A Beneficial Numerical Approach to Solve Systems of Linear

Integro-Differential Equations

NEŞE İŞLER ACAR 1, AYŞEGÜL DAŞCIOĞLU 2

1Department of Mathematics, Faculty of Arts and Science

Burdur Mehmet Akif Ersoy University

Burdur,

TURKEY

2Department of Mathematics, Faculty of Science

Pamukkale University

Denizli,

TURKEY

Abstract: The system of linear Fredholm-Volterra integro-differential equations (FVIDEs) has been solved in

this paper by an improved approximation method. Generalised Bernstein polynomials and collocation points

have been used to construct the theory of the method. The aim of the technique is to reduce systems of

integro-differential equations into an algebraic matrix equation, which corresponds to a linear algebraic equation

system, by means of Bernstein polynomials. In order to analyse the applicability of the method, some illustrative

examples have also been considered. It has been shown that the proposed method is faster and more effective

than the others when comparing the numerical results.

Key-Words: Bernstein polynomial approach; collocation method; system of integro-differential equations

Received: April 15, 2024. Revised: August 19, 2024. Accepted: September 11, 2024. Published: October 29, 2024.

1 Introduction

Systems of linear integro-differential equations

(IDEs) have a major role in the fields of natural

science, engineering, chemistry, physics, biology,

astronomy, potential theory, electrostatics, and

financial mathematics, etc. Many problems, such

as dynamic and genetic structures, risky businesses

(e.g. assurance companies), and neural networks

with time-varying delays, can be modelled by IDEs.

Therefore, numerical solutions of IDEs have become

a remarkable study both in the fields of mathematics

and physical science.

To date, many papers have been published related

to numerical methods for solutions of linear and

nonlinear IDEs systems. Studies on linear systems of

IDEs are collocation methods based on the Bernstein

operational matrix, [1], Bessel polynomials, [2],

[3], Euler polynomials, [4], Taylor polynomials,

[5], [6], Chebyshev polynomials, [7], and Fibonacci

polynomials, [8]. Apart from the collocation

methods, a numerical method based on rationalized

Haar functions, [9], has been presented to solve a

system of linear Fredholm IDEs. In addition, a

spectral method, [10], has been developed for the

solution of a linear Volterra IDE system. Moreover,

the Chebyshev collocation method has been used to

solve a system of second order IDEs modeling the

Markow-modulated jump-diffusion process, [11].

Considering the above promising studies

associated with collocation method, in this study

an alternative collocation method has been revealed

regarding to derivability property of the generalized

Bernstein polynomials to solve the system of linear

Fredholm-Volterra integro-differential equations

(FVIDEs). Basis of the developed method depends

on the definitions and the matrix relations of

Bernstein polynomials and their derivatives, [12],

[13], [14].

Definition 1.1: The generalized Bernstein basis

polynomials of Nth degree are defined by

pr,N (x) = 1

(b−a)NN

r(x−a)r(b−x)N−r;r=0,1, ..., N

on the interval [a, b]. For convenience, pr,N (x)=0

is accepted for r < 0and r > N . Besides, pr,N (a) =

pr,N (b) = 0 are verified for 0< r < N, and

p0,N (b) = pN,N (a) = 0, p0,N (a) = pN,N (b) = 1.

Definition 1.2: Let y: [a, b]→Ris

continuous function. Then the generalized Bernstein

polynomials of Nth degree are defined by

BN(y;x) =

r=0

ya+(b−a)r

Npr,N (x)

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on the interval [a, b].

Theorem 1.1: There is a relation between the

generalized Bernstein basis polynomials matrix and

their derivatives in the form

p(k)(x) = p(x)dk;k= 0,1, ..., m

such that p(x) = [ p0,N (x)p1,N (x). . . pN,N (x)]

and the elements of matrix d= (drs);r, s =

0,1, ..., N is as follows:

drs =1

b−a









N−r;s=r+ 1

2r−N;s=r

−r;s=r−1

0 ; otherwise

The rest of the paper is structured as follows:

In Section 2, the theory of the method has been

explained, and the solution algorithm has been given.

In Section 3, the application of the method to the

system of FVIDEs has been demonstrated on three

problems. Besides, the numerical results of the

proposed method have been compared with the other

methods. Some conclusions have been given in the

last section.

2 Description of the Method Based

on Bernstein Polynomials

A projection method was given by [15], for mth-order

linear FVIDE in the most general form. The main

idea of this method is applied and improved to

approximate the solution of the following FVIDEs

system:

k=0

qk(x)y(k)(x)=g(x) +

f(x, t)y(t)dt +

v(x, t)y(t)dt;a≤x, t ≤b

(1)

under the mixed conditions

Pm−1

k=0 Aky(k)(a)+Bky(k)(b) + Cky(k)(c)=λ, a < c < b

(2)

by the generalized Bernstein polynomials as follows:

y(k)

i(x)∼

=B(k)

N(yi;x)=PN

r=0 ya+(b−a)r

Np(k)

r,N (x) ; i=1,2, ..., n.

(3)

Here qk(x) = hqk

ij (x)i,f(x, t) = [fij (x, t)],

v(x, t)=[vij (x, t)] are n×nmatrices; g(x) =

[gi(x)] and y(x) = [yi(x)]Tare n×1matrices for

i, j = 1, ..., n.Ak=αk

l,Bk=βk

l,Ck=γk

l

are m×nmatrices; and λ= [λl]is m×1matrix for

l= 1, ..., m.

Theorem 2.1: Let xs∈[a, b]be collocation

points. If system (1) has a generalized Bernstein

polynomial solution (3), linear FVIDEs system with

nunknowns and mixed conditions have following

matrix relations:

k=0

QkPDk−F−V#Y=G,(4)

m−1

k=0 hAkp(a)dk+Bkp(b)dk+Ckp(c)dkiY=λ.

(5)

Here p(x)is n×n(N+ 1) matrix, dkis n(N+ 1)×

n(N+ 1) matrix and Yis n(N+ 1) ×1matrix,

Qk=diag [qk(xs)] ,P= [p(xs)] ,Dk=hdki,

F= [F(xs)] and V= [V(xs)] are n(N+ 1) ×

n(N+ 1) matrices. G= [g(xs)] and Y=Yare

n(N+ 1) ×1matrices.

Proof. Since system (1) has a generalized Bernstein

polynomial solution (3), unknown functions and their

derivatives can be written as

y(k)

i(x)∼

=p(k)(x)Yi=p(x)dkYi;i= 1, ..., n.

Here p(x)is 1×(N+ 1) matrix, dis (N+ 1) ×

(N+ 1) matrix defined in Theorem 1.1, and

Yi=yi(a)yia+b−a

N.. . yi(b)T;i= 1, ..., n

is (N+ 1) ×1matrix. Compactly, the unknow

functions and their derivatives can be restated by

y(k)(x)∼

=p(x)dkY;k= 0,1, ..., m, (6)

where the elements of matrices are defined as follows:

y(k)(x) =







y(k)

1(x)

y(k)

2(x)

y(k)

n(x)







,p(x) =







p(x) 0 ... 0

0p(x)... 0

.....

0 0 ... p(x)





n×n

dk=







dk0... 0

0dk. . . 0

.....

0 0 ... dk





n×n

,Y=













Substituting relation (6) into Eq. (1) yields

k=0

qk(x)p(x)dkY∼

=g(x) + F(x)Y+V(x)Y.

(7)

Here, the explicit forms of the above matrices are as

follows:

qk(x) = 





11 (x)qk

12 (x). . . qk

1n(x)

21 (x)qk

22 (x). . . qk

2n(x)

.....

n1(x)qk

n2(x). . . qk

nn (x)







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g(x) = 





g1(x)

g2(x)

gn(x)





,

F(x)=







f11 (x, t)p(t)dt

f12 (x, t)p(t)dt . . .

f1n(x, t)p(t)dt

f21 (x, t)p(t)dt

f22 (x, t)p(t)dt . . .

f2n(x, t)p(t)dt

.....

fn1(x, t)p(t)dt

fn2(x, t)p(t)dt . . .

fnn (x, t)p(t)dt







V(x)=







v11 (x, t)p(t)dt

v12 (x, t)p(t)dt . . .

v1n(x, t)p(t)dt

v21 (x, t)p(t)dt

v22 (x, t)p(t)dt . . .

v2n(x, t)p(t)dt

.....

vn1(x, t)p(t)dt

vn2(x, t)p(t)dt . . .

vnn (x, t)p(t)dt







Since y(k)

i(xs) = B(k)

N(yi;xs); i= 1, ..., n is valid on

the collocation points xs∈[a, b]for s= 0,1, ..., N ,

Equation (7) becomes

k=0

qk(xs)p(xs)dkY−F(xs)Y−V(xs)Y=g(xs).

This system of equations can also be

written compactly WY =Gsuch that

k=0

QkPDk−F−V, where

Qk=





qk(x0) 0 . . . 0

0qk(x1).. . 0

.....

0 0 . . . qk(xN)





,P=





p(x0)

p(x1)

p(xN)





,Dk=hdki,

F=





F(x0)

F(x1)

F(xN)





,V=





V(x0)

V(x1)

V(xN)





,G=





g(x0)

g(x1)

g(xN)





.

Similarly, substituting x=a, x =b, and x=c

into Eq. (6), given conditions is written in the form

UY =λsuch that

m−1

k=0

Akp(a)dk+Bkp(b)dk+Ckp(c)dk.(8)

Thus, the proof is completed.

The following steps are applied to solve the system of

FVIDEs (1) under the mixed conditions (2):

Step 1. First, the matrices Qk,P,D,F,Vdefined

in Theorem 2.1 are computed depending on the

collocation points, and then the fundamental matrix

relation belonging to (4) is obtained, it can be stated

WY =Gor [W;G].(9)

This matrix equation corresponds to an n(N+ 1)

dimensional system of linear algebraic equations with

unknown coefficients matrix Y.

Step 2. By calculating the matrices in Eq. (8) at the

given points, the augmented matrix form of the mixed

conditions can be expressed as

[U;λ].(10)

Step 3. There are two techniques available for

obtaining the solution of Eq. (9) under conditions

(10). Initially, the arrays of the row matrices (10) can

be added under the matrix (9). This gives the new

augmented rectangular matrix he

W;e

Gi, where e

Wis a

matrix of dimensions n(N+m+ 1)-by-n(N+ 1).

This can be called an addition technique. On the

other hand, the augmented square matrix hb

W;b

Giis

produced by replacing some of the rows of matrix

(9) with rows of matrix (10) by removing the first,

middle or last rows of matrix (9) and writing the

rows of matrix (10) in their place. This can also

be called displacement technique. The number of

collocation points, the given conditions and the order

of the equations all affect these strategies.

Step 4. If rank(e

W) = rank he

W;e

Gi=n(N+ 1)

or rank(b

W) = rank hb

W;b

Gi=n(N+ 1), the

unknown coefficient matrix Yis uniquely determined

by Y=e

W−1e

Gor Y=b

W−1b

G. This system

can be solved easily by the standard methods.

3 Illustrations of the Method

In this section, three numerical examples are provided

to support the suggested approach. The first example

demonstrates how simple it is to implement the

method. Moreover, the success of the method

is shown by the numerical results. Furthermore,

the numerical results of the proposed method have

been compared with those obtained using different

methods. Accordingly, the suggested method’s

outcomes demonstrated its importance, significant,

remarkable, and effective in solving the system

of FVIDEs. These numerical results have been

calculated using the MATLAB programme. The

absolute and maximum errors listed in the tables are

defined by:

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ei,N =|yi,N (x)−yi(x)|and ∥ei,N ∥∞=max

xs∈[a,b]|ei,N (xs)|.

Here yi,N (x)is the Bernstein approximation and

yi(x)is the exact solution of the system.

Example 3.1: The second-order system of FIDEs

defined on −1≤x, t ≤1given in [5], is as follows:

−y′′

1+y′

2+xy1+y2=−x4+ 6x2+ 7x−2

−1hty1(t)−t3y2(t)idt +

−1

xy2(t)dt

xy′′

1−y′′

2+y1−x2y2=−3x4

−x3

−x2+ 3x−14

−1

[(t+ 2) y1(t)−ty2(t)] dt +

−1

ty1(t)dt

y1(0) = 2, y′

1(1) = 3, y2(0) = 0, y′

2(−1) = −6.

Let us approach the solution of the system using

Bernstein polynomials for m= 2 and N= 2, and

then show step by step the application of the method

to this problem. Then, the collocation points are of

the form

x0=−1, x1= 0, x2= 1.

According to the matrices provided in Section 2,

the matrices in Eq. (1) for this system are as follows:

q0(x) = x1

1−x2,q1(x) = 0 1

0 0 ,

q2(x) = −1 0

x−1,v(x, t) = 0x

t0,

f(x, t) = t−t3

t+ 2 −t,

g(x) = −x4+ 6x2+ 7x−2

−3x4−x3−x2+ 3x−14 ,

y(x) = y1(x)

y2(x),y′(x) = y′

1(x)

y′

2(x),

y′′ (x) = y′′

1(x)

y′′

2(x), λ = [ 203−6]T.

Then the fundamental matrix of the system can be

written as

(Q2PD2+Q1PD +Q0P−V−F)Y=G.

Here the entries of this matrices are as follows:

Q2=





−100000

−1−10000

0 0 −1 0 0 0

0 0 0 −1 0 0

0000−1 0

0 0 0 0 1 −1







Q1=





010000

000000

000100

000000

000001

000000







Q0=





−1 1 0 0 0 0

1−1000 0

0 0 0 1 0 0

0 0 1 0 0 0

0 0 0 0 1 1

0 0 0 0 1 −1







P=





100000

000100

1/4 1/2 1/4 0 0 0

0 0 0 1/4 1/2 1/4

001000

000001













−1 1 0 0 0 0

−1/2 0 1/2 0 0 0

0−1 1 0 0 0

0 0 0 −1 1 0

0 0 0 −1/2 0 1/2

0 0 0 0 −1 1







,G=







−4

−20

−2

−14

−16







V=





0 0 0 0 0 0

−17/48 −1/8 −1/48 0 0 0

0 0 0 2/3 2/3 2/3

−1/3 0 1/3 0 0 0







F=





−1/3 0 1/3 1/5 0 −1/5

1 4/3 5/3 1/3 0 −1/3

−1/3 0 1/3 1/5 0 −1/5

1 4/3 5/3 1/3 0 −1/3

−1/3 0 1/3 1/5 0 −1/5

1 4/3 5/3 1/3 0 −1/3







Finally, by considering Equations (9) and (10), the

augmented matrix and conditions become

[W;G] =







−7/6 1 −5/6 −1/5 1 1/5 ; −4

−1/2 −1/3 −13/6 −11/6 1 −1/6 ; −20

−1/6 1 −5/6 −9/20 1/2 19/20 ; −2

−19/48 −17/24 −67/48 −5/6 1 −1/6 ; −14

−1/6 1 1/6 −13/15 −5/3 23/15 ; 10

−1/6 −7/3 −1/2 −5/6 1 −7/6 ; −16







and

[U;λ] = 





1/4 1/2 1/4 0 0 0 ; 2

0 0 0 1/4 1/2 1/4 ; 0

0−1 1 0 0 0 ; 3

000−1 1 0 ; −6





.

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Utilizing the addition technique mentioned in Step 3,

the augmented matrix for this system occurs as

W;e

Gi=







−7/6 1 −5/6 −1/5 1 1/5 ; −4

−1/2 −1/3 −13/6 −11/6 1 −1/6 ; −20

−1/6 1 −5/6 −9/20 1/2 19/20 ; −2

−19/48 −17/24 −67/48 −5/6 1 −1/6 ; −14

−1/6 1 1/6 −13/15 −5/3 23/15 ; 10

−1/6 −7/3 −1/2 −5/6 1 −7/6 ; −16

1/4 1/2 1/4 0 0 0 ; 2

0 0 0 1/4 1/2 1/4 ; 0

0−1 1 0 0 0 ; 3

0 0 0 −1 1 0 ; −6







Solving this linear system gives the exact solution of

the problem as

y1(x) = 3x+ 2 and y2(x) = 3x2.

Although, [5], found exact solutions for N= 3 in

their study, we have found exact solutions for N= 2.

Thus, the proposed method is faster than the Taylor

collocation method. It also shows that the exact

solution can be found in cases where the solution is

polynomial if the value Nis taken as the degree of

the polynomial.

Example 3.2: Let us consider a second-order system

of VIDEs, [1], [10] :

y′′

1(x)+2xy′

1(x)−y1(x)−

(y1(t)−y2(t)) dt

= 2 + x−ex+ 2xex−cos x,

y′′

2(x) + y′

2(x)−2xy2(x)−

(y1(t) + y2(t)) dt

= 2 cos x−3x−(1 + 2x)sin x−ex,

with the initial conditions

y1(0) = 1, y′

1(0) = 1, y2(0) = 1, y′

2(0) = 1,

and exact solutions y1(x) = ex,y2(x) = 1 + sin x.

By the proposed method, fundamental matrix

equation and its conditionals become

PD2+Q1PD +Q0P−VY=G,

p(0) + p(0) dY=λ.

In Table 1 and Table 2, the absolute errors are

compared with those of the other methods, [1], [10].

The numerical results of the proposed method have

been calculated using the displacement technique. In

our method, collocation points have been considered

as xs=s/N;s= 0,1, ..., N , and Newton-Cotes

points have been considered as xs= (2s−1)/(2(N+

1)); s= 1,2, ..., 2N−2for Bernstein operational

matrix method. Although the tables indicate that the

results of the proposed method are close to the other

methods, the values of the proposed method are better

than the others as move away from the initial point.

Besides, the absolute errors of the proposed method

approach zero with increasing Nvalues.

Table 1. The Comparison of the Absolute Errors for y1(x).

xsProposed Method Bernstein Operational

Matrix Method [1]

Spectral Method [10]

N= 5 N= 10 N= 15 N= 5 N= 10 N= 5

0.0 0 0 0 8.9e−016 8.9e−016 0.0e−009

0.1 2.5e−007 6.7e−014 2.2e−015 6.7e−007 1.5e−013 1.0e−009

0.2 1.2e−006 1.7e−013 4.8e−015 1.4e−006 2.9e−013 9.1e−008

0.3 2.1e−006 2.6e−013 4.1e−015 1.8e−006 4.3e−013 1.1e−006

0.4 2.5e−006 3.5e−013 6.8e−015 2.4e−006 5.6e−013 6.0e−006

0.5 3.1e−006 4.4e−013 8.1e−015 3.0e−006 7.0e−013 2.3e−005

0.6 4.3e−006 5.2e−013 9.2e−015 3.3e−006 8.3e−013 7.0e−005

0.7 4.9e−006 6.1e−013 1.1e−014 4.3e−006 9.5e−013 1.8e−004

0.8 2.1e−006 6.6e−013 1.2e−014 1.2e−005 1.1e−012 4.1e−004

0.9 3.5e−005 1.7e−012 1.3e−014 4.2e−005 4.4e−013 8.5e−004

1.0 1.3e−004 2.8e−011 1.3e−013 1.3e−004 3.1e−011 1.6e−003

Table 2. The Comparison of the Absolute Errors for y2(x).

xiProposed Method Bernstein Operational

Matrix Method [1]

Spectral Method [10]

N= 5 N= 10 N= 15 N= 5 N= 10 N= 5

0.0 0 0 0 7.8e−016 8.9e−016 0.0e−009

0.1 4.6e−008 4.2e−014 1.3e−015 1.4e−007 9.0e−014 0.0e−009

0.2 2.1e−007 9.9e−014 2.7e−015 2.9e−007 1.6e−013 2.0e−009

0.3 3.6e−007 1.5e−013 6.3e−015 3.6e−007 2.4e−013 4.3e−008

0.4 4.1e−007 1.9e−013 6.9e−015 4.5e−007 3.0e−013 3.3e−007

0.5 4.7e−007 2.4e−013 8.3e−015 5.5e−007 3.6e−013 1.5e−006

0.6 6.9e−007 2.8e−013 9.7e−015 5.6e−007 4.2e−013 5.5e−006

0.7 7.7e−007 3.2e−013 1.1e−014 7.7e−007 4.7e−013 1.6e−005

0.8 1.1e−006 3.4e−013 1.2e−014 2.7e−006 5.4e−013 4.1e−005

0.9 9.8e−006 9.5e−013 1.4e−014 1.1e−005 3.4e−013 9.4e−005

1.0 3.5e−005 1.6e−011 5.5e−013 3.3e−005 1.7e−011 2.0e−004

Example 3.3: Let us consider following system of

FIDEs given in 0≤x≤1:

y′′

1−xy′

2−y1= (x−2) sin x

(xcos t y1(t)−xsin t y2(t)) dt

y′′

2−2xy′

1+y2=−2xcos x

(sin xcos t y1(t)−sin xsin t y2(t)) dt

y1(0) = 0,y′

1(0) = 1,y2(0) = 1,y′

2(0) = 0.

Exact solution of this problem is y1(x) = sin x,

y2(x) = cos x.

Table 3. The Comparison of the Maximum Errors for y1(x).

NProposed Method Bessel Collocation Method [2] Fibonacci Collocation Method [8]

3 4.4e−003 5.0e−003 5.3e−003

7 9.9e−008 5.0e−007 5.0e−007

9 1.9e−010 4.0e−009 4.0e−009

10 1.6e−011 2.7e−010 2.7e−010

11 2.3e−013 2.5e−011 2.5e−011

12 1.9e−014 1.2e−012 1.1e−012

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Table 4. The Comparison of the Maximum Errors for y2(x).

NProposed Method Bessel Collocation Method, [2] Fibonacci Collocation Method, [8]

3 1.3e−002 1.4e−002 1.4e−002

7 2.2e−007 6.3e−007 6.3e−007

9 4.0e−010 4.2e−009 4.2e−009

10 7.6e−010 3.0e−010 3.0e−010

11 4.9e−013 2.6e−011 2.6e−011

12 3.2e−014 1.6e−012 1.5e−012

In Table 3 and Table 4, the maximum errors are

compared with those of the others, [2], [8]. For

all methods in the tables, the numerical results have

been calculated on the collocation points xs=s/N.

The tables indicate that the proposed method yields

better results than the others for increasing Nvalues.

Moreover, the numerical results that computed by

replacing the last rows of the augmented matrix

with the conditions are more effective than the other

displacement and addition techniques. Thus, one

reason for the effective results is that the augmented

matrix has been obtained by using the replacement

technique.

4 Conclusions and Inferences

In this study, a collocation method is improved by

considering the fundamental properties of Bernstein

polynomials to solve a system of linear IDEs. The

proposed method transforms a system of linear

FVIDEs into a system of linear algebraic equations

due to the matrix forms of the Bernstein polynomials

and their derivatives. To demonstrate the applicability

and efficiency of the method, three examples are

considered. Example 3.1 illustrates how this method

is applied to the problem and shows that it is faster

than the Taylor collocation method. In Examples

3.2 and 3.3, the results of the errors are compared

with those of other methods, which are Bernstein

operational matrix, Spectral, Bessel, and Fibonacci

collocation methods. In both examples, better results

are obtained as the value of Nincreases, by using

the proposed method. Moreover, when the solution

is polynomial, the exact solution can be found if

Nis chosen as the degree of the polynomial. The

use of the displacement technique to obtain the

augmented matrix proves advantageous for achieving

more effective results. Considering all aspects of

the study, the proposed method is a suitable and

rigorous numerical approach for solving various

linear systems, including differential, integral, and

integro-differential equations. Therefore, this method

could be applied to models like the Markow

modulated jump diffusion process, [11], in future

studies.

5HIHUHQFHV

[1] K. Maleknejad, B. Basirat, E. Hashemizadeh,

A Bernstein operational matrix approach

for solving a system of high order linear

Volterra-Fredholm integro-differential

equations, Mathematical and Computer

Modelling, Vol.55, 2012, pp.1363-1372.

[2] S. Yüzbaşı, N. Şahin, M. Sezer, Numerical

solutions of systems of linear Fredholm

integro-differential equations with Bessel

polynomial basis, Computers and Mathematics

with Applications, Vol.61, 2011, pp.3079-3096.

[3] S. Yüzbaşı, Numerical solutions of system

of linear Fredholm-Volterra integro-differential

equations by the Bessel collocation method

and error estimation, Applied Mathematics and

Computation, Vol.250, 2015, pp.320-338.

[4] F. Mirzaee, S. Bimes, Numerical solutions

of systems of high-order Fredholm

integro-differential equations using Euler

polynomials, Applied Mathematical Modelling,

Vol.39, 2015, pp.6767-6779.

[5] M. Gülsu, M. Sezer, Taylor collocation

method for solution of systems of high-order

linear Fredholm-Volterra integro-differential

equations, International Journal of Computer

Mathematics, Vol.83, No.4, 2006, pp.429-448.

[6] Y. Jafarzadeh, B. Keramati, Numerical method

for a system of integro-differential equations

and convergence analysis by Taylor collocation,

Ain Shams Engineering Journal, Vol.9, 2018,

pp.1433-1438.

[7] A. Akyüz-Daşcıoğlu, M. Sezer, Chebyshev

polynomial solutions of systems of higher-order

linear Fredholm-Volterra integro-differential

equations, Journal of the Franklin Institute,

Vol.342, 2005, pp.688-701.

[8] F. Mirzaee, S.F. Hoseini, Solving systems of

linear Fredholm integro-differential equations

with Fibonacci polynomials, Ain Shams

Engineering Journal, Vol.5, 2014, pp.271-283.

[9] K. Maleknejad, F. Mirzaee, S. Abbasbandy,

Solving linear integro-differential equations

system by using rationalized Haar functions

method, Applied Mathematics and Computation,

Vol.155, 2004, pp.317-328.

[10] O.M.A. Al-Faour, R.K. Saeed, Solution of

a system of linear Volterra integral and

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integro-differential equations by spectral

method, Al-Nahrain University Journal for

Science, Vol.6, No.2, 2006, pp.30-46.

[11] P. Diko, M. Usábel, A numerical method

for the expected penalty-reward function in

a Markov-modulated jump-diffusion process,

Insurance: Mathematics and Economics, Vol.49,

2011, pp.126-131.

[12] G.G. Lorentz, Bernstein Polynomials, Chelsea

Publishing Company, 1986.

[13] M.G. Phillips, Interpolation and Approximation

by Polynomials, Springer-Verlag, 2003.

[14] R.T. Farouki, The Bernstein polynomial basis:

A centennial retrospective, Computer Aided

Geometric Design, Vol.29, 2012, pp.379-419.

[15] N. İşler Acar, A. Daşcıoğlu, A projection method

for linear Fredholm–Volterra integro-differential

equations, Journal of Taibah University for

Science, Vol.13, No.1, 2019, pp.644-650.

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