Ain Shams Engineering Journal (2014) 5, 585–594

Ain Shams University

Ain Shams Engineering Journal

www.elsevier.com/locate/asejwww.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

A novel matrix approach to fractional finite

difference for solving models based on nonlinear

fractional delay differential equations

* Corresponding author. Tel.: +98 9113359210.

E-mail address: Parsa.math@gmail.com (B. Parsa Moghaddam).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

2090-4479 � 2013 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.

http://dx.doi.org/10.1016/j.asej.2013.11.007

B. Parsa Moghaddam *, Z. Salamat Mostaghim

Department of Mathematics, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran

Received 18 May 2013; revised 27 October 2013; accepted 17 November 2013Available online 2 January 2014

KEYWORDS

Finite difference method;

Caputo derivative;

Fractional integral;

Fractional delay differential

equations

Abstract Many real-life phenomena in physics, engineering, biology, medicine, economics, etc. can

bemodeled by fractional delay differential equations and having inmind that thesemodeling interpret

phenomena better, they are noticed by researchers, engineers, and mathematicians. In this paper the

method of fractional finite differences has been presented and used for numerical solution of such

models and used for solving a number of famous fractional order version of models such as the frac-

tional order version of Hutchinson model is related to rate of population growth, the fractional order

version of Verhulst Pearl model is related to the impact of a specific factor on the changes of popula-

tion in an areamodel, fractional order version of the negative impact of population growth in a specific

time andmodel of fractional order version of the four years life cycle of a population of lemmings. The

proposed method besides being simple is so exact which is sensible in the solved problems.� 2013 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.

1. Introduction

Fractional Calculus is a field of applied mathematics that deals

with derivatives and integrals of arbitrary orders (includingComplex Order). During the last decades, Fractional Calculushas been applied to almost every field of science, engineering

and mathematics. Some of the areas where Fractional Calculus

has made a profound impact include viscoelasticity andrheology, electrical engineering, electrochemistry, biology, bio-

physics and bioengineering, signal and image processing,mechanics, mechatronics, physics and control theory [1,2].

For completeness, in this section, we recall some definitions

and fundamental facts of fractional calculus theory, which canbe found in [3,4].

Definition 1.1. The Riemann–Liouville fractional integraloperator of order a on the usual Lebesgue space L1[a, b] is

defined as

aIax fðxÞ ¼ 1

CðaÞR x

aðx� tÞa�1fðtÞdt; a > 0;

aI0x fðxÞ ¼ fðxÞ

586 B. Parsa Moghaddam, Z. Salamat Mostaghim

It is worth mentioning that the Riemann–Liouville derivative

has certain disadvantages for describing some natural phenom-ena with fractional differential equations. Thus, we introduceCaputo’s definition [5] of fractal derivative operator C

0 Dax,

which is a modification of Riemann–Liouville definition.

Definition 1.2. The Caputo fractional derivative of order a of afunction f e L1[a, b] is defined as follows:

C0 D

ax fðxÞ ¼

1Cðn�aÞ

R x

0ðx� sÞn�a�1 dn

dsn fðsÞds; n� 1 < a < n;

fðnÞðxÞ; a ¼ n:

(

where n is a positive integer.

Remark 1.2. The following properties are useful for ourdiscussion:

For Riemann–Liouville fractional integration and the Cap-uto fractional derivative, we have the following properties:

1. For real values of a > 0, the Caputo fractional derivativereverses the operation to the Riemann–Liouville integrationfrom the left

C0 D

ax0I

ax fðxÞ ¼ fðxÞ; a > 0;

1. If f(x) e C[a][0, 1], then

0IaxC0 D

axfðxÞ ¼ fðxÞ �

X½a��1j¼0

xj

j!

dj

dxj f

� �ð0Þ; n� 1 < a 6 n;

where C[a][0, 1] is the space of [a] times, continuously differen-tiable functions.

Someproperties of the operator 0Iax, which are need here, we

mention only the following:For f e L1[a, b], l P �1, a, b P 0 and c > �1:

1. 0Iax0Ib

x f ðxÞ ¼0 Iaþbx f ðxÞ;

2. 0Iax0Ib

x f ðxÞ ¼0 Ibx0Ia

xf ðxÞ;3. 0Ia

xxc ¼ CðCþ1ÞCðaþcþ1Þ x

aþc:

Ordinary differential equations (ODEs) and delay differen-tial equations (DDEs) are used to describe many phenomena

of physical interest. While ODEs contain derivatives which de-pend on the solution at the present value of the independent var-iable (‘‘time’’), DDEs also contain derivatives which depend onthe solution at previous times.DDEs arise inmodels throughout

the sciences. For example, consist of the books by Pinney [6] andDriver [7] contains a large compilation of DDEs that have ap-peared in the literature. The more recent books by MacDonald

[8], Stepan [9],Kuang [10], Fowler [11], Epstein andPojman [12],Murray [13], Fall et al. [14], Beuter et al. [15], and Britton [16]emphasize particular DDE problems appearing in mechanical

engineering, chemistry, and biology.Mathematically, the subse-quent research of Mishkis [17], Bellman and Danskin [18], Bell-man and Cooke [19], and Krasovskii [20] set the stage for the

monumental work of Hale and Verduyn Lun [21] and studentsat Brown. Advanced mathematical issues on DDEs, functionalequations, and robust control are treated in the books by Haleand Verduyn Lunel [21], Diekmann et al. [22], Kolmanovskii

and Myshkis [23], and Michiels and Niculescu [24].

In recent years, Fractional Delay differential equations be-gin to arouse the attention of many researchers. These equa-tions have many application in various areas like control

theory [25], biology [26], economy [27], chaos [28–30], Agricul-ture [31] and so on.

Bhalekar and Daftardar [32] recently solved FDDEs as

follows:

0DaxyðxÞ¼ fðx;yðxÞ;yðx�dÞÞ; x2 ½0;T�; 0< a 6 1 ð1:1Þ

yðxÞ ¼ /ðxÞ; x 2 ð�d; 0� ð1:2Þ

by extending the Adams–Bashforth–Moulton predictor-

corrector method (EABMPC method) where f is in general anonlinear function of its arguments. It is worth mentioningthat the examples discussed in this paper are also solved byEABMPC method and compared with suggested method.

A stability analysis of the solution of fractional differentialdynamic system is considered by many researchers [33,34]. Inthis regard, Bhalekar et al. provided valuable results by using

the Adams–Bashforth–Moulton predictor-corrector scheme[35,36,29,30].

Besides, Wang [37] lately approximated the delayed frac-

tional order differential equation by combining the generalAdams–Bashforth–Moulton method with the linear interpola-tion method as:

0DaxyðxÞ¼ fðx;yðxÞ;yðx�dÞÞ; x P 0; m�1< a 6 m ð1:3Þ

yðxÞ ¼ uðxÞ; x 6 0; ð1:4Þ

where a is the order of the differential equations, u(x) is theinitial value, and m is an integer.

In addition, Wang et al. [38], based on Grunwald–Letnikovdefinition, introduced a numerical method for nonlinear func-tional order differential equations with constant time varyingdelay:

0DaxyðxÞ ¼ fðx; yðxÞ; yðx� dÞÞ; x 2 ½a; b�;

m� 1 < a 6 m ð1:5Þ

yðxÞ ¼ uðxÞ; x 6 a; ð1:6Þ

where a is the order of the differential equations, u(x) is theinitial value, and m is an integer.

Finally, Morgado et al. [39] concentrated on the following

initial value problem for a linear fractional differential equa-tion with finite delay d > 0

0DaxyðxÞ ¼ ayðx� dÞþ byðxÞþ fðxÞ; x> 0; 0< a 6 1; ð1:7Þ

yðxÞ ¼ uðxÞ; x 2 ½�d; 0� ð1:8Þ

and was solved by using adaptation of a fractional backwarddifference method where a and b are constant, f is a continuousfunction on [0, d], T > 0, the initial function u(x) is continu-ous on [�d, 0].

The goal of this paper is to provide a numerical scheme forsolving fractional delay differential equations of the general

form

0DaxyðxÞ¼ fðx;yðxÞ;yðx�dÞÞ; x2 ½a;b�; 0< a 6 1 ð1:9Þ

A novel matrix approach to fractional finite difference 587

under the following initial conditions

yðxÞ ¼ /ðxÞ � d 6 x 6 a ð1:10Þ

where 0Dax is the standard Caputo fractional derivative and d is

a constant delay.Throughout this paper, they will be referred to as fractional

order functional differential equations where the existence ofsolutions for Eq. (1.9) has been considered by Benchohraet al. in [40]. In this regard, the authors in sources [41–43] haveinvestigated the existence and uniqueness of the solutions for

fractional delay differential equations.In this paper, we generalize the finite difference method and

use it for solving FDDEs. This numerical method includes fi-

nite differences without preserving the delay index. In this pro-posed method we will analysis the errors. Also the adaptationof a variety of differential equations in the mathematical mod-

eling process of difference applications will be considered; forexample, the problem of the rated of population grown [44],the problem of the impact of a specific operating on the

changes in population [45], negative impact of populationgrown in a specific time [46] and four years life cycle of a pop-ulation of lemmings [32]; the point is that these fractional ver-sion of models are so similar to real phenomena.

2. Description of the problem

Consider a modeling problem for boundary value problems for

fractional delay differential equations as follows:

0DaxyðxÞ ¼ cðxÞyðxÞ þ dðxÞyðxÞyðx� dÞ ð2:1Þ

Such that if a < x < b and 0 6 a < 1, subject to the inter-val and boundary conditions we have:

yðxÞ ¼ /ðxÞ � d 6 x 6 a ð2:2Þ

where c(x), d(x) and /(x) are the smooth function and d is theamount of delay. If y(x) is to have smooth solution in prob-

lems (2.1) and (2.2), it should be satisfied in the boundaryproblems of (2.1) and (2.2); also it should be continuous on[a,b] and continuously differentiable on (a,b).

3. Numerical method

In this section, we will present a numerical method for solving

the boundary value problem (2.1) and (2.2) based on finite dif-ferences. However, the researchers firstly introduce y(b), be-cause two boundary conditions for finite differences method

are needed.

yðbÞ ¼0 IaxC0 D

axyðxÞjx¼b þ yð0Þ

¼0 IaxfcðxÞyðxÞ þ dðxÞyðxÞyðx� dÞgjx¼bþyð0Þ; x 2 ½0; b�; 0 < a 6 1

From Definition 1.2, for n= 1, we get

DaxyðxÞ ¼

1

Cð1� aÞ

Z x

0

ðx� sÞ�a d

dsyðsÞds; 0 < a 6 1:

To establish the numerical approximation scheme, letxi = ih, i = 0, . . . ,N be the integration time0 6 xi 6 b; h ¼ b

N. As usual, we take the following finite dif-

ference approximation for time fractional derivative:

DayðxiÞ ¼ 1Cð1�aÞ

Xi

j¼0

yðxjþ1Þ�yðxjÞh

R ðjþ1Þhjh

dsðxiþ1�sÞa þOðhÞ

¼ h1�a

Cð2�aÞ

Xi

j¼0

yðxi�jþ1Þ�yðxi�jÞh

½ðjþ 1Þ1�a � j1�a� þOðhÞ

¼ h1�a

Cð2�aÞ

Xi

j¼0½ðjþ 1Þ1�a � j1�a�Dþyi�j þOðhÞ

where Dþyi�j ¼yi�jþ1�yi�j

h.

This numerical method includes the finite difference opera-tor on a specific consistent mesh. In order to save the delayterms we can choose a mesh parameter such as h ¼ d

mwhere

m= pq, p is a positive integer and q is the Mantissa of d.This different method for the boundary value problem (2.1)

and (2.2) is given by

LNy ¼ h1�a

Cð2� aÞXi

j¼0½ðjþ 1Þ1�a � j1�a�Dþyi�j

¼ cðxiÞyi þ dðxiÞyiyi�m; i ¼ 1; . . . ;N ð3:1Þ

yi ¼ /i i ¼ �m;�mþ 1; . . . ; 0: ð3:2Þ

On simplification the discrete problem (3.1), (3.2) reduces toa system of (N+ 1) linear difference equations given by

Ay ¼ f; ð3:3Þ

where y = Æy0, . . . ,yNæt is an unknown vector and f= Æf0, -. . . , fNæt and A= [ai,k] are known factors. The nonzero entriesof the system matrix are given by:

a0;0 ¼ 1

For i ¼ 1

a1;0 ¼ h�a

Cð2�aÞ ð1� 21�aÞa1;1 ¼ h�a

Cð2�aÞ ð21�a � 2Þ � cðx1Þ

a1;2 ¼ h�a

Cð2�aÞFor i ¼ 2; . . . ;m; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a þ ði� pÞ1�a�ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � cðxiÞ

ai;iþ1 ¼ h�a

Cð2�aÞFor i ¼ 0; . . . ;m

fi ¼ dðxiÞyiyi�mwhere yi�m = /i�m.

For i¼mþ1; . . . ;N; p¼ 1; . . . ; i�1

ai;0 ¼ h�a

Cð2�aÞ ði1�a�ðiþ1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði�pþ2Þ1�a�2ði�pþ1Þ1�aþði�pÞ1�a��dðxiÞyiyi�mai;i ¼ h�a

Cð2�aÞ ð21�a�2Þ� cðxiÞ

ai;iþ1 ¼ h�a

Cð2�aÞ :

To calculate unknown vector y = Æy0, . . . ,ym, ym+1 . . . ,yN-

æt = ÆY1, Y2æt, the system matrix (3.3) can be rewritten asfollow:

A1 0

A1 A2

� �ðNþ1Þ�ðNþ1Þ

Y1

Y2

� �ðNþ1Þ�1

¼f1

f2

� �ðNþ1Þ�1

ð3:4Þ

At first, the following submatrix equation can be solved toobtain Y1

588 B. Parsa Moghaddam, Z. Salamat Mostaghim

A1Y1 ¼ f1 ð3:5Þ

where A1 is a matrix with (m+ 1) · (m+ 1)entries. The resultis used in the next submatrix equation:

A1Y1 þ A2Y2 ¼ f2 ð3:6Þ

giving Y2, where A2 is a matrix with (m + 1) · (m + 1) entries.Finally, y= ÆY1, Y2æt can be calculated.

4. Illustrative examples

To illustrate the effectiveness of the proposed method in thispaper, some test examples are in this section. The results ob-

tained by the present methods reveal that the present methodis very effective and convenient for FDDEs. It is worth men-tioning that in order to analyze the suggested method care-fully, it is compared with the method suggested in article [32]

and the amount of errors from those two is brought in Tables1–4. The algorithm programs presented is also written by Ma-ple software.

Remark 4.1. The maximum absolute error for the consideredexamples is calculated using the double mesh principle [47]because the exact solution for the considered examples is notavailable.

Table 4 The maximum absolute error of methods in Example

4.4.

a = 0.97 FFD method EABMPC method

N 100 200

d = 0.47 0.0000140 0.00347

d = 0.74 0.0000025 0.00174

Table 1 The maximum absolute error of methods in Example

4.1.

a = 0.98 FFD method EABMPC method

N 100 200

d = 8 0.0000995203 0.035237086

d = 11 0.000043096 0.0478095970

Table 2 The maximum absolute error of methods in Example

4.2.

a = 0.98 FFD method EABMPC method

N 100 200

d = 1 0.0000748422 0.016694116

d = 3 0.0000427045 0.061209148

Table 3 The maximum absolute error of methods in Example

4.3.

a = 0.98 FFD method EABMPC method

N 100 100

d = 1 0.006479257 0.020835822

d = 3 0.007163797 0.076282093

Maximum absolute error ¼ Max06i 6N

yNi � y2N2i�� ��

Example 4.1. In Hutchinson model which is related to the rate

of population growth, the rate of population growth in eachspecific time is dependent on some unique relationships on thattime, and its equation is given as [44]:

0DaxyðxÞ ¼ ryðxÞ� r

kyðxÞyðx� dÞ; x2 ½0;b�; 0< a 6 1 ð4:1Þ

under the following interval and boundary condition:

yðxÞ ¼ 0:5 � d 6 x 6 0

To obtain y(b), by applying Rieman–Liouville fractional inte-gration with respect x from the left on both sides of Eq. (4.1),

we yield:

yðbÞ ¼ 0:5r� ð0:5Þ2 rk

h i Cð1ÞCðaþ 1Þ b

a þ 0:5:

The nonzero entries of A1 submatrix are given by:

a0;0 ¼ 1

For i ¼ 1

a1;0 ¼ h�a

Cð2�aÞ ð1� 21�aÞa1;1 ¼ h�a

Cð2�aÞ ð21�a � 2Þ � r

a1;2 ¼ h�a

Cð2�aÞFor i ¼ 2; . . . ;m; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a þ ði� pÞ1�a�ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � r

ai;iþ1 ¼ h�a

Cð2�aÞFor i ¼ 0; . . . ;m

fi ¼ �0:5 rkyiyi�m

the nonzero entries of A2 submatrix are given by:

Figure 1 The numerical solution of Example 4.1 (d = 8,

r = 0.15, k = 1).

Figure 2 The numerical solution of Example 4.1 (d = 11,

r = 0.15, k = 1).

Figure 4 Comparison of numerical solutions for Example 4.1

(d = 11, r = 0.15, k= 1, a = 0.98).

Figure 3 Comparison of numerical solutions for Example 4.1

(d = 8, r= 0.15, k = 1, a = 0.98).

A novel matrix approach to fractional finite difference 589

For i¼mþ1; . . . ;N; p¼ 1; . . . ; i�1

ai;0¼ h�a

Cð2�aÞ ði1�a�ðiþ1Þ1�aÞ

ai;p¼ h�a

Cð2�aÞ ½ði�pþ2Þ1�a�2ði�pþ1Þ1�aþði�pÞ1�a�þ rkyiyi�m

ai;i¼ h�a

Cð2�aÞ ð21�a�2Þ� r

ai;iþ1¼ h�a

Cð2�aÞ

The results are shown in Table 1 and Figs. 1–4.

Example 4.2. The Verhulst–Pearl model is related to theimpact of a specific factor on the changes of population inan area whose equation is as [45]:

0DaxyðxÞ¼ cyðxÞ� cyðxÞyðx�dÞ; x2 ½0;b�; 0< a 6 1 ð4:2Þ

Under the following interval and boundary condition:

yðxÞ ¼ 0:1 � d 6 x 6 0

Similar to Example 4.1, we have

yðbÞ ¼ 0:09Cð1Þ

Cðaþ 1Þ cba þ 0:1:

The nonzero entries of A1 submatrix are given by:

a0;0 ¼ 1

For i ¼ 1

a1;0 ¼ h�a

Cð2�aÞ ð1� 21�aÞa1;1 ¼ h�a

Cð2�aÞ ð21�a � 2Þ � a

a1;2 ¼ h�a

Cð2�aÞFor i ¼ 2; . . . ;m; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a þ ði� pÞ1�a�ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � a

ai;iþ1 ¼ h�a

Cð2�aÞFor i ¼ 0; . . . ;m

fi ¼ �0:1cyiyi�m

Other nonzero entries of the main system are given by:

For i ¼ mþ 1; . . . ;N; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a

þði� pÞ1�a� þ yiyi�m

ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � a

ai;iþ1 ¼ h�a

Cð2�aÞ

The results are shown in Table 2 and Figs. 5–8.

Figure 5 The numerical solution of Example 4.2 (d = 1,

a= 0.3).

Figure 7 Comparison of numerical solutions for Example 4.2

(d = 1, a= 0.3, a = 0.98).

Figure 6 The numerical solution of Example 4.2 (d = 3,

a= 0.3).

Figure 8 Comparison of numerical solutions for Example 4.2

(d = 3, a= 0.3, a = 0.98).

590 B. Parsa Moghaddam, Z. Salamat Mostaghim

Example 4.3. The following model is related to the negativeimpact of population growth in a specific time, this model isbased on Hutchinson model when is rather than one, and itsequation is as [46]:

0DaxyðxÞ ¼ cyðxÞ� cyðxÞyðx� dÞ; x2 ½0;b�; 0< a 6 1 ð4:3Þ

Under the following interval and boundary condition:

yðxÞ ¼ 0:5� d 6 x 6 0

and

yðbÞ ¼ 0:25Cð1Þ

Cðaþ 1Þ cba þ 0:5:

The nonzero entries of A1 submatrix are given by:

a0;0 ¼ 1

For i ¼ 1

a1;0 ¼ h�a

Cð2�aÞ ð1� 21�aÞa1;1 ¼ h�a

Cð2�aÞ ð21�a � 2Þ � a

a1;2 ¼ h�a

Cð2�aÞFor i ¼ 2; . . . ;m; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a þ ði� pÞ1�a�ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � a

ai;iþ1 ¼ h�a

Cð2�aÞFor i ¼ 0; . . . ;m

fi ¼ �0:5cyiyi�m

Figure 11 Comparison of numerical solutions for Example 4.3

(d = 1, a= 1.8, a = 0.98).Figure 9 The numerical solution of Example 4.3 (d = 1,

a= 1.8).

Figure 10 The numerical solution of Example 4.3 (d = 3,

a= 1.8).

Figure 12 Comparison of numerical solutions for Example 4.3

(d = 3, a= 1.8, a = 0.98).

A novel matrix approach to fractional finite difference 591

the nonzero entries of A2 submatrix are given by

For i ¼ mþ 1; . . . ;N; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a

þði� pÞ1�a� � cyiyi�m

ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � c

ai;iþ1 ¼ h�a

Cð2�aÞ

The results are shown in Table 3 and Figs. 9–12.

Example 4.4. In this example we consider a model of the infa-mous fractional order version of the four years life cycle of apopulation of lemmings [32]:

DaxyðxÞ¼ cyðxÞ�dyðxÞyðx�dÞ; x2 ½0;b�; 0< a 6 1 ð4:4Þ

Under the following interval and boundary condition:

yðxÞ ¼ 19; �d 6 x 6 0

and

yðbÞ ¼ ½19c� ð19Þ2d� Cð1ÞCðaþ 1Þ b

a þ 19:

Figure 13 The numerical solution of Example 4.4 (d = 0.74,

c= 3.5, d= 0.18).

Figure 15 Comparison of numerical solutions for Example 4.4

(d = 0.47, a= 3.5, b= 0.18, a = 0.97).

Figure 14 The numerical solution of Example 4.4 (d = 0.74,

c= 3.5, d= 0.18). Figure 16 Comparison of numerical solutions for Example 4.4

(d = 0.74, c = 3.5, d= 0.18, a = 0.97).

592 B. Parsa Moghaddam, Z. Salamat Mostaghim

The nonzero entries of A1 submatrix are given by:

a0;0 ¼ 1

For i ¼ 1

a1;0 ¼ h�a

Cð2�aÞ ð1� 21�aÞa1;1 ¼ h�a

Cð2�aÞ ð21�a � 2Þ � c

a1;2 ¼ h�a

Cð2�aÞFor i ¼ 2; . . . ;m; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a þ ði� pÞ1�a�ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � a

ai;iþ1 ¼ h�a

Cð2�aÞFor i ¼ 0; . . . ;m

fi ¼ �19dyiyi�m:

The nonzero entries of A2 submatrix are given by

For i ¼ mþ 1; . . . ;N; p ¼ 1; . . . ; i� 1

ai;0 ¼ h�a

Cð2�aÞ ði1�a � ðiþ 1Þ1�aÞ

ai;p ¼ h�a

Cð2�aÞ ½ði� pþ 2Þ1�a � 2ði� pþ 1Þ1�a þ ði� pÞ1�a�þdyiyi�m

ai;i ¼ h�a

Cð2�aÞ ð21�a � 2Þ � c

ai;iþ1 ¼ h�a

Cð2�aÞ

The results are shown in Table 4 and Figs. 13–16.

5. Conclusion and discussion

In this paper, a boundary value problem for nonlinear frac-tional delay differential equation is considered. With this ma-

A novel matrix approach to fractional finite difference 593

trix scheme which is based on fractional finite differences, anapproximate solution for solving different kinds of boundaryproblems with fractional order is obtained. There is not an ex-

act solution for such examples, so in tables the maximum abso-lute error is calculated using the double mesh principle [47].The effects of delay on the value of error were also studied.

It was shown that by increasing the delay, the value of errorwas also increased. The graphs of the solutions of the consid-ered examples for different value of delay d and a are plotted in

Figs. 1–16 to examine the effect of delay in producingfluctuations.

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594 B. Parsa Moghaddam, Z. Salamat Mostaghim

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Behrouz Parsa Moghaddam received his M.S

and Ph.D in applied mathematics from the

University of Guilan (2005, 2009) under the

supervision of Professor Arman Aghili. He is

currently an assistant professor of applied

mathematics in Islamic Azad University,

Lahijan Branch. His research interests include

integral transforms, orthogonal function and

boundary value problems.

Z. Salamat Mostaghim received her M.Sc in

applied mathematics in 2012 from the Lahijan

Islamic Azad University (2012) under the

supervision of Dr. Parsa Moghaddam. Her

main research interests include numerical

methods for solving fractional delay differen-

tial equations and Boundary value problems.