Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 678174, 11 pages
doi:10.1155/2012/678174
Research Article
Numerical Solution for Complex Systems of
Fractional Order
Rabha W. Ibrahim
Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia
Correspondence should be addressed to Rabha W. Ibrahim, rabhaibrahim@yahoo.com
Received 18 October 2012; Accepted 4 December 2012
Academic Editor: Turgut Öziş
Copyright q 2012 Rabha W. Ibrahim. 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.
By using a complex transform, we impose a system of fractional order in the sense of RiemannLiouville fractional operators. The analytic solution for this system is discussed. Here, we
introduce a method of homotopy perturbation to obtain the approximate solutions. Moreover,
applications are illustrated.
1. Introduction
Fractional models have been studied by many researchers to sufficiently describe the operation of variety of computational, physical, and biological processes and systems. Accordingly,
considerable attention has been paid to the solution of fractional differential equations,
integral equations, and fractional partial differential equations of physical phenomena.
Most of these fractional differential equations have analytic solutions, approximation,
and numerical techniques 1–3. Numerical and analytical methods have included finite
difference methods such as Adomian decomposition method, variational iteration method,
homotopy perturbation method, and homotopy analysis method 4–7.
The idea of the fractional calculus i.e., calculus of integrals and derivatives of any
arbitrary real or complex order was planted over 300 years ago. Abel in 1823 investigated the
generalized tautochrone problem and for the first time applied fractional calculus techniques
in a physical problem. Later Liouville applied fractional calculus to problems in potential
theory. Since that time the fractional calculus has drawn the attention of many researchers in
all areas of sciences see 8–10.
One of the most frequently used tools in the theory of fractional calculus is furnished
by the Riemann-Liouville operators. It possesses advantages of fast convergence, higher
stability, and higher accuracy to derive different types of numerical algorithms. In this
2
Journal of Applied Mathematics
paper, we will deal with scalar linear time-space fractional differential equations. The time
is taken in sense of the Riemann-Liouville fractional operators. Also, This type of differential
equation arises in many interesting applications. For example, the Fokker-Planck partial
differential equation, bond pricing equations, and the Black-Scholes equations are in this class
of differential equations partial and fractional.
In 11, the author used complex transform to obtain a system of fractional order
nonhomogeneous keeping the equivalency properties. By employing the homotopy
perturbation method, the analytic solution is presented for coupled system of fractional order.
Furthermore, applications are imposed such as wave equations of fractional order.
2. Fractional Calculus
This section concerns with some preliminaries and notations regarding the fractional calculus.
Definition 2.1. The fractional arbitrary order integral of the function f of order α > 0 is defined by
Iaα ft t
a
t − τα−1
fτ dτ.
Γα
2.1
When a 0, we write Iaα ft ft ∗ φα t, where ∗ denoted the convolution product see
12, φα t tα−1 /Γα, t > 0 and φα t 0, t ≤ 0 and φα → δt as α → 0 where δt is the
delta function.
Definition 2.2. The fractional arbitrary order derivative of the function f of order 0 ≤ α < 1
is defined by
Daα ft d
dt
t
a
d
t − τ−α
fτ dτ Ia1−α ft.
Γ1 − α
dt
2.2
Remark 2.3. From Definitions 2.1 and 2.2, a 0, we have
Γ μ
1
D t tμ−α , μ > −1; 0 < α < 1,
Γ μ−α
1
Γ μ
1
I α tμ tμ
α , μ > −1; α > 0.
Γ μ
α
1
α μ
2.3
The Leibniz rule is
∞ ∞ α
α
Daα ftgt Daα−k ftDak gt Daα−k gtDak ft,
k
k
k0
k0
2.4
Journal of Applied Mathematics
3
where
Γα 1
α
.
k
Γk 1Γα 1 − k
2.5
Definition 2.4. The Caputo fractional derivative of order μ > 0 is defined, for a smooth
function ft by
c
1
D ft : Γ n−μ
μ
t
0
f n ζ
t − ζμ−n
1
dζ,
2.6
where n μ 1, the notation μ stands for the largest integer not greater than μ.
Note that there is a relationship between Riemann-Liouville differential operator and
the Caputo operator
fa
1
μ
μ
Da ft c Da ft,
Γ 1 − μ t − aμ
2.7
and they are equivalent in a physical problem i.e., a problem which specifies the initial
conditions.
In this paper, we consider the following fractional differential equation:
Dα ut, z at, zuzz bt, zuz ct, zu ft, z,
2.8
where a /
0, b, c, u, f are complex valued functions, analytic in the domain D : J × U; J 0, T , T ∈ 0, ∞ and U : {z ∈ C, |z| ≤ 1}.
The above equation involves well-known time fractional diffusion equations.
3. Complex Transforms
In this section, we will transform the fractional differential equation 2.8 into a coupled
nonlinear system of fractional order has similar form. It was shown in 11 that the complex
transform
ut, z σzut, z,
3.1
where σ /
0 is a complex valued function of complex variable z ∈ U, reduces 2.8 into the
system
Dα v a1 vzz − a2 wzz b1 vz − b2 wz c1 v − c2 w
Dα w a1 wzz a2 vzz b1 wz b2 vz c1 w c2 v,
3.2
4
Journal of Applied Mathematics
where
a1 a1 ,
a2 a2
b 1 b1 2σ1z a1 σ1 a2 σ2 2σ2z a1 σ2 − a2 σ1 σ12 σ22
b 2 b2 −
2σ1z a1 σ2 − a2 σ1 2σ2z a2 σ2 − a1 σ1 σ12 σ22
c 1 c1 σ1zz a1 σ1 a2 σ2 σ2zz a1 σ2 − a2 σ1 σ1z b1 σ1 b2 σ2 σ2z b1 σ2 − b2 σ1 σ12 σ22
c 2 c2 −
σ1zz a1 σ2 − a2 σ1 σ2zz a1 σ1 − a2 σ2 σ1z b1 σ2 − b2 σ1 − σ2z b1 σ1 b2 σ2 .
σ12 σ22
σz : σ1 z iσ2 z,
at, z a1 t, z ia2 t, z,
ct, z c1 t, z ic2 t, z,
ut, z vt, z iwt, z
bt, z b1 t, z ib2 t, z
ut, z vt, z iwt, z.
3.3
Also, it was shown that the complex transform
ut, z ρt, zut, z,
3.4
reduces the nonhomogenous equation
Dα ut, z at, zuzz bt, zuz ct, zu ft, z,
3.5
into the system
Dα v a1 vzz − a2 wzz b1 vz − b2 wz c1 v − c2 w f 1
Dα w a1 wzz a2 vzz b1 wz b2 vz c1 w c2 v f 2 ,
where
b1 b1 b2 b2 −
c 1 c1 a1 a1 ,
a2 a2
2ρ1z a1 ρ1 a2 ρ2 2ρ2z a1 ρ2 − a2 ρ1
ρ12 ρ22
2ρ1z a1 ρ2 − a2 ρ1 2ρ2z a2 ρ2 − a1 ρ1
ρ12 ρ22
ρ1zz a1 ρ1 a2 ρ2 ρ2zz a1 ρ2 − a2 ρ1 ρ1z b1 ρ1 b2 ρ2 ρ2z b1 ρ2 − b2 ρ1
ρ12 ρ22
3.6
Journal of Applied Mathematics
ρ1zz a1 ρ2 − a2 ρ1 ρ2zz a1 ρ1 − a2 ρ2 ρ1z b1 ρ2 − b2 ρ1 − ρ2z b1 ρ1 b2 ρ2
c 2 c2 −
ρ12 ρ22
ρ1 f1 − h1 ρ2 f2 − h2
f1 ρ12 ρ22
ρ2 f1 − h1 ρ1 f2 − h2
f2 ,
ρ12 ρ22
f
5
f αρt 1−α
−
I u f 1 if 2
ρ
ρ
h1 ρ1t I 1−α v − ρ2t I 1−α w
h2 ρ2t I 1−α v ρ1t I 1−α w,
0.
ρt, z : ρ1 t, z iρ2 t, z /
3.7
4. Numerical Solution
Let us put
F1 t, z, v, w φ1 t, z − L1 v, w − N1 v, w
F2 t, z, v, w φ2 t, z − L2 v, w − N2 v, w,
4.1
where φ1 t, z and φ2 t, z are arbitrary functions;
L1 v, w −1 v 1 w − a1 vzz b1 vz c1 v a2 wzz b2 wz c2 w ,
L2 v, w −2 v 2 w − a2 vzz b2 vz c2 v a1 wzz b1 wz c1 w
4.2
are the linear parts of F1 and F2 , respectively. While N1 and N2 are the nonlinear parts of F1
and F2 , respectively. Moreover, let us set the homotopy system
1 − p Dα vt, z pDα vt, z − φ1 t, z L1 v, w N1 v, w 0, p ∈ 0, 1
1 − p Dα wt, z pDα wt, z − φ2 t, z L2 v, w N2 v, w 0,
4.3
where
vt, z ∞
vn t, zpn ,
n0
N1 v, w wt, z ∞
wn t, zpn ,
n0
∞
k0
k
Nk p ,
N2 v, w ∞
k0
4.4
k pk .
N
6
Journal of Applied Mathematics
Hence we obtain the following system:
⎛
⎛
⎛
⎞
⎞
⎞
0
0
v0 t, z
⎜ v t, z ⎟
⎜ v t, z ⎟
⎜ w t, z ⎟
⎜ 1
⎜ 0
⎜ 0
⎟
⎟
⎟
⎜
⎜
⎜
⎟
⎟
⎟
α ⎜ v 2 t, z ⎟
v
w
z
z
t,
t,
⎜
⎜
⎟
⎟
1
1
1 ⎜
− 1 ⎜
D ⎜
⎟
⎟
⎟
..
..
⎜ .. ⎟
⎜
⎜
⎟
⎟
⎝ . ⎠
⎝
⎝
⎠
⎠
.
.
vn t, z
vn−1 t, z
wn−1 t, z
⎛
⎞
⎞ ⎛
0
0
⎜
⎟ ⎜φ t, z⎟
N0 v0 t, z
⎜
⎟
⎟ ⎜ 1
⎜
⎟
⎟ ⎜
N1 v0 t, z, v1 t, z
⎟ ⎜ 0 ⎟,
−⎜
⎜
⎜
⎟
⎟
..
⎜
⎟ ⎜ .. ⎟
⎝
⎠ ⎝ . ⎠
.
0
Nn−1 v0 t, z, v1 t, z, . . . , vn−1 t, z
⎛
⎛
⎛
⎞
⎞
⎞
0
0
w0 t, z
⎜ w t, z ⎟
⎜ v t, z ⎟
⎜ w t, z ⎟
⎜ 1
⎜ 0
⎜ 0
⎟
⎟
⎟
⎜
⎜
⎜
⎟
⎟
⎟
w2 t, z ⎟ 2 ⎜ v1 t, z ⎟ 2 ⎜ w1 t, z ⎟
Dα ⎜
⎜
⎜
⎜
⎟
⎟
⎟
..
..
..
⎜
⎜
⎜
⎟
⎟
⎟
⎝
⎝
⎝
⎠
⎠
⎠
.
.
.
wn t, z
vn−1 t, z
wn−1 t, z
⎛
⎞ ⎛
⎞
0
0
⎜
⎟ ⎜φ t, z⎟
0 w0 t, z
N
⎜
⎟ ⎜ 2
⎟
⎜
⎟ ⎜
⎟
1 w0 t, z, w1 t, z
N
⎟ ⎜ 0 ⎟,
−⎜
⎜
⎟ ⎜ . ⎟
..
⎜
⎟ ⎜ . ⎟
⎝
⎠ ⎝ . ⎠
.
n−1 w0 t, z, w1 t, z, . . . , wn−1 t, z
N
4.5
0
where
v0 t, z k−1
j
j t
,
νzv0
j!
j0
α ∈ k − 1, k, νz ∞
νn zn ,
n0
v1 t, z − I α L1 v0 t, z − I α N0 v0 t, z I α φ1 t, z,
..
.
vn t, z −I α L1 vn−1 t, z − I α Nn−1 v0 t, z, . . . , vn−1 t, z,
w0 t, z k−1
j0
j t
zw0
j
j!
,
∞
α ∈ k − 1, k, z n zn ,
n0
0 w0 t, z I α φ2 t, z,
w1 t, z − I α L2 w0 t, z − I α N
..
.
n−1 w0 t, z, . . . , wn−1 t, z.
wn t, z − I α L2 wn−1 t, z − I α N
4.6
Journal of Applied Mathematics
7
Consequently, we have the approximate solution
⎞
⎛
∞
∞
∞
j
j t
α⎝
−I
vt, z νzv0
L1 vj t, z Nj − φ1 t, z⎠
j!
j0
j0
j0
⎞
⎛
∞
∞
∞
j
j t
α⎝
j − φ2 t, z⎠.
−I
wt, z zw0
L2 wj t, z N
j!
j0
j0
j0
4.7
Thus, we impose a nonlinear integral equation in the following formula:
vt, z wt, z t
∞
j
t − τα−1
j t
F1 τ, ζ, u dτ
νzv0
j!
Γα
0
j0
∞
j0
j t
zw0
j
j!
t
0
t − τα−1
F2 τ, ζ, u dτ.
Γα
4.8
Now we can sake the main result of this section.
Theorem 4.1. Consider the fractional differential system 3.6 subject to the initial conditions
m
m
vm 0, z v0 z, wm 0, z w0 z, m 0, 1, 2, . . . , k − 1 .
4.9
The homotopy perturbation technique implies that the initial value problem (3.6–4.9) can
be expressed as a nonlinear integral equation of the form 4.8.
We proceed to prove the analytical convergence of our solution.
vn t,z
of the homotopy series vt, z Theorem 4.2. Suppose the sequence un t, z wn t,z
∞
∞
n
n
v
t,
zp
and
wt,
z
w
t,
zp
is
defined
for
p ∈ 0, 1.
Assume
n0 n
n0 n
the initial
v0 t,z
vt,z
approximation u0 t, z w0 t,z inside the domain of the solution ut, z wt,z . If un
1 ≤
ρun for all n, where 0 < ρ < 1, then the solution is absolutely convergent when p 1.
Proof. Let Cn t, z be the sequence of partial sum of the homotopy series. Our aim is to show
that Cn t, z is a Cauchy sequence. Consider
Cn
1 t, z − Cn t, z un
1 t, z
≤ ρun t, z ≤ ρ2 un−1 t, z
≤ · · · ≤ ρn
1 u0 t, z.
4.10
8
Journal of Applied Mathematics
For n ≥ m, n ∈ N, we have
Cn t, z − Cm t, z Cn t, z − Cn−1 t, z Cn−1 t, z − Cn−2 t, z
· · ·
Cm
1 t, z − Cm t, z
≤ Cn t, z − Cn−1 t, z Cn−1 t, z − Cn−2 t, z
· · · Cm
1 t, z − Cm t, z
≤
1 − ρn−m m
1
ρ u0 t, z.
1−ρ
4.11
Hence
lim Cn t, z − Cm t, z 0;
n,m → ∞
4.12
therefore, Cn t, z is a Cauchy sequence in the complex Banach space and consequently yields
that the series solution is convergent. This completes the proof.
Recently the homotopy methods are used to obtain approximate analytic solutions
of the time-fractional nonlinear equation and time-space-fractional nonlinear equation see
12–17.
5. Applications
In this section, we will consider the pump wave equations along the fiber Schrödinger
equations. These types of equations are the fundamental equations for describing nonrelativistic quantum mechanical behavior taking the form
1
iDα ut, z − uzz t, z − |u|2 ut, z.
2
5.1
Under the transform u u v iw such that either |u|2 |v|2 or |u|2 |w|2 , we have
the uncoupled system
1
iDα vt, z − vzz t, z − |v|2 vt, z
2
1
iD wt, z − wzz t, z − |w|2 wt, z,
2
5.2
α
where 0 < α ≤ 1. Subject to the initial conditions
v0 0, z eiz ,
w0 0, z 1.
5.3
Journal of Applied Mathematics
9
Operating 5.2 by I α , we have
ivt, z v0 0, z I
α
1
− vzz t, z − |v|2 vt, z
2
1
2
iwt, z w0 0, z I − wzz t, z − |w| wt, z .
2
5.4
α
By the same computation as in Section 5, we receive
v0 eiz ,
itα
eiz ,
2Γα 1
w1 itα
Γα 1
itα 2
eiz ,
22 Γ2α 1
w2 itα 2
Γ2α 1
wn itα n
.
Γnα 1
v1 v2 w0 1
5.5
..
.
vn itα n
eiz ,
2n Γnα 1
Thus the solution u is given by
ut, z ∞
∞
itα n
itα n
iz
e
,
2n Γnα 1
Γnα 1
n0
n0
T
.
5.6
Moreover, under the same transform, 5.1 reduces to coupled system
1
iDα vt, z − vzz t, z − |v|2 |w|2 vt, z
2
1
iDα wt, z − wzz t, z − |v|2 |w|2 wt, z,
2
5.7
Operating 5.7 by I α , we have
1
ivt, z v0 0, z I α − vzz t, z − |v|2 |w|2 vt, z
2
1
2
2
α
iwt, z w0 0, z I − wzz t, z − |v| |w| wt, z .
2
5.8
Therefore,
ut, z ∞
∞
itα n
itα n
iz
e
,
2n Γnα 1
Γnα 1
n0
n0
T
,
iz e 1.
5.9
10
Journal of Applied Mathematics
a
c
e
b
d
f
Figure 1: a–d The solution v when α 0.5, α 0.75, α 0.9, and α 1, respectively. e,f the
solution u, v when α 0.5 and α 1.
6. Conclusion
We suggested two types of complex transforms for systems of fractional differential equations. We concluded that the complex fractional differential equations can be transformed into
coupled and uncoupled system of homogeneous and nonhomogeneous types. Moreover, we
employed the homotopy perturbation scheme for solving the nonlinear complex fractional
differential systems. The convergence of the method is discussed in a domain that contains
the initial solution. The Schrödinger equation is illustrated as an application. This type of
equation is used in the quantum mechanics, which describes how the quantum state of a
physical system changes with time. In the standard quantum mechanics, the wave function
is the most complete explanation that can be specified to a physical system. Solutions of the
Schrdinger’s equation describe not only molecular, atomic, and subatomic systems, but also
macroscopic systems see Figure 1.
Journal of Applied Mathematics
11
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