# A Reproducing Kernel Hilbert Space Method for Solving Systems of Fractional Integrodifferential Equations

Authors: Samia Bushnaq; Banan Maayah; Shaher Momani; Ahmed Alsaedi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We present a new version of the reproducing kernel Hilbert space method (RKHSM) for the solution of systems of fractional integrodifferential equations. In this approach, the solution is obtained as a convergent series with easily computable components. Several illustrative examples are given to demonstrate the effectiveness of the present method. The method described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.


---

## Body

## 1. Introduction

In this paper, we consider the following system of fractional integrodifferential equations:(1)
D
α
i
x
i
(
t
)
=
F
i
(
t
,
x
1
(
t
)
,
…
,
x
1
(
k
)
(
t
)
,
…
,
x
i
-
1
(
t
)
,
…
,
x
i
-
1
(
k
)
(
t
)
,
1111111
x
i
+
1
(
t
)
,
…
,
x
i
+
1
(
k
)
(
t
)
,
…
,
x
n
(
t
)
,
…
,
x
n
(
k
)
(
t
)
)
+
∫
0
t
G
i
(
t
,
τ
,
x
1
(
τ
)
,
…
,
x
1
(
k
)
(
τ
)
,
…
,
x
n
(
τ
)
,
11111111111
…
,
x
n
(
k
)
(
τ
)
)
d
τ
,
where i
=
1
,
…
,
n, k
=
0,1
,
…
,
m, 0
≤
t
≤
1, and D
α
i is derivative of order α
i in the sense of Caputo and m
-
1
<
α
i
≤
m, subject to the initial conditions:
(2)
x
i
(
j
)
(
a
)
=
a
j
i
,
j
=
0,1
,
…
,
m
-
1
,
i
=
1,2
,
…
,
n
,
a
≥
0
.In the last two decades, fractional calculus has found diverse applications in various scientific and technological fields [1, 2], such as thermal engineering, acoustics, electromagnetism, control, robotics, viscoelasticity, diffusion, edge detection, turbulence, signal processing, and many other physical and biological processes. Fractional differential equations have also been applied in modeling many physical and engineering problems. Most systems of fractional integrodifferential equations do not have exact solutions, so numerical techniques are used to solve such systems. The homotopy perturbation method, the Adomian decomposition method, and other methods are used to give an approximate solution to linear and nonlinear problems; see [3–13] and the references therein.In our previous work [14], we proposed a reproducing kernel Hilbert space method for solving integrodifferential equations of fractional order based on the reproducing kernel theory [14, 15]. In this paper, we will generalize the idea of the RKHSM to provide a numerical solution for systems of fractional integrodifferential equations (1). To demonstrate the effectiveness of the RKHSM algorithm, several numerical experiments of linear and nonlinear systems of fractional equations (1) will be presented.This paper is organized as follows. An introduction of the algorithm for solving systems of fractional integrodifferential equations is given in Section2. In Section 3, we introduce several examples to show the efficiency of the method. Finally, a conclusion is given in Section 4.


## 2. The Algorithm

After homogenizing the initial conditions (2), we apply the operator I
α
i, the Riemann-Liouville fractional integral of order α
i [2, 16–20], to both sides of (1) to have
(3)
x
i
(
t
)
=
M
i
(
t
)
,
i
=
1,2
,
…
,
n
,
where
(4)
M
i
(
t
)
=
I
α
i
(
∫
0
t
G
i
(
t
,
τ
,
x
1
(
τ
)
,
…
,
x
1
(
k
)
(
τ
)
,
…
,
x
n
(
τ
)
,
…
,
x
n
(
k
)
(
τ
)
)
F
i
(
t
,
x
1
(
t
)
,
…
,
x
1
(
k
)
(
t
)
,
…
,
x
i
-
1
(
t
)
,
…
,
x
i
-
1
(
k
)
(
t
)
,
x
i
+
1
(
t
)
,
…
,
x
i
+
1
(
k
)
(
t
)
,
…
,
x
n
(
t
)
,
…
,
x
n
(
k
)
(
t
)
)
11111111
+
∫
0
t
G
i
(
t
,
τ
,
x
1
(
τ
)
,
…
,
x
1
(
k
)
(
τ
)
,
…
,
x
n
(
τ
)
,
…
,
x
n
(
k
)
(
τ
)
)
d
τ
∫
0
t
G
i
(
t
,
τ
,
x
1
(
τ
)
,
…
,
x
1
(
k
)
(
τ
)
,
…
,
x
n
(
τ
)
,
…
,
x
n
(
k
)
(
τ
)
)
)
,
i
=
1
,
2
,
…
,
n
,
k
=
0
,
1
,
…
,
m
.It is clear that (3) is equivalent to (1), so every solution of the integral equation (3) is also a solution of our original problem (1) and vice versa.To solve (3) by means of the reproducing kernel Hilbert space method, first, we need to construct a reproducing kernel of certain spaces W
2
m
+
1
[
a
,
b
]
≔
{
u
∣
u
(
j
) is absolutely continuous, j
=
1
,
2
,
…
,
m
-
1, and u
(
m
)
∈
L
2
[
a
,
b
]
} in which every function satisfies the homogenous initial conditions of (1).(i)
The inner product of the spaceW
2
1
[
0,1
]
=
{
u
∣
u is absolutely continuous real value function, u
′
∈
L
2
[
0,1
]
} is given by
(5)
〈
u
,
v
〉
W
2
1
≔
∫
0
1
(
u
(
t
)
v
(
t
)
+
u
′
(
t
)
v
′
(
t
)
)
d
t
and norm|
|
u
|
|
W
2
1
=
〈
u
,
u
〉
W
2
1.
In [21], Li and Cui proved that W
2
1
[
0,1
] is a reproducing kernel Hilbert space and its reproducing kernel is given by
(6)
R
(
x
,
y
)
≔
1
2
sinh
⁡
⁡
1
×
[
cosh
⁡
⁡
(
x
+
y
-
1
)
+
cosh
⁡
⁡
|
x
-
y
|
-
1
]
.
(ii)
The inner product of the spaceW
2
2
[
0,1
]
=
{
u
∣
u
,
u
′ are absolutely continuous real value functions, u
′
′
∈
L
2
[
0,1
]
,
u
(
0
)
=
0
} is given by
(7)
〈
u
,
v
〉
W
2
2
≔
u
(
0
)
v
(
0
)
+
u
′
(
0
)
v
′
(
0
)
+
∫
0
1
u
′′
(
t
)
v
′′
(
t
)
d
t
and norm|
|
u
|
|
W
2
2
=
〈
u
,
u
〉
W
2
2. W
2
2
[
0,1
] is a reproducing kernel Hilbert space and its reproducing kernel is given by
(8)
S
(
x
,
y
)
:
=
{
1
6
y
(
-
y
2
+
3
x
(
2
+
y
)
)
,
y
≤
x
1
6
x
(
-
x
2
+
3
y
(
2
+
x
)
)
,
y
>
x
.
(iii)
The inner product of the spaceW
2
3
[
0,1
]
=
{
u
∣
u
,
u
′
,
u
′
′ are absolutely continuous real value functions, u
′
′
′
∈
L
2
[
0,1
]
,
u
(
0
)
=
u
′
(
0
)
=
0
} is given by
(9)
〈
u
,
v
〉
W
2
3
≔
∑
i
=
0
2
u
(
i
)
(
0
)
v
(
i
)
(
0
)
+
∫
0
1
u
′′′
(
t
)
v
′′′
(
t
)
d
t
and norm|
|
u
|
|
W
2
3
=
〈
u
,
u
〉
W
2
3. W
2
3
[
0,1
] is a reproducing kernel Hilbert space and its reproducing kernel is given by
(10)
M
(
x
,
y
)
≔
{
f
(
x
,
y
)
,
y
≤
x
f
(
y
,
x
)
,
y
>
x
,where f
(
x
,
y
)
=
(
1
/
120
)
y
2
(
-
x
2
(
-
126
+
10
x
-
5
x
2
+
x
3
)
+
5
(
-
1
+
x
)
x
y
2
-
(
-
1
+
x
2
)
y
3
).The method of obtaining the reproducing kernel can be found in [15].LetL
i
:
W
2
m
+
1
[
0,1
]
→
W
2
1
[
0,1
] such that L
i
x
i
(
t
)
=
x
i
(
t
). Then L
i, i
=
1
,
2
,
…
,
n, are bounded linear operators.Let{
t
j
}
j
=
1
∞ be a countable dense set in [
0,1
]. Let φ
j
i
(
t
)
=
R
(
t
j
,
t
) and ψ
j
i
(
t
)
=
L
i
*
φ
j
i
(
t
), where L
i
* is the adjoint operator of L
i.By Gram-Schmidt process we can construct an orthonormal system{
ψ
-
j
i
(
t
)
}
j
=
1
∞ of W
2
m
+
1
[
0,1
], where
(11)
ψ
-
j
i
(
t
)
=
∑
k
=
1
j
β
j
k
i
ψ
k
i
(
t
)
,
β
j
j
i
>
0
,
∀
j
=
1
,
2
,
…
,
i
=
1
,
2
,
…
,
n
.Theorem 1.
Let{
t
j
}
j
=
1
∞ be a dense set in [
0,1
]. Then {
ψ
j
i
(
t
)
}
j
=
1
∞ is a complete system of W
2
m
+
1
[
0,1
].For the proof, see [14].Theorem 2.
Let{
t
j
}
j
=
1
∞ be a dense set in [
0,1
] and the solution of (3) is unique on W
2
m
+
1
[
0,1
]. Then the solution of (3) is given by x
i
(
t
)
=
∑
j
=
1
∞
A
j
ψ
-
j
i
(
t
), where A
j
=
∑
k
=
1
j
β
j
k
i
M
i
(
t
k
).For the proof, see [14].One can get an approximate solutionx
i
n
(
t
) by taking finitely many terms in the series representation of x
i
(
t
) and x
i
n
(
t
)
=
∑
j
=
1
n
A
j
ψ
-
j
i
(
t
).SinceW
2
m
+
1
[
0,1
] is a Hilbert space, then ∑
j
=
1
∞
∑
k
=
1
∞
β
j
k
i
M
i
(
t
k
)
<
∞.Theorem 3.
The approximate solutionx
i
n
(
t
) and its derivatives x
i
n
(
j
) are uniformly convergent to x
i
(
j
)
(
t
), i
=
1,2
,
…
,
n, j
=
0
,
1
,
….Proof.
By the reproducing kernel property ofK
(
x
,
y
) and Schwarz inequality, we can obtain
(12)
|
x
i
n
(
t
)
-
x
i
(
t
)
|
=
|
〈
x
i
n
(
t
)
-
x
i
(
t
)
,
K
(
x
,
y
)
〉
W
2
m
+
1
|
≤
∥
K
(
x
,
y
)
∥
W
2
m
+
1
∥
x
i
n
(
t
)
-
x
i
(
t
)
∥
W
2
m
+
1
≤
c
0
∥
x
i
n
(
t
)
-
x
i
(
t
)
∥
W
2
m
+
1
,
where c
0 is a constant.
By the representation ofK
(
x
,
y
) we can obtain
(13)
|
x
i
n
(
j
)
(
t
)
-
x
i
(
j
)
(
t
)
|
=
|
〈
x
i
n
(
j
)
(
t
)
-
x
i
(
j
)
(
t
)
,
K
(
j
)
(
x
,
y
)
〉
W
2
m
+
1
|
≤
∥
K
(
j
)
(
x
,
y
)
∥
W
2
m
+
1
∥
x
i
n
(
j
)
(
t
)
-
x
i
(
j
)
(
t
)
∥
W
2
m
+
1
.
Since K
(
j
)
(
x
,
y
), j
=
1
,
2
,
…, is uniformly bounded about x and y, we have
(14)
∥
K
(
j
)
(
x
,
y
)
∥
W
2
m
+
1
≤
c
j
,
j
=
1
,
2
,
…
,
and so
(15)
|
x
i
n
(
j
)
(
t
)
-
x
i
(
j
)
(
t
)
|
≤
c
j
∥
x
i
n
(
j
)
(
t
)
-
x
i
(
j
)
(
t
)
∥
W
2
m
+
1
,
1111111111111111111111111
.
1111
j
=
1
,
2
,
…
.
Thus x
i
(
t
) and its derivatives x
i
n
(
j
)
(
t
) are uniformly convergent to x
i
(
j
)
(
t
)
,
j
=
1
,
2
,
….


## 3. Numerical Results

In this paper, three numerical examples are given to show the accuracy of this method. The computations are performed by Mathematica 8.0. We compare the results by this method with the exact solution of each example.Example 1.
Consider the following linear system of fractional integrodifferential equations:(16)
D
α
1
x
(
t
)
=
1
+
t
+
t
2
-
y
(
t
)
-
∫
0
t
(
x
(
τ
)
+
y
(
τ
)
)
d
τ
,
D
α
2
x
(
t
)
=
-
1
-
t
+
x
(
τ
)
-
∫
0
t
(
x
(
τ
)
-
y
(
τ
)
)
d
τ
,
x
(
0
)
=
1
,
y
(
0
)
=
-
1
,
0
<
α
1
,
α
2
≤
1
.
The exact solution for α
1
=
α
2
=
1 is x
(
t
)
=
t
+
e
t, y
(
t
)
=
t
-
e
t.
After homogenizing the initial conditions and using this method, takingt
i
=
i
/
n, i
=
1
,
2
,
…
,
n, and n
=
20, the graphs of the approximate solutions for different values of α
1 and α
2 are plotted in Figure 1. From Figure 1, it is clear that the approximate solutions are in good agreement with the exact solutions when α
1
=
α
2
=
1, and the solution continuously depends on the fractional derivative.Graphical results for Example1 when α
1
=
α
2
=
α
=
1
,
0.9
,
0.8, and 0.7.
(a)
(b)
(c)
(d)Example 2.
Consider the following nonlinear system of fractional integrodifferential equations:(17)
D
α
1
x
(
t
)
=
1
-
1
2
y
′
2
(
t
)
+
∫
0
t
[
(
t
-
τ
)
y
(
τ
)
+
x
(
τ
)
y
(
τ
)
]
d
τ
,
D
α
2
x
(
t
)
=
2
t
+
∫
0
t
[
(
t
-
τ
)
x
(
τ
)
-
y
2
(
τ
)
+
x
2
(
τ
)
]
d
τ
,
x
(
0
)
=
0
,
y
(
0
)
=
1
,
0
<
α
1
,
α
2
≤
1
.
The exact solution for α
1
=
α
2
=
1 is x
(
t
)
=
sinh
⁡
⁡
t, y
(
t
)
=
cosh
⁡
⁡
t.
After homogenizing the initial conditions and using this method, takingt
i
=
i
/
n, i
=
1
,
2
,
…
,
n, and n
=
30, the graphs of the approximate solutions for different values of α
1 and α
2 are plotted in Figure 2.Graphical results for Example2 when α
1
=
α
2
=
α
=
1
,
0.9
,
0.8, and 0.7.
(a)
(b)
(c)
(d)Example 3.
Consider the following nonlinear system of fractional integrodifferential equations:(18)
D
α
1
x
(
t
)
=
1
-
t
3
3
-
1
2
y
′
2
(
t
)
+
1
2
∫
0
t
(
x
2
(
τ
)
+
y
2
(
τ
)
)
d
τ
,
D
α
2
x
(
t
)
=
-
1
+
t
2
-
t
x
(
t
)
+
1
4
∫
0
t
(
x
2
(
τ
)
-
y
2
(
τ
)
)
d
τ
,
x
(
0
)
=
1
,
x
′
(
0
)
=
2
,
y
(
0
)
=
-
1
,
y
′
(
0
)
=
0
,
1
<
α
1
,
α
2
≤
2
.
The exact solution for α
1
=
α
2
=
2 is x
(
t
)
=
t
+
e
3, y
(
t
)
=
t
-
e
t.
After homogenizing the initial conditions and using this method, takingt
i
=
i
/
n, i
=
1
,
2
,
…
,
n, and n
=
20, the graphs of the approximate solutions for different values of α
1 and α
2 are plotted in Figure 3.Graphical results for Example3 when α
1
=
α
2
=
α
=
2
,
1.9
,
1.8, and 1.7.
(a)
(b)
(c)
(d)


## 4. Conclusion

In this paper, we introduce a new algorithm for solving systems of fractional integrodifferential equations. The approximate solution obtained by this method and its derivative are both uniformly convergent. The obtained results demonstrate the reliability of the algorithm and its wider applicability to linear and nonlinear systems of fractional differential equations.



---

*Source: 103016-2014-03-24.xml*