addition, we apply thepresent method for a stable population model in continuous time. In this model, we will only include females. The men-tioned model [1,2] is given by

BðtÞ ¼ gðtÞ þZ t

0Kðt; xÞBðxÞdx ð1Þ

where

K(t,x) = K(t � x): net maternity function of females class age x at time t.g(t): contribution of birth due to female already present at time t.B(t): the number of female births.

In recent years, several authors for the integral and integro-differential equations have worked semi-analytical methodssuch as the Taylor-series expansion method [5], the Taylor collocation method [6], the homotopy perturbation method [7,8],the Haar functions method, [9,10], the He’s variational iteration technique [11], the differential transformation method [12],the Legendre-spectral method [13], the Tau method [14], the Legendre multi wavelets method [15], the finite-difference

. All rights reserved.

fax: +90 252 211 14 72., [email protected] (M. Sezer), [email protected] (B. Kemancı).

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S�. Yüzbas�ı et al. / Applied Mathematical Modelling 37 (2013) 2086–2101 2087

scheme [16], the variational iteration method [17], the trigonometric wavelets method [18], the Legendre matrix method[19,20], the Adomian method [21], the Galerkin method [22], the modified homotopy perturbation method [23] and themoving least square method [24,25].

Orthogonal polynomials are widely used in applications in mathematics, mathematical physics, engineering, and com-puter science [26,27]. One of the most common set of orthogonal polynomials is the set of the Legendre polynomials{P0(x),P1(x), . . . ,Pn(x)} which are orthogonal on [�1,1] with respect to the weight function w(x) = 1 [28–30]. The Legendrepolynomials Pn(x) satisfy the Legendre differential equation [31]

ð1� x2Þd2y

dx2 � 2xdydxþ nðnþ 1Þy ¼ 0; �1 6 x 6 1; n P 0

where

PnðxÞ ¼12n

Xn2½ �

k¼0

n

k

� �2n� 2k

n

� �xn�2k; n 2 N;

n2

h i¼

n2 ; if n is evenn�1

2 ; if n is odd

( ): ð2Þ

Legendre polynomials are examples of Eigen functions of singular Strum–Liouville problems and have been used extensivelyin the solution of the boundary value problems and in computational fluid dynamics [32–34]. The Legendre wavelets methodbased on the Legendre polynomials has been used for differential-difference equations [35,36]. On the other hand, the meth-ods based on Legendre polynomials may be more suitable for solving differential, linear and nonlinear integro-differentialequations, and integro-differential-difference equations [28,29,37,38,5].

Since the beginning of the 1994s, Taylor and Chebyshev methods to solve linear differential, integral, integro-differential,difference, integro-difference and systems of integro-differential equations have been used by Sezer et al. [6,39–44].

Also, Yüzbas�ı et al. [45–48] have studied the Bessel matrix and collocation methods for Fredholm integro-differentialequations, Volterra integro-differential equations, system of Fredholm integro-differential equations and system of Volterraintegral equations. Moreover, Yüzbas�ı et al. [49,50] have developed the Bessel collocation approach for continuous popula-tion models for single and interacting species, and the pollution model of a system of lakes.

In this paper, as a more general structure of the model (1), we consider the m-th order linear integro-differential equationwith variable coefficients in form

L½yðxÞ� ¼Xm

k¼0

FkðxÞyðkÞðxÞ þ k1

Z 1

�1Kf ðx; tÞyðtÞdt þ k2

Z x

�1Kvðx; tÞyðtÞdt ¼ gðxÞ; �1 6 x; t 6 1 ð3Þ

under the mixed conditions

Xm�1

k¼0

ajkyðkÞð�1Þ þ bjkyðkÞð1Þ þ cjkyðkÞð0Þ� �

¼ lj; j ¼ 0;1; . . . ;m� 1 ð4Þ

where ajk, bjk, cjk, k1, k2 and lj are suitable constants, y(0)(x) = y(x) is a unknown function, Fk(x), g(x), Kf(x, t) and Kv(x, t) are thefunctions defined on interval �1 6 x, t 6 1 and also, Kf(x, t) and Kv(x, t) can be represented by Maclaurin series.

In this study, by improving the Legendre collocation method [43] with the aid of the residual error function used in [51–54], we will obtain an improved approximate solution of (3) with the conditions (4) in the form

yN;MðxÞ ¼ yNðxÞ þ eN;MðxÞ ð5Þ

where

yNðxÞ ¼XN

n¼0

anPnðxÞ ð6Þ

is the Legendre polynomial solution and the approximation eN,M(x) to error function eN(x),

eN;MðxÞ ¼XM

n¼0

a�nPnðxÞ;

is the Legendre polynomial solution of the error problem obtained by aid of the residual error function. Here,an; a�n; n ¼ 0;1;2; . . . ;N, are the unknown Legendre coefficients; N and M are chosen any positive integers such thatM P N P m; and Pn(x), n = 0,1,2, . . . ,N are the Legendre polynomials defined by Eq. (2).

This paper is organized as follows:The required matrix relations for solution are given in Section 2. In Section 3, the Legendre collocation method is

presented for Fredholm–Volterra integro-differential equations. In Section 4, we improve the Legendre polynomial solution,given in Section 3, by using the residual error function. Also, we give an error estimation in Section 4. In Section 5, we illus-trate some numerical examples to clarify the method. We apply the present method for model (1) in Section 6. Section 7concludes this article with a brief summary.

2088 S�. Yüzbas�ı et al. / Applied Mathematical Modelling 37 (2013) 2086–2101

2. Basic matrix relations

Let us write Eq. (3) in the form

DðxÞ þ k1If ðxÞ þ k2IvðxÞ ¼ gðxÞ ð7Þ

where the differential part

DðxÞ ¼Xm

k¼0

FkðxÞyðkÞðxÞ;

the Fredholm integral part

If ðxÞ ¼Z 1

�1Kf ðx; tÞyðtÞdt

and the Volterra integral part

IvðxÞ ¼Z x

�1Kvðx; tÞyðtÞdt:

We convert the approximate solution y(x) and its kth order derivative y(k)(x), the parts D(x) and If(x), and the mixed condi-tions in (4) to the matrix forms.

2.1. Matrix Relations for y(x) and y(k)(x)

Let us assume that the function y(x) can be expanded to the truncated Taylor series

yðxÞ ffiXN

n¼0

ynxn; yn ¼yðnÞð0Þ

n!ð8Þ

and the truncated Legendre series

yðxÞ ffiXN

n¼0

anPnðxÞ; �1 6 x 6 1: ð9Þ

Then the solution expressed by (8) and (9) and their derivatives can be written in the matrix forms, respectively,

½yðxÞ� ¼ XðxÞY; ½yðkÞðxÞ� ¼ XðkÞðxÞY ð10Þ

and

½yðxÞ� ¼ PðxÞA; ½yðkÞðxÞ� ¼ PðkÞðxÞA ð11Þ

where

XðxÞ ¼ ½1 x x2 . . . xN �; Y ¼ ½ y0 y1 . . . yN �T

PðxÞ ¼ ½ P0ðxÞ P1ðxÞ . . . PNðxÞ � and A ¼ ½ a0 a1 . . . aN �T :

On the other hand, by using the Legendre recursive formula (2) and taking n = 0,1,2, . . . ,N, we can obtain the matrix equation

PTðxÞ ¼ DXTðxÞ or PðxÞ ¼ XðxÞDT ð12Þ

where, for odd values of N

D ¼

ð�1Þ0

20

00

� �00

� �0 . . . 0

0ð�1Þ0

21

10

� �21

� �. . . 0

..

. ... ..

. ...

ð�1ÞN�1

2

2N�1

N � 1N�1

2

!0 . . . 0

0ð�1Þ

N�12

2N

NN�1

2

!N þ 1

N

� �. . .

ð�1Þ0

2N

N

0

� �2N

N

� �

26666666666666666664

37777777777777777775


and for even values of N

D ¼

ð�1Þ0

20

00

� �00

� �0 . . . 0

0ð�1Þ0

21

10

� �21

� �. . . 0

..

. ... ..

. ...

0ð�1Þ

N�22

2N�1

N � 1N�2

2

!N

N � 1

� �. . . 0

ð�1ÞN=2

2N

NN2

!N

N

� �0 . . .

ð�1Þ0

2N

N

0

� �2N

N

� �

26666666666666666664

37777777777777777775

:

Also, the relation between the matrix X(x) and its derivative X(1)(x) is

Xð1ÞðxÞ ¼ XðxÞBT ð13Þ

where

B ¼

0 0 . . . 0 01 0 . . . 0 00 2 . . . 0 0... ..

. ... ..

. ...

0 0 . . . N 0

26666664

37777775:

From the matrix Eq. (13), we can write the recurrance relation

XðkÞðxÞ ¼ XðxÞðBTÞk: ð14Þ

By using the relations (12) and (14), we have the recurrence relations

PðkÞðxÞ ¼ XðkÞðxÞDT ¼ XðxÞðBTÞkDT ; k ¼ 0;1; . . . ;m: ð15Þ

Consequently, by substituting the matrix relations (15) into Eq. (11), we obtain the matrix relations for y(x) and y(k) as

½yðkÞðxÞ� ¼ XðxÞðBTÞkDT A; k ¼ 0;1;2; . . . ;m: ð16Þ

Note that, from Eqs. (10)–(12), we have the relation

DT A ¼ Y or A ¼ ðDTÞ�1Y: ð17Þ

2.2. Matrix representations based on collocation points

To obtain an approximate solution in the form (6) of the problem (3) and (4), we can use a matrix method based on thecollocation points defined by

xi ¼ �1þ 2N

i; i ¼ 0;1; . . . ;N ð18Þ

which is a Legendre collocation method. Now, let us substitute the colocation points (18) into Eq. (7) and thus, we obtain thesystem

DðxiÞ ¼ gðxiÞ þ k1If ðxiÞ þ k2IvðxiÞ; i ¼ 0;1; . . . ;N

or the matrix equation

D ¼ Gþ k1If þ k2Iv ð19Þ

where

D ¼

Dðx0ÞDðx1Þ

..

.

DðxNÞ

0BBBB@1CCCCA; G ¼

gðx0Þgðx1Þ

..

.

gðxNÞ

0BBBB@1CCCCA; If ¼

If ðx0ÞIf ðx1Þ

..

.

If ðxNÞ

0BBBB@1CCCCA and Iv ¼

Ivðx0ÞIvðx1Þ

..

.

IvðxNÞ

0BBBB@1CCCCA:


2.3. Matrix relation for the differential part D(x)

To reduce the part D(x) to the matrix form by means of the collocation points (18), we first write the matrix D defined inEq. (19) as

D ¼Xm

k¼0

FkYðkÞ ð20Þ

where

Fk ¼

Fkðx0Þ 0 . . . 00 Fkðx1Þ . . . 0

. . . . . . . . . . . .

0 0 FkðxNÞ

2666437775; YðkÞ ¼

yðkÞðx0ÞyðkÞðx1Þ

..

.

yðkÞðxNÞ

266664377775 and G ¼

gðx0Þgðx1Þ

..

.

gðxNÞ

266664377775:

By putting the collocation points xi, (i = 1,2, . . . ,N) into the relation (16) we have the system of matrix equations as

½yðkÞðxiÞ� ¼ XðxiÞðBTÞkDT A; k ¼ 0;1;2; . . . ;m

or briefly

YðkÞ ¼

yðkÞðx0ÞyðkÞðx1Þ

..

.

yðkÞðxNÞ

266664377775 ¼

Xðx0ÞXðx1Þ

..

.

XðxNÞ

266664377775½ðBTÞkDT A� ¼ XðBTÞkDT A ð21Þ

where

X ¼

Xðx0ÞXðx1Þ

..

.

XðxNÞ

266664377775 ¼

1 x0 . . . xN0

1 x1 . . . xN1

..

. ... ..

. ...

1 xN . . . xNN

266664377775:

Consequently, from the matrix forms (20) and (21), we obtain the fundamental matrix relation for the differential part D(x)

D ¼Xm

k¼0

FkXðBTÞkDT A: ð22Þ

2.4. Matrix relations for the Fredholm integral part If(x)

Let us now form the matrix relation for Fredholm integral part If (x) in Eq. (7). The kernel function Kf(x, t) can be approx-imated by the truncated Legendre series

Kf ðx; tÞ ¼XN

m¼0

XN

n¼0

kL;fmnPmðxÞPnðtÞ ð23Þ

and the truncated Maclaurin series

Kf ðx; tÞ ¼XN

m¼0

XN

n¼0

kT;fmnxmtn ð24Þ

where

kT;fmn ¼

1m!n!

@mþnKf ð0;0Þ@xm@tn ; m;n ¼ 0;1; . . . ;N:

Let us first convert the expressions (23) and (24) to matrix forms and then equalize;

½Kf ðx; tÞ� ¼ PðxÞKLf PTðtÞ ¼ XðxÞKT

f XTðtÞ ð25Þ

where

PðxÞ ¼ ½ P0ðxÞ P1ðxÞ . . . PNðxÞ �; XðxÞ ¼ ½1 x . . . xN �; KLf ¼ kL;f

mn

h i; KT

f ¼ kT;fmn

h i; m;n ¼ 0;1; . . . ;N:


By using the relations (12) and (25), we get the relation, in the similar way to (17),

KTf ¼ DT KL

f D or KLf ¼ ðD

�1ÞT KTf D�1: ð26Þ

Substituting the matrix forms (25) and (11) corresponding to the functions Kf(x,t) and y(t) into the Fredholm integral partIf(x), we have the matrix relation

½If ðxÞ� ¼ PðxÞKLf QA ¼ XðxÞKT

f D�1QA ð27Þ

where

Q ¼R 1�1 PTðtÞPðtÞdt ¼ ½qmn�; m;n ¼ 0;1; . . . ;N;

qmn ¼2

2mþ1 ; m ¼ n;

0; m – n;

(
and the matrix KL
f is defined in (24).By putting the collocation points xi, (i = 0,1, . . . ,N) defined in (18), into the relation (27) we obtain the system of matrix

equations,

If ðxiÞ ¼ XðxiÞKTf D�1QA; i ¼ 0;1; . . . ;N

or briefly the matrix equation is

If ¼ XKTf D�1QA: ð28Þ

which is the fundamental matrix relation for the Fredholm integral part If(x).

2.5. Matrix relations for the Volterra integral part Iv(x)

Following the given way for the Fredholm part If(x), we have the matrix relation

½IvðxÞ� ¼ XðxÞKTvHðxÞDT A ð29Þ

where

HðxÞ ¼ ½hnmðxÞ� ¼R x�1 XTðtÞXðtÞdt; hnmðxÞ ¼ xmþnþ1�ð�1Þmþnþ1

mþnþ1 ; m;n ¼ 0;1; . . . ;N; KTv ¼ kT;v

mn

h i; kT;v

mn ¼ 1m!n!

@mþnKv ð0;0Þ@xm@tn :

For the collocation points xi, (i = 0,1, . . . ,N), the matrix relation (29) becomes the system of matrices

½IvðxiÞ� ¼ XðxiÞKTvHðxiÞDT A; i ¼ 0;1; . . . ;N

or briefly the matrix equation is

Iv ¼ XKHeDA ð30Þ

where

X ¼

Xðx0Þ 0 . . . 00 Xðx1Þ . . . 0

..

. ... ..

. ...

0 0 . . . XðxNÞ

266664377775; K ¼

KTv 0 . . . 0

0 KTv . . . 0

..

. ... ..

. ...

0 0 . . . KTv

2666664

3777775; eD ¼DT

DT

..

.

DT

266664377775; H ¼

Hðx0Þ 0 . . . 00 Hðx1Þ . . . 0

..

. ... ..

. ...

0 0 . . . HðxNÞ

266664377775:

The matrix relation (30) is the fundamental matrix relation for the Volterra integral part Iv(x).

2.6. Matrix relations for the mixed conditions

We obtain the corresponding matrix form for the conditions (4), by means of the relation (16), as
Xm�1
k¼0

½ajhXð�1Þ þ bjkXð1Þ þ cjkXð0Þ�ðBTÞkDT A ¼ lj; j ¼ 0;1; :::;m� 1: ð31Þ

3. Legendre collocation method

We now ready to construct the fundamental matrix equation correponding to Eq. (3). For this purpose, substituting thematrix relations (22), (28) and (30) into Eq. (19) and simplifying, we gain the fundamental matrix equation


Xm

k¼0

FkXðBTÞkDT þ k1XKTf D�1Q þ k2XKHeD( )

A ¼ G ð32Þ

which corresponds to a system of (N + 1) algebraic equations with the (N + 1) unknown Legendre coefficients a0,a1, . . . ,aN.Briefly, we can write Eq. (32) in the form

WA ¼ G or ½W; G� ð33Þ

where

W ¼ ½wpq� ¼Xm

k¼0

FkXðBTÞkDT þ k1XKTf D�1Q þ k2XKHeD; p; q ¼ 0;1; . . . ;N

and G ¼ ½ gðx0Þ gðx1Þ . . . gðxNÞ �T . On the other hand, the matrix form (31) for the conditions (4) can be written as

UiA ¼ li or ½Ui;li� ð34Þ

where

Ui ¼Xm�1

k¼0

½ajkXð�1Þ þ bjkXð1Þ þ cjkXð0Þ�ðBTÞkDT A ¼ ½ui0 ui1 . . . uiN �; j ¼ 0;1; . . . ;m� 1:

To obtain the solution of Eq. (3) under conditions (4), by replacing the rows matrices (34) by the last m rows of the matrix(33), we have the new augmented matrix

½fW; eG� ¼

w0 0 w0 1 . . . w0 N ; gðx0Þw1 0 w1 1 . . . w1 N ; gðx1Þ

..

. ... . .

. ... ..

. ...

wN�m 0 wN�m 1 . . . wN�m N ; gðxN�mÞu0 0 u0 1 . . . u0 N ; l0

u1 0 u1 1 . . . u1 N ; l1

..

. ... . .

. ... ..

. ...

um�1 1 um�1 2 . . . um�1 N...

lm�1

266666666666666664

377777777777777775: ð35Þ

However, we do not have to replace the last rows. For example, if the matrix W is singular, then the rows that have the samefactor or all zeros are replaced.

If rank fW ¼ rank ½fW; eG� ¼ N þ 1, then we can write

A ¼ fW�1eG:
Thus the coefficients an, (n = 0,1, . . . ,N) are uniquely determined by Eq. (35).
As a result, by substituting the determined coefficients into Eq. (6), we get the Legendre polynomial solution

yNðxÞ ¼XN

k¼0

anPnðxÞ: ð36Þ

On the other hand, when jfWj ¼ 0, if rank fW ¼ rank ½fW; eG� < N þ 1, then we may find a particular solution. Otherwise ifrank fW – rank ½fW; eG� < N þ 1, then there is not a solution.

The computational errors may be big for large values of N in the process due to rounding errors in computing. Therefore,the results may not be accurate enough for large values of N such as (N� 20). By assuming that the functions Fk(x), g(x),Kf(x, t) and Kv(x, t) in Eq. (1) have not any flops, we compute the total number of flops required to find the approximate solu-tion of problem (1) and (2) by applying the present method with truncated limited N as

39þ 1493

N4 þ 3913

N2 þ 72

m2 þ N3m2 þ 5N3mþ 8N2m2 þ 20N2mþ 212

Nm2 þ 452

Nmþ 3193

N þ 152

mþ 2843

N3 þ 2N6

þ 16N5 þ 4XNþ1

p¼1

p!

where m denotes order of the derivative in Eq. (1). Hence, to estimate complexity of the method: express number of flops as a(polynomial) function of the problem dimensions, and simplify by keeping only the leading terms.


Also, by means of the system (33) we may obtain some particular solutions. If k1 = k2 = 0 in Eq. (3), the equation becomesthe higher-order linear differential equation and also, if Fk(x) = 0 for k – 0, the equation becomes the Volterra–Fredholm inte-gral equation.

4. Residual correction and error estimation

In this section, an error estimation will be given for the Legendre polynomial solution (36) with the residual error function[51–54]. In addition, we improve the Legendre polynomial solution (36) with the aid of the residual error function. Firstly,we consider the residual function of the Legendre polynomial approximation as

RNðxÞ ¼ L½yNðxÞ� � gðxÞ: ð37Þ

Here, yN(x) is the Legendre polynomial solution given by (36) of the problem (3) and (4). Thus, yN(x) satisfies the problem

L½yNðxÞ� ¼Xm

k¼0

FkðxÞyðkÞN ðxÞ þ k1R 1�1 Kf ðx; tÞyNðtÞdt þ k2

R x�1 Kvðx; tÞyNðtÞdt ¼ gðxÞ þ RNðxÞ; �1 6 x; t 6 1;

Xm�1

k¼0

ajkyðkÞN ð�1Þ þ bjkyðkÞN ð1Þ þ cjkyðkÞN ð0Þ� �

¼ lj; j ¼ 0;1; . . . ;m� 1:

8>>>><>>>>:
Now, let us define the error function by
eNðxÞ ¼ yðxÞ � yNðxÞ ð38Þ

such that y(x) is the exact solution of the problem (3) and (4).By using Eqs. (3), (4), (37) and (38), we have the error differential equation

L½eNðxÞ� ¼ L½yðxÞ� � L½yNðxÞ� ¼ �RNðxÞ:

Since the approximate solution (36) provides the conditions (4), we can write
Xm�1
k¼0

ajkyðkÞN ð�1Þ þ bjkyðkÞN ð1Þ þ cjkyðkÞN ð0Þ� �

¼ lj; j ¼ 0;1; . . . ;m� 1: ð39Þ

Hence, the conditions (4) and (39) are reduced to the homogenous conditions
Xm�1
k¼0

ajkeðkÞN ð�1Þ þ bjkeðkÞN ð1Þ þ cjkeðkÞN ð0Þ� �

¼ 0; j ¼ 0;1; . . . ;m� 1:

Briefly, the error problem is
Xm
k¼0

FkðxÞeðkÞN ðxÞ þ k1R 1�1 Kf ðx; tÞeNðtÞdt þ k2

R x�1 Kvðx; tÞeNðtÞdt ¼ �RNðxÞ;

Xm�1

k¼0

ajkeðkÞN ð�1Þ þ bjkeðkÞN ð1Þ þ cjkeðkÞN ð0Þ� �

¼ 0; j ¼ 0;1; . . . ;m� 1:

8>>>><>>>>: ð40Þ

By solving the error problem (40) with the presented method in Section 3, we get the approximation eN,M(x) to eN(x) asfollows

eN;MðxÞ ¼XM

n¼0

a�nLnðxÞ; ðM P NÞ:

Consequently, by means of the polynomials yN(x) and eN,M (x), (M P N), we obtain the corrected Legendre polynomial solu-tion yN,M(x) = yN(x) + eN,M(x). Here, eN(x) = y(x) � yN (x), EN,M(x) = eN(x) � eN,M(x) = y(x) � yN,M(x) and eN,M (x) denote the errorfunction, the corrected error function and the estimated error function, respectively.

If the exact solution of Eq. (3) is not known, then the absolute errors jeN(xi)j = jy(xi) � yN(xi)j, (�1 6 xi 6 1) are not com-puted. However, the absolute errors jeN(xi)j = jy(xi) � yN (xi)j, (�1 6 xi 6 1) can be estimated by using the absolute error func-tion jeN,M(x)j.

5. Illustrations examples

In this section, we show the efficiency of the presented method by solving the following examples. In tables and figures,we give the values of the exact solution y(x), the Legendre polynomial solution yN(x), the corrected Legendre polynomialsolution yN,M(x) = yN(x) + eN,M(x), the absolute error function jeN(x)j = jy(x) � yN(x)j, the corrected absolute error functionjEN,M(x)j = jy(x) � yN,M(x)j and the estimated absolute error function jeN,M(x)j at the selected points of the given interval.We have made all numerical computations in Matlab.


Example 1. Let us first consider the problem

yð3ÞðxÞ þ xyð2ÞðxÞ � sinðxÞyðxÞ þZ 1

�1cosðxþ tÞyðtÞdt � 1

2

Z x

�1sinðxþ tÞyðtÞdt ¼ gðxÞ; �1 6 x; t 6 1 ð41Þ

with the initial conditions y(0) = 0, y0(0) = 1, y00(0) = 0. Here,

gðxÞ ¼ cosðxÞ �54þ 1

8sinð2Þ � 1

4xþ 1

2sinðxÞ cosðxÞ

� �þ sinðxÞ �x� sinðxÞ þ 1

2sinð2Þ � 9

8� 1

8cosð2Þ

� �:

The exact solution of the problem is y(x) = sin (x) and the approximate solution for N = 3 is given by

yðxÞ ¼X3

n¼0

anJnðxÞ:

Now, let us apply the procedure in Section 3 to obtain this approximate solution. Firstly, we note that


8sinð2Þ � 1

4xþ 1

2sinðxÞ cosðxÞ


2sinð2Þ � 9

8� 1

8cosð2Þ

� �;

P0ðxÞ ¼ � sinðxÞ; P2ðxÞ ¼ x; P3ðxÞ ¼ 1; m ¼ 3; kf ¼ 1; kv ¼ �12; Kf ðx; tÞ ¼ cosðxþ tÞ

and Kv(x, t) = sin (x + t). The set of collocation points (18) for N = 3 is computed as

x0 ¼ �1; x1 ¼ �13; x2 ¼

13; x3 ¼ 1

�
and the fundamental matrix equation of the problem from Eq. (32) is
F0XDT þ F2XðBTÞ2DT þ F3XðBTÞ3DT þ k1XKTf D�1Q þ k2XKHeDn o

A ¼ G:

The augmented matrix for this fundamental matrix equation is calculated as

½W; G� ¼

1715=1024 �1105=3236 �1996=897 20633=708 ; 362=3191894=871 �275=2718 �2051=1765 5696=339 ; 768=577

1739=1394 �560=1399 678=629 4605=274 ; 1925=1653�789=1123 �1399=938 2287=1065 7003=240 ; 416=2287

2666437775:

From Eq. (34), the matrix forms for the initial conditions are written as

½U0;l0� ¼ 1 0 �1=2 0 ; 0½ �;½U1;l1� ¼ 0 1 0 �3=2 ; 1½ �

and

½U2;l2� ¼ 0 0 3 0 ; 0½ �:

The new augmented matrix based on the conditions from Eq. (35) becomes

½fW; eG� ¼1715=1024 �1105=3236 �1996=897 20633=708 ; 362=319

1 0 �1=2 0 ; 00 1 0 �3=2 ; 10 0 3 0 ; 0

2666437775:

By solving this system, the Legendre coefficients matrix is gained as

A ¼ �1=15085296206198680 1325=1421 0 �64=1421½ �T :

Now, let us substitute the determined Legendre coefficients a0; a1; . . . ; aN A ¼ a0 a1 a2 a3½ �T� �

into Eq. (6). Hence, theLegendre polynomial solution for N = 3 of Eq. (41) is obtained as

y3ðxÞ ¼ �0:662897e� 16þ 0:999999999999999947x� 0:112596768604286048x3:

Now, let us find the improved Legendre polynomial solution for M = 5. For this purpose, let us first consider the errorproblem

eð3Þ3 ðxÞ þ xeð2Þ3 ðxÞ � sinðxÞe3ðxÞ þR 1�1 cosðxþ tÞe3ðtÞdt � 1

2

R x�1 sinðxþ tÞe3ðtÞdt ¼ �R3ðxÞ

e3ð0Þ ¼ 0; eð1Þ3 ð0Þ ¼ 0; eð2Þ3 ð0Þ ¼ 0

(; �1 6 x; t 6 1 ð42Þ


where the residual error function is

Table 1Numeri

xi

00.20.40.60.81

00.20.40.60.81

R3ðxÞ ¼ yð3Þ3 ðxÞ þ xyð2Þ3 ðxÞ � sinðxÞy3ðxÞ þZ 1

�1cosðxþ tÞy3ðtÞdt � 1

2

Z x

�1sinðxþ tÞy3ðtÞdt � gðxÞ

such that


8sinð2Þ � 1

4xþ 1

2sinðxÞ cosðxÞ


2sinð2Þ � 9

8� 1

8cosð2Þ

� �:

By solving the error problem (42) for M = 5 with the method introduced in Section 3, the Legendre error function approxi-mation e3,5(x) to e3(x) is found as

e3;5ðxÞ ¼ 0:431796220687e� 16� ð0:815862134795e� 16Þx� ð0:162170397690e� 15Þx2

� ð0:521065714805e� 1Þx3 þ ð0:139362780371e� 4Þx4 þ ð0:744750174272e� 2Þx5:

As a result, we have the improved Legendre polynomial solution

y3;5ðxÞ ¼ y3ðxÞ þ e3;5ðxÞ¼ �0:231100938e� 16þ 0:999999999999999866x� ð0:164703340084820606Þx3

þ ð0:139362780371400562e� 4Þx4 þ ð0:744750174272197895e� 2Þx5:

Table 1 shows some numerical values of the exact solution, the Legendre polynomial solution and the improved Legendrepolynomial solutions. In Table 2, the actual absolute errors are compared with the absolute errors estimated by the presentedmethod for N = 3, 6 and M = 5, 8, 10, 12, and also the absolute error functions are compared in Figs. 1a and 1b. We see fromthese comparisons that the estimated absolute errors are quite close to the actual absolute errors. Table 3 denotes the absoluteerrors of the improved Legendre polynomial solutions for N = 3, 6 and M = 5, 8, 10, 12. The improved absolute error functionsare given in Figs. 1c and 1d. It is seen from Tables 2 and 3 and Fig. 1 that the errors decrease when N and M are increased.

Example 2. [55] We consider the Volterra integro-differential equation

y0ðxÞ þ yðxÞ ¼ 1þ 2xþZ x

0xð1þ 2xÞetðx�tÞyðtÞdt; 0 6 x; t 6 1 ð43Þ

with the initial conditions y(0) = 1. The exact solution of the problem is yðxÞ ¼ ex2 .While we apply the present method in here, we note that we have used the collocation points

xi ¼ aþ b� aN

i; i ¼ 0;1; . . . ;N for a ¼ 0 and b ¼ 1:

Then, the matrix H(x) in Eq. (29) becomes

HðxÞ ¼ ½hnmðxÞ� ¼Z x

0XTðtÞXðtÞdt; hnmðxÞ ¼

xmþnþ1

mþ nþ 1; m;n ¼ 0;1; . . . ;N:

By applying for N = 10, M = 12 and M = 15 the procedure in Section 3, we obtain the approximate solutions. We compute theapproximate solution for N = 10 and M = 12 as follows

cal results of the exact and the approximate solutions for N = 3, 6 and M = 5, 8, 10, 12 of Eq. (41).

Exact solution Legendre polynomial solution Improved Legendre polynomial solution

y(xi) = sin (xi) y3(xi) y3,5(xi) y3,8(xi)

0 �0.6628971591e�16 �0.2311009384e�16 0.83009748760e�170.19866933079506 0.19909922585117 0.19868477877792 0.198669333435600.38941834230865 0.39279380680933 0.38953560542113 0.389418364805820.56464247339504 0.57567909798147 0.56500500241883 0.564642575962420.71735609089952 0.74235045447461 0.71811799554711 0.717356367789550.84147098480790 0.88740323139571 0.84275809793594 0.84147155694017

y(xi) = sin (xi) y6(xi) y6,10(xi) y6,12(xi)0 �0.2502855304e�16 �0.2771992209e�16 �0.1986684310e�160.19866933079506 0.19866991127742 0.19866933186047 0.198669330794070.38941834230865 0.38943617585994 0.38941834976166 0.389418342303450.56464247339504 0.56477768835375 0.56464249424702 0.564642473382000.71735609089952 0.71795243996293 0.71735613162333 0.717356090874750.84147098480790 0.84342306450779 0.84147105142825 0.84147098476731

Table 2Comparison of the absolute error functions for N = 3, 6 and M = 5, 8, 10, 12 of Eq. (41).

xi Absolute errors for Legendre polynomialsolution

Estimated absolute errors for Legendre polynomial solution

je3(xi)j = jy(xi) � y3(xi)j je3,5(xi)j je3,8(xi)j

0 6.6290e�017 4.3180e�017 7.4591e�0170.2 4.2990e�004 4.1445e�004 4.2989e�0040.4 3.3755e�003 3.2582e�003 3.3754e�0030.6 1.1037e�002 1.0674e�002 1.1037e�0020.8 2.4994e�002 2.4232e�002 2.4994e�0021 4.5932e�002 4.4645e�002 4.5932e�002

xi je6(xi)j = jy(xi) � y6(xi)j je6,10(xi)j je6,12(xi)j0 2.5029e�017 1.1745e�017 4.9136e�0190.2 5.8048e�007 5.7942e�007 5.8048e�0070.4 1.7834e�005 1.7826e�005 1.7834e�0050.6 1.3521e�004 1.3519e�004 1.3521e�0040.8 5.9635e�004 5.9631e�004 5.9635e�0041 1.9521e�003 1.9520e�003 1.9521e�003

Table 3Numerical results of the corrected error functions for N = 3, 6 and M = 5, 8, 10, 12 of Eq. (41).

xi Improved absolute errors jEN,M (xi)j = jy(xi) � yN,M(xi)j

jE3,5(xi)j jE3,8(xi)j jE6,10(xi)j jE6,12(xi)j

0 2.3110e�017 8.3010e�018 1.3284e�017 2.4537e�0170.2 1.5448e�005 2.6405e�009 1.0654e�009 9.8863e�0130.4 1.1726e�004 2.2497e�008 7.4530e�009 5.1985e�0120.6 3.6253e�004 1.0257e�007 2.0852e�008 1.3041e�0110.8 7.6190e�004 2.7689e�007 4.0724e�008 2.4772e�0111 1.2871e�003 5.7213e�007 6.6620e�008 4.0591e�011

Fig. 1a.(41).


y10;12ðxÞ ¼ 1þ ð0:855137252615e� 15Þxþ 1:00000128108x2 � ð0:308175968273e� 4Þx3 þ ð0:500341561093Þx4

� ð0:222950077045e� 2Þx5 þ ð0:176074274121Þx6 � ð0:267813065739e� 1Þx7

þ ð0:938907818985e� 1Þx8 � ð0:692591978932e� 1Þx9 þ ð0:686750725063e� 1Þx10

� ð0:317178628171e� 1Þx11 þ ð0:931755003590e� 2Þx12:

In Table 4, we compare the absolute errors obtained by the present method, the Tau method [55], the Makroglou [56] and theBessel collocation method [47]. In addition, the absolute error functions are compared in Fig. 2.

Comparison of the absolute error functions jeN (x)j = jy(x) � yN(x)j and the estimated absolute error functions jeN,M(x)j for N = 3 and M = 5, 8 of Eq.

Fig. 1b. Comparison of the absolute error functions jeN (x)j = jy(x) � yN(x)j and the estimated absolute error functions jeN,M(x)j for N = 6 and M = 10, 12 of Eq.(41).

Fig. 1c. Comparison of the improved absolute error functions jEN,M(x)j = jy(x) � yN,M(x)j for N = 3 and M = 5, 8 of Eq. (41).

Fig. 1d. Comparison of the improved absolute error functions jEN,M(x)j = jy(x) � yN,M(x)j for N = 6 and M = 10, 12 of Eq. (41).


Table 4Comparison of the absolute errors for N = 10 and M = 12, M = 15 of Eq. (43).

xi Tau method [55] Makroglou [56] Bessel collocation method [47] Present method

N = 10, je10(xi)j N = 10, je10(xi)j N = 10, je10(xi)j jE10,12(xi)j jE10,15(xi)j

0 0 0 0 4.4409e�016 5.5511e�0160.2 5.72156e�12 3.63e�8 4.2104e�008 8.4080e�010 3.0314e�0120.4 2.38451e�08 1.60e�7 3.6931e�008 7.2627e�010 2.5462e�0120.6 3.18608e�06 4.45e�7 7.4295e�008 6.7522e�010 2.3772e�0120.8 1.04921e�04 1.11e�6 3.7483e�006 3.8974e�010 2.4036e�0121 1.61516e�03 2.75e�6 1.1979e�004 6.6206e�009 1.3332e�012

Fig. 2. Comparison of the absolute error functions jeN (x)j = jy(x) � yN(x)j and jEN,M(x)j = jy(x) � yN,M(x)j for N = 10 and M = 12, 15 of Eq. (43).


6. Application of the population model (1)

In this section, we will apply the method presented for a stable population model (1). In this application, we investigatethe number of female births for g(t) = et, k(t,x) = t � x and t 2 [0,1]. Then, the model (1) converts to the problem [1,2]

Table 5Compar

xi

00.20.40.60.81

BðtÞ ¼ et þZ t

0ðt � xÞBðxÞdx: ð44Þ

The exact solution of problem is BðtÞ ¼ 12 ½et þ cosðtÞ þ sinðtÞ�.

By applying the present method in similar way to Example 1 and Example 2, we compute the approximate solutions forN = 3, 5 and M = 7, 8, 10. We tabulate the absolute errors of the improved Legendre solutions of the model problem (44) inTable 5. Fig. 3a shows graph of the population in the interval 0 6 t 6 1. Fig. 3b denotes the absolute error functions obtainedby the present method for N = 3, 5 and M = 7, 8, 10 of the population model. It is seen from Figs. 3a and 3b that the accuracyof the solutions increases as N is increased.

ison of the absolute errors for N = 3, 5 and M = 7, 8, 10 of the population model (44).

jE3,5(xi)j jE3,7(xi)j jE5,8(xi)j jE5,10(xi)j

0 0 0 02.5398e�009 8.6475e�010 1.5064e�011 7.2164e�0166.0574e�009 8.2890e�011 4.9200e�012 1.2490e�0159.2081e�009 3.4984e�011 8.4055e�013 9.9920e�0161.1702e�008 1.0530e�009 9.6488e�012 9.9920e�0161.5658e�008 2.4439e�010 5.0843e�012 2.1094e�015

Fig. 3b. Comparison of the absolute errors functions of the population model (44).

Fig. 3a. Plot of the numerical and the exact solutions of the population model (44).


7. Conclusions

In this article, we have improved the Legendre collocation method, based on Legendre polnomials, for Fredholm–Volterraintegro-differential equations. This improvement is based on the residual error function. In addition, an error estimation isgiven with the residual error function. Moreover, if the exact solution of the problem is unknown, then the absolute errorsjeN(xi)j = jy(xi) � yN(xi)j, (�1 6 xi 6 1) can be estimated by the approximation jeN,M(x)j. It is seen from Tables 1–3 that the esti-mated absolute errors jeN,M(xi)j are quite close to the actual absolute errors jeN(xi)j = jy(xi) � yN (xi)j. We see from tables andfigures that the errors decrease when N and M are increased. The comparisons of the present method (ILCM) by the othermethods show that our method is very effective. The present method is proposed to find approximate solution and also ana-lytical solution, and is valid when Fk(x), g(x), Kf(x, t) and Kv(x, t) are the functions defined on �1 6 x, t 6 1. When the problemis defined in a finite range [a,b], by means of the linear transformation

x ¼ ðb� aÞt=2þ ðbþ aÞ=2;

this range can be convert to the range [�1,1]. Then any integro-differential equation can be solved by the present method. Ifequation has an exact solution that is a polynomial degree N or less than N, the exact solution is obtained by the suggested


method. A considerable advantage of the method is that the approximate solutions are computed very easily by using a well-known symbolic software such as Matlab, Maple and Mathematica.

Acknowledgement

The authors thank the editor and the referees for their many constructive comments and suggestions to improve thepaper.

References

[1] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, 2001.[2] D.-B. Pougaza. The Lotka integral equation as a stable population model. Postgraduate Essay, African Institute for Mathematical Sciences (AIMS), 2007.[3] I.D. Kopeikin, V.P. Shishkin, Integral form of the general solution of equations of steady-state thermoelasticity, J. Appl. Math. Mech. (PMM U.S.S.R.). 48

(1) (1984) 117–119.[4] A.J. Lotka, On an integral equation in population analysis, Ann. Math.Stat 10 (1939) 144–161.[5] K. Maleknejad, N. Aghazadeh, Numerical solutions of Volterra integral equations of the second kind with convolution kernel by using Taylor-series

expansion method, Appl. Math. Comput. 161 (3) (2005) 915–922.[6] M. Gülsu, M. Sezer, Taylor collocation method for solution of systems of high-order linear Fredholm–Volterra integro-differential equations, Intern. J.

Comput.Math. 83 (4) (2006) 429–448.[7] E. Yusufoglu, A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations, Math. Comput. Model. 47 (2008)

1099–1107.[8] M. Dehghan, F. Shakeri, Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method,

Prog. Electromagnetics Res. PIER 78 (2008) 361–376.[9] K. Maleknejad, F. Mirzaee, Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method, Int. J. Comput.

Math. 80 (2003) 1397–1405.[10] K. Maleknejad, F. Mirzaee, S. Abbasbandy, Solving linear integro-differential equations system by using rationalized Haar functions method, Appl.

Math. Comput. 155 (2004) 317–328.[11] M. Dehghan, F. Shakeri, Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He’s variational

iteration technique, Int. J. Numer. Methods Biomed. Eng. 26 (2010) 705–715.[12] A. Arikoglu, I. Ozkol, Solutions of integral and integro-differential equation systems by using differential transform method, Comput. Math. Appl. 56

(2008) 2411–2417.[13] F. Fakhar-Izadi, M. Dehghan, The spectral methods for parabolic Volterra integro-differential equations, J. Comput. Appl. Math. 235 (2011) 4032–4046.[14] J. Pour-Mahmoud, M.Y. Rahimi-Ardabili, S. Shahmorad, Numerical solution of the system of Fredholm integro-differential equations by the Tau

method, Appl. Math. Comput. 168 (2005) 465–478.[15] M. Lakestani, B.N. Saray, M. Dehghan, Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre

multiwavelets, J. Comput. Appl. Math. 235 (2011) 3291–3303.[16] M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83 (2006) 123–129.[17] J. Saberi-Nadjafi, M. Tamamgar, The variational iteration method: a highly promising method for solving the system of integro differential equations,

Comput. Math. Appl. 56 (2008) 346–351.[18] M. Lakestani, M. Jokar, M. Dehghan, Numerical solution of nth-order integro-differential equations using trigonometric wavelets, Math. Methods Appl.

Sci. 34 (2011) 1317–1329.[19] H.H. Sorkun, S. Yalçinbas, Approximate solutions of linear Volterra integral equation systems with variable coefficients, Appl. Math. Model. 34 (11)

(2012) 3451–3464.[20] F. Kabakçı, Legendre polynomial solutions of linear differential, integral and integro-differential equations, M.Sc. Thesis, Graduate School of Natural

and Applied Sciences, Mugla University, 2007.[21] J. Biazar, E. Babolian, R. Islam, Solution of a system of Volterra integral equations of the first kind by Adomian method, Appl. Math. Comput. 139 (2003)

249–258.[22] K. Maleknejad, M. Tavassoli Kajani, Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math.

Comput. 159 (2004) 603–612.[23] M. Javidi, Modified homotopy perturbation method for solving system of linear Fredholm integral equations, Math. Comput. Model. 50 (2009) 159–

165.[24] M. Dehghan, R. Salehi, The numerical solution of the non-linear integro-differential equations based on the meshless method, J. Comput. Appl. Math.

236 (2012) 2367–2377.[25] D. Mirzaei, M. Dehghan, A meshless based method for solution of integral equations, Appl. Numer. Math. 60 (2010) 245–262.[26] M.E.A. El-Mikkawy, G.S. Cheon, Combinatorial and hypergeometric identities via the Legendre polynomials – a computational approach, Appl. Math.

Comput. 166 (2005) 181–195.[27] H.R. Marzban, M. Razzaghi, Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials, J. Franklin Inst. 341 (2004)

279–293.[28] M. Sezer, M. Gülsu, Solving high-order linear differential equations by a Legendre matrix method based on Hybrid Legendre and Taylor polynomials,

Numer. Meth. Part. Diff. Equ. 26 (3) (2010) 647–661.[29] K. Maleknejad, M. Tavassoli Kajani, Solving second kind integral equation by Galerkin methods with hybrid Legendre and Block–Pulse functions, Appl.

Math. Comput. 145 (2003) 623–629.[30] W.N. Everitt, L.L. Litlejohn, R. Wellman, Legendre polynomials, Legendre–Stirling numbers and the left-definite spectral analysis of the Legendre

differential expressions, J. Comput. Appl. Math. 148 (2002).[31] M.R. Spiegel, Theory and Problems of Fourier Analysis, Schaum’s Outline Series, McGraw-Hill Inc., New York, 1994.[32] L. Fox, I. Parker, Chebyshev Polynomials in Numerical Analysis, Clarendon Press, Oxford, 1968.[33] A.G. Morris, T.S. Horner, Chebyshev Polynomials in Numerical Analysis, Clarendon Press, Oxford, 1968.[34] E.M.E. Elbarbary, Legendre expansion method for the solution of the second-and fourth-order elliptic equations, Math. Comput. Simulat. 59 (2002)

389–399.[35] S.K. Vanani, F. Soleymani, M. Avaji, Legendre wavelet method for solving differential algebraic equations, Aust. J. Basic Appl. Sci. 5 (9) (2011) 2105–

2110.[36] S.K. Vanani, J.S. Hafshejani, F. Soleymani, M. Khan, Numerical solution of functional differential equations using Legendre wavelets method, World

Appl. Sci. J. 13 (12) (2011) 2522–2525.[37] E.M.E. Elbarbary, Legendre expansion method for the solution of the second-and fourth-order elliptic equations, Math. Comput. Simulat. 59 (2002)

389–399.[38] I.P. Streltsov, Approximation of Chebyshev and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral

equations, Comput. Phys. Commun. 126 (2000) 178–181.


[39] S. Nas, M. Sezer, A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations, Int. J. Math. Educ. Sci. Technol. 31(2) (2000) 213–225.

[40] S. Yalçınbas, M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials,Appl. Math. Comput. 112 (2002) 291–308.

[41] M. Sezer, Taylor polynomial solutions of Volterra integral equations, Int. J. Math. Educ. Sci. Technol. 25 (5) (1994) 625–633.[42] M. Sezer, M. Kaynak, Chebyshev Series Solutions of Fredholm Integral Equations, Int. J. Math. Educ. Sci. Technol. 27 (5) (1996) 649–657.[43] S. Yalçinbas, M. Sezer, H.H. Sorkun, Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Appl. Math. Model.

210 (2009) 334–349.[44] A. Karamete, M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79 (9) (2002) 987–

1000.[45] S�. Yüzbas�ı, N. S�ahin, M. Sezer, Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases, Comput.

Math. Appl. 61 (10) (2011) 3079–3096.[46] S�. Yüzbas�ı, N. S�ahin, M. Sezer, Bessel matrix method for solving high-order linear Fredholm integro-differential equations, J. Adv. Res. Appl. Math. 3 (2)

(2011) 23–47.[47] S�. Yüzbas�ı, N. S�ahin, M. Sezer, Bessel polynomial solutions of high-order linear Volterra integro-differential equations, Comput. Math. Appl. 62 (4)

(2011) 1940–1956.[48] N. S�ahin, S�. Yüzbas�ı, M. Gülsu, A collocation approach for solving systems of linear Volterra integral equations with variable coefficients, Comput. Math.

Appl. 62 (2) (2011) 755–769.[49] S�. Yüzbas�ı, Bessel collocation approach for solving continuous population models for single and interacting species, Appl. Math. Model. 36 (8) (2012)

3787–3802.[50] S�. Yüzbas�ı, N. S�ahin, M. Sezer, A collocation approach for solving modeling the pollution of a system of lakes, Math. Comput. Model. 55 (3-4) (2012)

330–341.[51] F.A. Oliveira, Collocation and residual correction, Numer. Math. 36 (1980) 27–31.[52] _I. Çelik, Approximate calculation of eigenvalues with the method of weighted residuals–collocation method, Appl. Math. Comput. 160 (2005) 401–410.[53] S. Shahmorad, Numerical solution of the general form linear Fredholm–Volterra integro-differential equations by the Tau method with an error

estimation, Appl. Math. Comput. 167 (2005) 1418–1429.[54] _I. Çelik, Collocation method and residual correction using Chebyshev series, Appl. Math. Comput. 174 (2006) 910–920.[55] S.M. Hosseini, S. Shahmorad, Numerical solution of a class of integro-differential equations by the Tau method with an error estimation, Appl. Math.

Comput. 136 (2003) 559–570.[56] A. Makroglou, Convergence of a block-by-block method for nonlinear Volterra integrodifferential equations, Math. Comput. 35 (1980) 783–796.

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