Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 434753, 7 pageshttp://dx.doi.org/10.1155/2013/434753
Research ArticleNumerical and Analytical Study for Fourth-OrderIntegro-Differential Equations Using a Pseudospectral Method
N. H. Sweilam,1 M. M. Khader,2 and W. Y. Kota3
1 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt2 Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt3 Department of Mathematics, Faculty of Science, Mansoura University, Damietta 35516, Egypt
Correspondence should be addressed to M. M. Khader; [email protected]
Received 16 July 2012; Accepted 2 December 2012
Academic Editor: Pedro Ribeiro
Copyright Ā© 2013 N. H. Sweilam et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement ofthe unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formulaof the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points tosystem of algebraic equations with shifted Chebyshev coefficients. Thus, by solving this system of equations, the shifted Chebyshevcoefficients are obtained. Special attention is given to study the convergence analysis and derive an upper bound of the error of thepresented approximate formula. Numerical results are performed in order to illustrate the usefulness and show the efficiency andthe accuracy of the present work.
1. Introduction
The integro-differential equation (IDE) is an equation thatinvolves both integrals and derivatives of an unknown func-tion. Mathematical modeling of real-life problems usuallyresults in functional equations, like ordinary or partialdifferential equations, and integral and integro-differentialequations, stochastic equations. Many mathematical formu-lations of physical phenomena contain integro-differentialequations; these equations arise in many fields like physics,astronomy, potential theory, fluid dynamics, biological mod-els, and chemical kinetics. Integro-differential equations;are usually difficult to solve analytically; so, it is requiredto obtain an efficient approximate solution [1ā5]. Recently,several numerical methods to solve IDEs have been givensuch as variational iterationmethod [6, 7], homotopy pertur-bation method [8, 9], spline functions expansion [10, 11], andcollocation method [12ā15].
Chebyshev polynomials arewell-known family of orthog-onal polynomials on the interval [ā1, 1] that have manyapplications [4, 6, 8, 13].They are widely used because of theirgood properties in the approximation of functions. However,with our best knowledge, very little work was done to
adapt these polynomials to the solution of integro-differentialequations. Orthogonal polynomials have a great variety andwealth of properties. Some of these properties take a veryconcise form in the case of the Chebyshev polynomials, mak-ing Chebyshev polynomials of leading importance amongorthogonal polynomials.The Chebyshev polynomials belongto an exclusive band of orthogonal polynomials, known asJacobi polynomials, which correspond to weight functions ofthe form (1 ā š„)š¼(1 + š„)š½ and which are solutions of Sturm-Liouville equations [16].
In this work, we derive an approximate formula of theintegral derivative š¦(š)(š„) and derive an upper bound of theerror of this formula, and then we use this formula to solvea class of two-point boundary value problems (BVPs) for thefourth-order integro-differential equations as
š¦(šš£)
(š„) = š (š„)+š¾š¦ (š„)+ā«š„
0
[š (š”) š¦ (š”)+š (š”) Ī (š¦ (š”))] šš”,
0ā¤š„, š”ā¤1,
(1)
2 Mathematical Problems in Engineering
under the boundary and initial conditions
š¦ (0) = š¼0, š¦
(0) = š¼1,
š¦ (1) = š½0, š¦
(1) = š½1,
(2)
where š(š„), š(š„), and š(š„) are known functions and š¾, š¼0,
š¼1, š½0, andš½
1are suitable constants. Several numericalmeth-
ods to solve the fourth-order integro-differential equationshave been given such as Chebyshev cardinal functions [17],variational iteration method [7], and others.
2. Some Basic Properties and Derivation of anApproximate Formula of the Derivative forChebyshev Polynomials Expansion
The Chebyshev polynomial of the first kind is a polynomialin š§ of degree š, defined by the relation
šš(š§) = cos šš, when š§ = cos š. (3)
The Chebyshev polynomials of degree š > 0 of the first kindhave precisely š zeros and š + 1 local extrema in the interval[ā1, 1]. The zeros of š
š(š§) are denoted by
š§š= cos (š ā 1/2) š
š, š = 1, 2, . . . , š. (4)
The Chebyshev polynomials can be determined with the aidof the following recurrence formula [18]:
šš+1
(š§) = 2š§šš(š§) ā š
šā1(š§) ,
š0(š§) = 1, š
1(š§) = š§, š = 1, 2, . . . .
(5)
The analytic form of the Chebyshev polynomials šš(š§) of
degree š is given by
šš(š§) = š
[š/2]
āš=0
(ā1)š
2šā2šā1 (š ā š ā 1)!
(š)! (š ā 2š)!š§šā2š
, (6)
where [š/2] denotes the integer part of š/2.The orthogonalitycondition is
ā«1
ā1
šš(š§) šš(š§)
ā1 ā š§2šš„ =
{{{{{{
{{{{{{
{
š, for š = š = 0;
š
2, for š = š Ģø= 0;
0, for š Ģø= š.
(7)
In order to use these polynomials on the interval [0, 1],we define the so called shifted Chebyshev polynomials byintroducing the change of variable š§ = 2š„ ā 1. The shiftedChebyshev polynomials are denoted by šā
š(š„) and defined as
šā
š(š„) = š
š(2š„ ā 1) = š
2š(āš„).
The function š¦(š„), which belongs to the space of squareintegrable in [0, 1], may be expressed in terms of shiftedChebyshev polynomials as
š¦ (š„) =
ā
āš=0
šššā
š(š„) , (8)
where the coefficients ššare given by
š0=
1
šā«1
0
š¦ (š„) šā
0(š„)
āš„ ā š„2šš„, š
š=
2
šā«1
0
š¦ (š„) šā
š(š„)
āš„ ā š„2šš„,
š = 1, 2, . . . .
(9)
In practice, only the first (š + 1) terms of shiftedChebyshev polynomials are considered. Then, we have that
š¦š(š„) =
š
āš=0
šššā
š(š„) . (10)
Lemma 1. The analytic form of the shifted Chebyshev polyno-mials šā
š(š„) of degree š is given by
šā
š(š„) = š
š
āš=0
(ā1)šāš
22š
(š + š ā 1)!
(2š)! (š ā š)!š„š
, š = 1, 2, . . . .
(11)
Proof. Since we have šāš(š„) = š
2š(āš„), then by substituting
in (6), we can obtain that
šā
š(š„) = 2š
š
āš=0
(ā1)š22šā2šā1
(2š ā š ā 1)!
(š)! (2š ā 2š)!š„šāš
,
š = 1, 2, . . . .
(12)
Now, we put š = š ā š in (12) we obtain the desired result(11).
Themain approximate formula of the derivative of š¦š(š„),
and is given in the following theorem.
Theorem 2. Let š¦(š„) be approximated by shifted Chebyshevpolynomials as (10), and also suppose that š is integer; then,
š·š
(š¦š(š„)) =
š
āš=š
š
āš=š
šššš,š,š
š„šāš
, (13)
where šš,š,š
is given by
šš,š,š
= (ā1)šāš
22š
š (š + š ā 1)!š!
(š ā š)! (2š)! (š ā š)!. (14)
Proof. Since the differential operator š·š is linear, we canobtain that
š·š
(š¦š(š„)) =
š
āš=0
ššš·š
(šā
š(š„)) . (15)
Sinceš·šš = 0, š is a constant, and
š·š
š„š
={
{
{
0, for š ā š, š < š,š!
(š ā š)!š„šāš
, for š ā š, š ā„ š. (16)
Then, we have that
š·š
šā
š(š„) = 0, š = 0, 1, . . . , š ā 1, (17)
Mathematical Problems in Engineering 3
and for š = š, š + 1, . . . , š, and by using (16), we get that
š·š
šā
š(š„) = š
š
āš=š
(ā1)šāš
22š
(š + š ā 1)!
(š ā š)! (2š)!š·š
š„š
= š
š
āš=š
(ā1)šāš
22š
(š + š ā 1)!š!
(š ā š)! (2š)! (š ā š)!š„šāš
.
(18)
A combination of (17), (18), and (14) leads to the desired resultand completes the proof of the theorem.
3. Error Analysis
In this section, special attention is given to study the conver-gence analysis and evaluate the upper bound of the error ofthe proposed formula.
Theorem 3 (Chebyshev truncation theorem; see [18]). Theerror in approximating š¦(š„) by the sum of its first š terms isbounded by the sum of the absolute values of all the neglectedcoefficients. If
š¦š(š„) =
š
āš=0
šššš(š„) , (19)
then
šøš(š) ā”
š¦ (š„) ā š¦š (š„) ā¤
ā
āš=š+1
šš , (20)
for all š¦(š„), allš, and all š„ ā [ā1, 1].
Theorem 4. The derivative of order š for the shifted Chebyshevpolynomials can be expressed in terms of the shifted Chebyshevpolynomials themselves in the following form:
š·š
(šā
š(š„)) =
š
āš=š
šāš
āš=0
Īš,š,š
šā
š(š„) , (21)
where
Īš,š,š
=(ā1)šāš
2š (š + š ā 1)!Ī (š ā š + 1/2)
āšĪ (š + 1/2) (š ā š)! (š ā š ā š)! (š + š ā š)!
,
ā0= 2, ā
š= 1, š = 0, 1, . . . .
(22)
Proof. We use the properties of the shifted Chebyshev poly-nomials [18] and expand š„šāš in (18) in the following form:
š„šāš
=
šāš
āš=0
ššššā
š(š„) , (23)
where šššcan be obtained using (9), andš¦(š„) = š„šāš; then,
ššš
=2
āšš
ā«1
0
š„šāš
šā
š(š„)
āš„ ā š„2šš„, ā
0= 2, ā
š= 1,
š = 1, 2, . . . .
(24)
At š = 0, we find that šš0
= (1/š) ā«10
(š„šāš
šā
0(š„)/
āš„ ā š„2)šš„ = (1/āš)(Ī(š ā š + 1/2)/(š ā š)!); also, at anyš and using the formula (10), we can find that
ššš
=š
āš
š
āš=0
(ā1)šāš
(š + š ā 1)!22š+1
Ī (š + š ā š + 1/2)
(š ā š)! (2š)! (š + š ā š)!,
š = 1, 2, 3, . . . ,
(25)
employing (18) and (23) gives
š·š
(šā
š(š„)) =
š
āš=š
šāš
āš=0
Īš,š,š
šā
š(š„) , š = š, š + 1, . . . , (26)
where
Īš,š,š =
{{{{{{{{{{{{
{{{{{{{{{{{{
{
š
(ā1)šāš(š + š ā 1)!2
2šš!Ī (š ā š + 1/2)
(š ā š)! (2š)!āš (Ī (š + 1 ā š))2, š = 0;
(ā1)šāššš (š + š ā 1)!2
2š+1š!
āš (š ā š)! (š ā š)! (2š)!
Ć
š
ā
š=0
(ā1)šāš(š + š ā 1)!2
2šĪ (š + š ā š + 1/2)
(š ā š)! (2š)! (š + š ā š)!
, š = 1, 2, 3, . . . .
(27)
After some lengthy manipulation, Īš,š,š
can be put in thefollowing form:
Īš,š,š
=(ā1)šāš
2š (š + š ā 1)!Ī (š ā š + 1/2)
āšĪ (š + 1/2) (š ā š)! (š ā š ā š)! (š + š ā š)!
,
š = 0, 1, . . . ,
(28)
and this completes the proof of the theorem.
Theorem 5. The error |šøš(š)| = |š·
š
š¦(š„) ā š·š
š¦š(š„)| in
approximatingš·šš¦(š„) byš·šš¦š(š„) is bounded by
šøš (š) ā¤
ā
āš=š+1
šš(
š
āš=š
šāš
āš=0
Īš,š,š
)
. (29)
Proof. A combination of (8), (10), and (21) leads to
šøš (š) =
š·š
š¦ (š„) ā š·š
š¦š(š„)
=
ā
āš=š+1
šš(
š
āš=š
šāš
āš=0
Īš,š,š
šā
š(š„))
,(30)
but |šāš(š„)| ā¤ 1; so, we can obtain that
šøš (š) ā¤
ā
āš=š+1
šš(
š
āš=š
šāš
āš=0
Īš,š,š
)
, (31)
and subtracting the truncated series from the infinite series,bounding each term in the difference, and summing thebounds complete the proof of the theorem.
4 Mathematical Problems in Engineering
4. Procedure Solution for the Fourth-OrderIntegro-Differential Equation
In this section, we will present the proposed method to solvenumerically the fourth-order integro-differential equation ofthe form in (1).The unknown functionš¦(š„)may be expandedby finite series of shifted Chebyshev polynomials as in thefollowing approximation:
š¦š(š„) =
š
āš=0
šššā
š(š„) , (32)
and approximated formula of its derivatives can be defined inTheorem 2. From (1), (32), andTheorem 2, we have that
š
āš=š
š
āš=š
šššš,š,š
š„šāš
= š (š„) + š¾
š
āš=0
šššā
š(š„)
+ ā«š„
0
[š (š”) (
š
āš=0
šššā
š(š”))
+ š (š”) Ī(
š
āš=0
šššā
š(š”))] šš”.
(33)
We now collocate (33) at (š ā 1 + š) points š„š , š =
0, 1, . . . , š ā š as
š
āš=4
š
āš=4
šššš,š,4
š„šā4
š
= š (š„š ) + š¾
š
āš=0
šššā
š(š„š )
+ ā«š„š
0
[š (š”) (
š
āš=0
šššā
š(š”))
+ š (š”) Ī(
š
āš=0
šššā
š(š”))] šš”.
(34)
For suitable collocation points, we use roots of shifted Cheby-shev polynomial šā
š+1āš(š„). The integral terms in (34) can be
found using composite trapezoidal integration technique as
ā«š„š
0
[š (š”) (
š
āš=0
šššā
š(š”)) + š (š”) Ī(
š
āš=0
šššā
š(š”))] šš”
ā āš
2(Ī© (š”
0) + Ī© (š”
šæ) + 2
šæā1
āš=1
Ī©(š”š)) ,
(35)
where Ī©(š”) = š(š”) āšš=0
šššā
š(š”) + š(š”)Ī(ā
š
š=0šššā
š(š”)), ā
š =
š„š /šæ, for an arbitrary integer šæ, š”
š+1= š”š+ āš , š = 0, 1, . . . ,
š ā š, and š = 0, 1, . . . , šæ. So, by using (34) and (35), weobtain
š
āš=4
š
āš=4
šššš,š,š
š„šāš
š
= š (š„š ) + š¾
š
āš=0
šššā
š(š„š )
+āš
2(Ī© (š”
0) + Ī© (š”
šæ) + 2
šæā1
āš=1
Ī©(š”š)) .
(36)
Also, by substituting (32) in the boundary conditions (2), wecan obtain š equations as follows:
š
āš=0
(ā1)š
šš= š¼0,
š
āš=0
šš= š½0,
š
āš=2
šššā
š
(0) = š¼1,
š
āš=2
šššā
š
(1) = š½1.
(37)
Equation (36), together with š equations of the bound-ary conditions (37), give (š + 1) of system of algebraicequations which can be solved, for the unknowns š
š, š =
0, 1, . . . , š, using conjugate gradient method or Newtoniteration method.
5. Numerical Results
In this section, to verify the validity and the accuracy andsupport our theoretical discussion of the proposed method,we give some computations results of numerical examples.
Example 6. Consider the nonlinear fourth-order integro-differential equation as in (1) and (2) with š(š„) = 1, š¾ =0, š(š”) = 0, š(š”) = š
āš”
, and Ī(š¦) = š¦2(š„); then, theintegro-differential equation will be
š¦(šš£)
(š„) = 1 + ā«š„
0
šāš”
š¦2
(š”) šš”, 0 ā¤ š„ ā¤ 1, (38)
subject to the boundary conditions
š¦ (0) = š¦
(0) = 1, š¦ (1) = š¦
(1) = š. (39)
The exact solution of this problem is š¦(š„) = šš„ [7].We apply the suggested method with š = 5 and
approximate the solution š¦(š„) as follows:
š¦5(š„) =
5
āš=0
šššā
š(š„) . (40)
From (38), (40), andTheorem 2, we have that5
āš=4
š
āš=4
šššš,š,4
š„šā4
= 1 + ā«š„
0
šāš”
(
5
āš=0
šššā
š(š”))
2
šš”. (41)
We now collocate (41) at points, š„š , š = 0, 1 as
5
āš=4
š
āš=4
šššš,š,4
š„šā4
š = 1 + ā«
š„š
0
šāš”
(
5
āš=0
šššā
š(š”))
2
šš”. (42)
Mathematical Problems in Engineering 5
For suitable collocation points we use roots of shifted Cheby-shev polynomial šā
2(š„). The integral terms in (42) can be
found using composite trapezoidal integration technique as
ā«š„š
0
šāš”
(
5
āš=0
šššā
š(š”))
2
šš”
=āš
2(Ī© (š”
0) + Ī© (š”
šæ) + 2
šæā1
āš=1
Ī©(š”š)) ,
(43)
where Ī©(š”) = šāš”(ā5š=0
šššā
š(š”))2, āš = š„š /šæ, for an arbitrary
integer šæ, š”š+1
= š”š+ āš , š = 0, 1, and š = 0, 1, . . . , šæ. So,
by using (43) and (42), we obtain
5
āš=4
š
āš=4
šššš,š,4
š„šā4
š
= 1 +āš
2(Ī© (š”
0) + Ī© (š”
šæ) + 2
šæā1
āš=1
Ī©(š”š)) .
(44)
Also, by substituting (40) in the boundary conditions (39), wecan obtain four equations as follows:
š0ā š1+ š2ā š3+ š4ā š5= 1,
š0+ š1+ š2+ š3+ š4+ š5= š,
š0š0+ š1š1+ š2š2+ š3š3+ š4š4+ š5š5= 1,
š 0š0+ š 1š1+ š 2š2+ š 3š3+ š 4š4+ š 5š5= š,
(45)
where šš= šā
š
(0) and š š= šā
š
(1).Equation (44), together with four equations of the bound-
ary conditions (45), represent, a nonlinear system of sixalgebraic equations in the coefficients š
š; by solving it using
the Newton iteration method, we obtain
š0= 1.75379, š
1= 0.85039, š
2= 0.10478,
š3= 0.00872, c
4= 0.00057, š
5= 0.00003.
(46)
The behavior of the approximate solution using the proposedmethod with š = 5, the approximate solution usingvariational iteration method (VIM), and the exact solutionare presented in Figure 1. Table 1 shows the behavior ofthe absolute error between exact solution and approximatesolution using the presented method at š = 6 and š = 8.FromFigure 1 andTable 1, it is clear that the proposedmethodcan be considered as an efficient method to solve the non-linear integro-differential equations. Table 1 indicates thatas š increases the errors decrease more rapidly; hence, forbetter results, using number š is recommended. Also, wecan conclude that the obtained approximated solution is inexcellent agreement with the exact solution.
Example 7. Consider the linear fourth-order integro-differential equation as in (1) and (2) withš(š„) = š„+(š„+3)šš„,
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.80 0.2 0.4 0.6 0.8 1
š¦(š„)
š„
Exact solution š¦(š„)Chebyshev solutionVIM solution
Figure 1: The behavior of the exact solution, the approximate solu-tion using VIM, and the approximate solution using the proposedmethod atš = 5.
š¾ = 1, š(š”) = ā1, ā(š”) = 0, and Ī(š¦) = š¦(š„); then, theintegro-differential equation will be
š¦(šš£)
(š„) = š„ + (š„ + 3) šš„
+ š¦ (š„)
ā ā«š„
0
š¦ (š”) šš”, 0 ā¤ š„ ā¤ 1,
(47)
subject to the boundary conditions
š¦ (0) = 1, (1) = 1 + š,
š¦
(0) = 2, š¦
(1) = 3š.(48)
The exact solution of this problem is š¦(š„) = 1 + š„šš„ [17].We apply the suggested method with š = 5 and
approximate the solution š¦(š„) as follows:
š¦ (š„) ā
5
āš=0
šššā
š(š„) . (49)
By the same procedure in the previous example, we have
5
āš=4
š
āš=4
šššš,š,4
š„šā4
š
= š (š„š ) +
5
āš=0
šššā
š(š„š ) +
āš
2
Ć (Ī© (š”0) + Ī© (š”
šæ) + 2
šæā1
āš=1
Ī©(š”š)) , š = 0, 1, 2,
(50)
6 Mathematical Problems in Engineering
Table 1: The behavior of the absolute error between the exactsolution and approximate solution atš = 6 andš = 8.
š„ |š¦ex. ā š¦ap.| atš = 6 |š¦ex. ā š¦ap.| atš = 80.0 2.2548š ā 10 2.0254š ā 100.2 2.3654š ā 04 1.2548š ā 060.4 3.5687š ā 04 3.2541š ā 060.6 0.1587š ā 04 5.2548š ā 060.8 9.2450š ā 04 7.2581š ā 061.0 1.2589š ā 10 2.2548š ā 10
Table 2: The behaviour of the absolute error between the exactsolution and approximate solution atš = 7 andš = 9.
š„ |š¦ex. ā š¦ap.| atš = 7 |š¦ex. ā š¦ap.| atš = 90.0 1.2587š ā 08 5.1236š ā 090.2 6.2548š ā 03 2.2258š ā 050.4 2.0254š ā 03 9.2154š ā 050.6 1.3654š ā 03 2.0054š ā 050.8 0.2540š ā 03 2.3690š ā 051.0 6.0254š ā 08 5.2478š ā 09
where Ī©(š”) = ā5š=0
šššā
š(š”), and the nodes š”
š+1= š”š+ āš , š =
0, 1, . . . , šæ, š”0
= 0, and āš = š„š /šæ. We can write the initi-
alboundary conditions in the form
š0ā š1+ š2ā š3+ š4ā š5+ š6= 1,
š0+ š1+ š2+ š3+ š4+ š5+ š6= 1 + š,
š0š0+ š1š1+ š2š2+ š3š3+ š4š4+ š5š5+ š6š6= 2,
š 0š0+ š 1š1+ š 2š2+ š 3š3+ š 4š4+ š 5š5+ š 6š6= 3š.
(51)
By using (50) and (51), we obtain a linear system of sevenalgebraic equations in the coefficients š
š; by solving it using
the conjugate gradient method, we obtain
š0= 2.09189, š
1= 1.32820, š
2= 0.26461,
š3= 0.03079, š
4= 0.00264, š
5= 0.00015.
(52)
The behavior of the approximate solution using the proposedmethod with š = 6, the approximate solution usingvariational iteration method (VIM) and the exact solutionare presented in Figure 2. Table 2 shows the behaviour ofthe absolute error between exact solution and approximatesolution using the presented method at š = 7 and š =9. From this figure, it is clear that the proposed methodcan be considered as an efficient method to solve the linearintegro-differential equations. Also, we can conclude that theobtained approximate solution is in excellent agreement withthe exact solution.
6. Conclusion and Discussion
Integro-differential equations are usually difficult to solveanalytically; so, it is required to obtain the approximate solu-tion. In this paper, we proposed the pseudospectral method
Exact solution š¦(š„)Chebyshev solutionVIM solution
4
3.5
3
2.5
2
1.5
1
0.50 0.2 0.4 0.6 0.8 1
š¦(š„)
š„
Figure 2: The behavior of the exact solution, the approximate solu-tion using VIM, and the approximate solution using the proposedmethod atš = 6.
using shifted Chebyshev method for solving the integro-differential equations. The Chebyshev method is useful foracquiring both the general solution and particular solutionas demonstrated in examples. Special attention is given tostudy the convergence analysis and derive an upper boundof the error of the derived approximate formula. From ourobtained results, we can conclude that the proposed methodgives solutions in excellent agreement with the exact solutionand better than the other methods. An interesting featureof this method is that when an integral system has linearlyindependent polynomial solution of degreeš or less thanš,themethod can be used for finding the analytical solution. Allcomputations are done using MATLAB 8.
References
[1] R. P. Agarwal, āBoundary value problems for higher orderintegro-differential equations,ā Nonlinear Analysis: Theory,Methods & Applications, vol. 7, no. 3, pp. 259ā270, 1983.
[2] E. Babolian, F. Fattahzadeh, and E. G. Raboky, āA Chebyshevapproximation for solving nonlinear integral equations ofHam-merstein type,āApplied Mathematics and Computation, vol. 189,no. 1, pp. 641ā646, 2007.
[3] A. H. Borzabadi, A. V. Kamyad, and H. H. Mehne, āA differentapproach for solving the nonlinear Fredholm integral equationsof the second kind,āAppliedMathematics and Computation, vol.173, no. 2, pp. 724ā735, 2006.
[4] M. M. Khader, āOn the numerical solutions for the fractionaldiffusion equation,ā Communications in Nonlinear Science andNumerical Simulation, vol. 16, no. 6, pp. 2535ā2542, 2011.
[5] N. H. Sweilam, M. M. Khader, and A. M. Nagy, āNumericalsolution of two-sided space-fractional wave equation usingfinite difference method,ā Journal of Computational and AppliedMathematics, vol. 235, no. 8, pp. 2832ā2841, 2011.
Mathematical Problems in Engineering 7
[6] M. M. Khader, āIntroducing an efficient modification of thevariational iteration method by using Chebyshev polynomials,āApplications and Applied Mathematics, vol. 7, no. 1, pp. 283ā299,2012.
[7] N. H. Sweilam, āFourth order integro-differential equationsusing variational iteration method,ā Computers & Mathematicswith Applications, vol. 54, no. 7-8, pp. 1086ā1091, 2007.
[8] M. M. Khader, āIntroducing an efficient modification of thehomotopy perturbation method by using Chebyshev polyno-mials,ā Arab Journal of Mathematical Sciences, vol. 18, no. 1, pp.61ā71, 2012.
[9] N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, āHomotopyperturbation method for linear and nonlinear system of frac-tional integro-differential equations,ā International Journal ofComputational Mathematics and Numerical Simulation, vol. 1,no. 1, pp. 73ā87, 2008.
[10] M. M. Khader and S. T. Mohamed, āNumerical treatmentfor first order neutral delay differential equations using splinefunctions,ā EngineeringMathematics Letters, vol. 1, no. 1, pp. 32ā43, 2012.
[11] S. T. Mohamed and M. M. Khader, āNumerical solutions to thesecond order Fredholm integro-differential equations using thespline functions expansion,ā Global Journal of Pure and AppliedMathematics, vol. 34, pp. 21ā29, 2011.
[12] M. M. Khader and A. S. Hendy, āThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre pseudo-spectral method,ā International Journalof Pure and Applied Mathematics, vol. 74, no. 3, pp. 287ā297,2012.
[13] M.M. Khader, N. H. Sweilam, and A.M. S. Mahdy, āAn efficientnumerical method for solving the fractional difusion equation,āJournal of Applied Mathematics and Bioinformatics, vol. 1, no. 2,pp. 1ā12, 2011.
[14] N. H. Sweilam, M. M. Khader, andW. Y. Kota, āOn the numeri-cal solution of Hammerstein integral equations using Legendreapproximation,ā International Journal of Applied MathematicalResearch, vol. 1, no. 1, pp. 65ā76, 2012.
[15] S. Yousefi and M. Razzaghi, āLegendre wavelets method for thenonlinear Volterra-Fredholm integral equations,ā Mathematicsand Computers in Simulation, vol. 70, no. 1, pp. 1ā8, 2005.
[16] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials,Chapman & Hall/CRC, Washington, DC, USA, 2003.
[17] M. Lakestani and M. Dehghan, āNumerical solution of fourth-order integro-differential equations using Chebyshev cardinalfunctions,ā International Journal of Computer Mathematics, vol.87, no. 6, pp. 1389ā1394, 2010.
[18] M. A. Snyder, Chebyshev Methods in Numerical Approximation,Prentice-Hall Inc., Englewood Cliffs, NJ, USA, 1966.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of