We investigate the numerical solution of linear fractional integro-differential equations by least squares method with aid of shiftedChebyshev polynomial. Some numerical examples are presented to illustrate the theoretical results.
1. Introduction
Many problems can be modeled by fractional Integro-differential equations from various sciences and engineeringapplications. Furthermore most problems cannot be solvedanalytically, and hence finding good approximate solutions,using numerical methods, will be very helpful.
Recently, several numerical methods to solve frac-tional differential equations (FDEs) and fractional Integro-differential equations (FIDEs) have been given. The authorsin [1, 2] applied collocation method for solving the follow-ing: nonlinear fractional Langevin equation involving twofractional orders in different intervals and fractional Fred-holm Integro-differential equations. Chebyshev polynomialsmethod is introduced in [3–5] for solving multiterm frac-tional orders differential equations and nonlinear Volterraand Fredholm Integro-differential equations of fractionalorder.The authors in [6] applied variational iterationmethodfor solving fractional Integro-differential equations withthe nonlocal boundary conditions. Adomian decompositionmethod is introduced in [7, 8] for solving fractional diffu-sion equation and fractional Integro-differential equations.References [9, 10] used homotopy perturbation method forsolving nonlinear Fredholm Integro-differential equations offractional order and system of linear Fredholm fractionalIntegro-differential equations. Taylor series method is intro-duced in [11] for solving linear integrofractional differentialequations of Volterra type. The authors in [12, 13] give an
application of nonlinear fractional differential equations andtheir approximations and existence and uniqueness theoremfor fractional differential equations with integral boundaryconditions.
In this paper least squares method with aid of shiftedChebyshev polynomial is applied to solving fractionalIntegro-differential equations. Least squaresmethodhas beenstudied in [14–18].
In this paper, we are concerned with the numericalsolution of the following linear fractional Integro-differentialequation:
𝐷𝛼
𝜑 (𝑥) = 𝑓 (𝑥) + ∫
1
0
𝐾 (𝑥, 𝑡) 𝜑 (𝑡) 𝑑𝑡, 0 ≤ 𝑥, 𝑡 ≤ 1, (1)
with the following supplementary conditions:
𝜑(𝑖)
(0) = 𝛿𝑖, 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ N, (2)
where 𝐷𝛼𝜑(𝑥) indicates the 𝛼th Caputo fractional derivativeof 𝜑(𝑥); 𝑓(𝑥), 𝐾(𝑥, 𝑡) are given functions, 𝑥 and 𝑡 are realvariables varying in the interval [0, 1], and 𝜑(𝑥) is theunknown function to be determined.
2. Basic Definitions of Fractional Derivatives
In this section some basic definitions and properties offractional calculus theory which are necessary for the formu-lation of the problem are given.
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 431965, 5 pageshttp://dx.doi.org/10.1155/2014/431965
2 Mathematical Problems in Engineering
Definition 1. A real function 𝑓(𝑥), 𝑥 > 0, is said to be in thespace 𝐶
𝜇, 𝜇 ∈ R, if there exists a real number 𝑝 > 𝜇 such that
𝑓(𝑥) = 𝑥𝑝
𝑓1(𝑥), where 𝑓
1(𝑥) ∈ 𝐶[0, 1).
Definition 2. A function𝑓(𝑥), 𝑥 > 0, is said to be in the space𝐶𝑚
𝜇, 𝑚 ∈ N ∪ {0}, if 𝑓(𝑚) ∈ 𝐶
𝜇.
Definition 3. The left sided Riemann-Liouville fractionalintegral operator of order 𝛼 ≥ 0 of a function𝑓 ∈ 𝐶
𝜇, 𝜇 ≥ −1,
is defined as [19]
𝐽𝛼
𝑓 (𝑥) =1
Γ (𝛼)∫
𝑥
0
𝑓 (𝑡)
(𝑥 − 𝑡)1−𝛼
𝑑𝑡, 𝛼 > 0, 𝑥 > 0, (3)
𝐽0
𝑓 (𝑥) = 𝑓 (𝑥) . (4)
Definition 4. Let 𝑓 ∈ 𝐶𝑚
−11, 𝑚 ∈ N ∪ {0}. Then the Caputo
fractional derivative of 𝑓(𝑥) is defined as [20–22]
𝐷𝛼
𝑓 (𝑥) =
{
{
{
𝐽𝑚−𝛼
𝑓𝑚
(𝑥) , 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ N,𝐷𝑚
𝑓 (𝑥)
𝐷𝑥𝑚, 𝛼 = 𝑚.
(5)
Hence, we have the following properties:
(1) 𝐽𝛼
𝐽]𝑓 = 𝐽𝛼+]
𝑓, 𝛼, ] > 0, 𝑓 ∈ 𝐶𝜇, 𝜇 > 0,
(2) 𝐽𝛼
𝑥𝛾
=Γ (𝛾 + 1)
Γ (𝛼 + 𝛾 + 1)𝑥𝛼+𝛾
, 𝛼 > 0, 𝛾 > −1, 𝑥 > 0,
(3) 𝐽𝛼
𝐷𝛼
𝑓 (𝑥) = 𝑓 (𝑥) −
𝑚−1
∑
𝑘=0
𝑓(𝑘)
(0+
)𝑥𝑘
𝑘!,
𝑥 > 0, 𝑚 − 1 < 𝛼 ≤ 𝑚,
(4)𝐷𝛼
𝐽𝛼
𝑓 (𝑥) = 𝑓 (𝑥) , 𝑥 > 0, 𝑚 − 1 < 𝛼 ≤ 𝑚,
(5)𝐷𝛼
𝐶 = 0, 𝐶 is a constant,
(6)𝐷𝛼
𝑥𝛽
=
{{
{{
{
0, 𝛽 ∈ N0, 𝛽 < [𝛼] ,
Γ (𝛽 + 1)
Γ (𝛽 − 𝛼 + 1)𝑥𝛽−𝛼
, 𝛽 ∈ N0, 𝛽 ≥ [𝛼] ,
(6)
where [𝛼] denoted the smallest integer greater than or equalto 𝛼 and N
0= {0, 1, 2, . . .}.
3. Solution of Linear FractionalIntegro-Differential Equation
In this section the least squares method with aid of shiftedChebyshev polynomial is applied to study the numericalsolution of the fractional Integro-differential (1).
This method is based on approximating the unknownfunction 𝜑(𝑥) as
𝜑𝑛(𝑥) ≅
𝑛
∑
𝑖=0
𝑎𝑖𝑇∗
𝑖(𝑥) , 0 ≤ 𝑥 ≤ 1, (7)
where𝑇∗𝑖(𝑥) is shifted Chebyshev polynomial of the first kind
which is defined in terms of the Chebyshev polynomial 𝑇𝑛(𝑥)
by the following relation [23]:
𝑇∗
𝑛(𝑥) = 𝑇
𝑛(2𝑥 − 1) , (8)
and the following recurrence formulae:
𝑇∗
𝑛(𝑥) = 2 (2𝑥 − 1) 𝑇
∗
𝑛−1(𝑥) − 𝑇
∗
𝑛−2(𝑥) , 𝑛 = 2, 3, . . . ,
(9)
with initial conditions
𝑇∗
0(𝑥) = 1, 𝑇
∗
1(𝑥) = 2𝑥 − 1, (10)
𝑎𝑖, 𝑖 = 0, 1, 2, . . ., are constants.Substituting (7) into (1) we obtain
𝐷𝛼
(
𝑛
∑
𝑖=0
𝑎𝑖𝑇∗
𝑖(𝑥)) = 𝑓 (𝑥) + ∫
1
0
𝐾 (𝑥, 𝑡) [
𝑛
∑
𝑖=0
𝑎𝑖𝑇∗
𝑖(𝑡)] 𝑑𝑡.
(11)
Hence the residual equation is defined as
𝑅 (𝑥, 𝑎0, 𝑎1, . . . , 𝑎
𝑛)
=
𝑛
∑
𝑖=0
𝑎𝑖𝐷𝛼
𝑇∗
𝑖(𝑥) − 𝑓 (𝑥) − ∫
1
0
𝐾 (𝑥, 𝑡) [
𝑛
∑
𝑖=0
𝑎𝑖𝑇∗
𝑖(𝑡)] 𝑑𝑡.
(12)
Let
𝑆 (𝑎0, 𝑎1, . . . , 𝑎
𝑛) = ∫
1
0
[𝑅 (𝑥, 𝑎0, 𝑎1, . . . , 𝑎
𝑛)]2
𝑤 (𝑥) 𝑑𝑥,
(13)
where 𝑤(𝑥) is the positive weight function defined on theinterval [0, 1]. In this work we take 𝑤(𝑥) = 1 for simplicity.Thus
𝑆 (𝑎0, 𝑎1, . . . , 𝑎
𝑛)
= ∫
1
0
{
𝑛
∑
𝑖=0
𝑎𝑖𝐷𝛼
𝑇∗
𝑖(𝑥) − 𝑓 (𝑥)
− ∫
1
0
𝐾 (𝑥, 𝑡) [
𝑛
∑
𝑖=0
𝑎𝑖𝑇∗
𝑖(𝑡)] 𝑑𝑡}
2
𝑑𝑥.
(14)
So, finding the values of 𝑎𝑖, 𝑖 = 0, 1, . . . , 𝑛, which minimize
𝑆 is equivalent to finding the best approximation for thesolution of the fractional Integro-differential equation (1).
The minimum value of 𝑆 is obtained by setting𝜕𝑆
𝜕𝑎𝑗
= 0, 𝑗 = 0, 1, . . . , 𝑛. (15)
Applying (15) to (14) we obtain
∫
1
0
{
𝑛
∑
𝑖=0
𝑎𝑖𝐷𝛼
𝑇∗
𝑖(𝑥) − 𝑓 (𝑥) − ∫
1
0
𝐾 (𝑥, 𝑡) [
𝑛
∑
𝑖=0
𝑎𝑖𝑇∗
𝑖(𝑡)] 𝑑𝑡}
×{𝐷𝛼
𝑇∗
𝑗(𝑥) − ∫
1
0
𝐾 (𝑥, 𝑡) 𝑇∗
𝑗(𝑡) 𝑑𝑡} 𝑑𝑥.
(16)
Mathematical Problems in Engineering 3
By evaluating the above equation for 𝑗 = 0, 1, . . . , 𝑛 we canobtain a system of (𝑛 + 1) linear equations with (𝑛 + 1)
unknown coefficients 𝑎𝑖’s. This system can be formed by
using matrices form as follows:
𝐴
=
(((
(
∫
1
0
𝑅 (𝑥, 𝑎0) ℎ0𝑑𝑥 ∫
1
0
𝑅 (𝑥, 𝑎1) ℎ0𝑑𝑥 . . . ∫
1
0
𝑅 (𝑥, 𝑎𝑛) ℎ0𝑑𝑥
∫
1
0
𝑅 (𝑥, 𝑎0) ℎ1𝑑𝑥 ∫
1
0
𝑅 (𝑥, 𝑎1) ℎ1𝑑𝑥 . . . ∫
1
0
𝑅 (𝑥, 𝑎𝑛) ℎ1𝑑𝑥
...... d
...
∫
1
0
𝑅 (𝑥, 𝑎0) ℎ𝑛𝑑𝑥 ∫
1
0
𝑅 (𝑥, 𝑎1) ℎ𝑛𝑑𝑥 . . . ∫
1
0
𝑅 (𝑥, 𝑎𝑛) ℎ𝑛𝑑𝑥
)))
)
,
𝐵 =((
(
∫
1
0
𝑓 (𝑥) ℎ0𝑑𝑥
∫
1
0
𝑓 (𝑥) ℎ1𝑑𝑥
...
∫
1
0
𝑓 (𝑥) ℎ𝑛𝑑𝑥
))
)
,
(17)
where
ℎ𝑗= 𝐷𝛼
𝑇∗
𝑗(𝑥) − ∫
1
0
𝐾 (𝑥, 𝑡) 𝑇∗
𝑗(𝑡) 𝑑𝑡, 𝑗 = 0, 1, . . . , 𝑛,
𝑅 (𝑥, 𝑎𝑖) =
𝑛
∑
𝑖=0
𝑎𝑖𝐷𝛼
𝑇∗
𝑖(𝑥) − ∫
1
0
𝐾 (𝑥, 𝑡) [
𝑛
∑
𝑖=0
𝑎𝑖𝑇∗
𝑖(𝑡)] 𝑑𝑡,
𝑖 = 0, 1, . . . , 𝑛.
(18)
By solving the above system we obtain the values of theunknown coefficients and the approximate solution of (1).
4. Numerical Examples
In this section, some numerical examples of linear fractionalIntegro-differential equations are presented to illustrate theabove results. All results are obtained by using Maple 15.
Example 1. Consider the following fractional Integro-differential equation:
𝐷1/2
𝜑 (𝑥) =(8/3) 𝑥
3/2
− 2𝑥1/2
√𝜋+𝑥
12+ ∫
1
0
𝑥𝑡𝜑 (𝑡) 𝑑𝑡,
0 ≤ 𝑥, 𝑡 ≤ 1,
(19)
subject to 𝜑(0) = 0 with the exact solution 𝜑(𝑥) = 𝑥2 − 𝑥.Applying the least squares method with aid of
shifted Chebyshev polynomial of the first kind 𝑇∗
𝑖(𝑥),
𝑖 = 0, 1, . . . , 𝑛 at 𝑛 = 5, to the fractional Integro-differential
Column Row
1 1
3 3
2 2
4 4
5 5
6 6
0.2
0.4
0.6
0.8
1.2
1.4
1.6
1.8
1
×106
Figure 1: The matrix inverse of Example 1.
0
10.2 0.4 0.6 0.8
ExactApproximate
−0.25
−0.20
−0.10
−0.15
−0.05
x
Figure 2: Numerical results of Example 1.
equation (19) we obtain a system of (6) linear equationswith (6) unknown coefficients 𝑎
𝑖, 𝑖 = 0, 1, . . . , 5. This system
can be transformed into a matrix equation and by solvingthis matrix equation we obtain the inverse which is givenin Figure 1 and we obtain the values of the coefficients.Substituting the values of the coefficients into (7) we obtainthe approximate solution which is the same as the exactsolution and the results are shown in Figure 2.
4 Mathematical Problems in Engineering
Column Row
1 1
3 3
2 2
4 4
5 5
6 6
100000
200000
300000
400000
500000
600000
Figure 3: The matrix inverse of Example 2.
0
0
0.1
0.2
0.3
0.2 0.4 0.6 0.8 1
ExactApproximate
x
Figure 4: Numerical results of Example 2.
Example 2. Consider the following fractional Integro-differential equation:
𝐷5/6
𝜑 (𝑥) = 𝑓 (𝑥) + ∫
1
0
𝑥𝑒𝑡
𝜑 (𝑡) 𝑑𝑡, 0 ≤ 𝑥, 𝑡 ≤ 1, (20)
subject to 𝜑(0) = 0, where
𝑓 (𝑥) = −3
91
𝑥1/6
Γ (5/6) (−91 + 216𝑥2
)
𝜋+ (5 − 2𝑒) 𝑥 (21)
with the exact solution 𝜑(𝑥) = 𝑥 − 𝑥3.
ColumnRow
11
33
0
22
4 4
5 5
6 6
−5
5
10
15
×109
Figure 5: The matrix inverse of Example 3.
Similarly as in Example 1 applying the least squaresmethod with aid of shifted Chebyshev polynomial of the firstkind 𝑇∗
𝑖(𝑥), 𝑖 = 0, 1, . . . , 𝑛 at 𝑛 = 5, to the fractional Integro-
differential equation (20) the numerical results are shown inFigures 3 and 4 andwe obtain the approximate solutionwhichis the same as the exact solution.
Example 3. Consider the following fractional Integro-differential equation:
𝐷5/3
𝜑 (𝑥) =3√3Γ (2/3) 𝑥
1/3
𝜋−1
5𝑥2
−1
4𝑥
+ ∫
1
0
(𝑥𝑡 + 𝑥2
𝑡2
) 𝜑 (𝑡) 𝑑𝑡, 0 ≤ 𝑥, 𝑡 ≤ 1,
(22)
subject to 𝜑(0) = ��(0) = 0 with the exact solution 𝜑(𝑥) = 𝑥2.Similarly as in Examples 1 and 2 applying the least squares
method with aid of shifted Chebyshev polynomial of the firstkind 𝑇∗
𝑖(𝑥), 𝑖 = 0, 1, . . . , 𝑛 at 𝑛 = 5, to the fractional Integro-
differential equation (22) the numerical results are shown inFigures 5 and 6 andwe obtain the approximate solutionwhichis the same as the exact solution.
5. Conclusion
In this paper we study the numerical solution of threeexamples by using least squares method with aid of shiftedChebyshev polynomial which derives a good approximation.We show that this method is effective and has high conver-gency rate.
Mathematical Problems in Engineering 5
0
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1
1
ExactApproximate
x
Figure 6: Numerical results of Example 3.
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper.
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