Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.13-21
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2012
www.i-csrs.org
Available free online at http://www.geman.in
Solution of Time-Fractional
Navier-Stokes Equation by Using
Homotopy Analysis Method
A.A. Ragab1 , K.M. Hemida2 , M.S. Mohamed3 and M.A. Abd El Salam4
Mathematics Department, Faculty of Science
Al-Azhar University
Nasr City (11884), Cairo, Egypt
1
E-mail: awaragab158@yahoo.com
2
E-mail: hemidakamal@yahoo.com
3
E-mail: m s mohamed200@yahoo.com
4
E-mail: mohamed salam1985@yahoo.com
(Received: 26-8-12 / Accepted: 18-10-12)
Abstract
The homotopy analysis method (HAM) is used to obtain an approximate
solution of the nonlinear time fractional Navier-Stokes equation by introducing
the fractional derivative in the caputo’s sense. Convergence of the solution and
effects for the method are discussed within comparing the obtained results with
exact solution of the corresponding nonlinear problem, which indicated that the
proposed method is very effective and simple. The HAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and control
the convergence region and rate of convergence of the series solution. It also
suggests that both the homotopy perturbation method (HPM), Adomian decomposition method (ADM) and variational iteration method (VIM) are special
cases of the HAM.
Keywords: Homotopy analysis method, fractional partial differential equation
14
1
A.A.Ragab et al.
Introduction
Nonlinear partial differential equations (NPDEs) are encountered in such various fields as physics, chemistry, biology, mathematics and engineering. Many
important phenomena in various field are will describe and generalize by an
ordinary or partial fractional differential equations.
Recently, El-Shahed and Salem [8] have generalized the classical Navier–
Stokes equations by replacing the first time derivative by a fractional derivative
of orderα ,0 < α ≤ 1. They used Laplace transform, Fourier sine transform
and finite Hankel transforms to obtain exact solutions for three different special
cases.
This model is generalized by replacing the first-time derivative by a fractional derivative of orderα ,0 < α ≤ 1. The time-fractional model for Navier–
Stokes equations has the following form
1
Dtα u + (u · ∇) u = − ∇P + ν∇2 u,
ρ
(1.1)
Where t is the time, u is the velocity vector, P is the pressure, ν is the kinematics viscosity and ρ is the density.
2
Basic Definitions
In this section we give some definitions and properties of the fractional calculus. The fractional calculus is a name for the theory of integrals and derivatives
of arbitrary order. Various definitions of fractional integration and differentiation are found in [1], [3], [7], and [12], such as Grunwald-Letnikov’s definition,
Riemann-Liouville definition, and Caputo’s definition and generalized function
approach. For the purpose of this paper, the Caputo’s definition of the fractional differentiation will be used, taking the advantage of Caputo’s approach
that the initial conditions for fractional differential equation with Caputo’s
derivatives take on the traditional form as for integer-order differential equation.
Definition 2.1. Areal function h (t), t > 0, is said to be in the space Cµ , µ ∈
R , if there exists a real number p > µ , such that h (t) = tp h1 (t) ,
where h1 (t) ∈ C (0, ∞) , and it is said to be in the space Cµn if and only
if h(n) ∈ Cµ , n ∈ N
Definition 2.2. The Riemann-Liouville fractional integral operator (J α ) of
order α ≥ 0 , of a function h ∈ Cµ , µ ≥ −1 is defined as
Z t
1
α
(t − τ )α−1 h (τ ) dτ ]
(2.1)
J h (t) =
Γ (α) 0
15
Solution of Time-Fractional...
J 0 h (t) = h (t)
Γ (α) is the well known gamma function. Some of the J α properties of the
operator which we will need here are as follows:
J α J β h (t) = J α+β h (t) , J α J β h (t) = J β J α h (t) , J α tγ =
Γ (γ + 1)
tγ+α .
Γ (α + γ + 1)
Definition 2.3. The fractional derivative (Dα ) of h(t) in the Caputo’s sense
is defined as follows
Z t
1
α
D h (t) =
(t − τ )n−α−1 h (τ ) dτ
(2.2)
Γ (n − α) 0
for
n − 1 < α < n,
n ∈ N , t > 0 , h ∈ C n−1
The following are two basic properties of Caputo’s fractional Derivative [?]
Let h ∈ C n−1 , n ∈ N then Dα h , 0 ≤ α ≤ n is well defined and
Dα h ∈ C−1
Let −1 < α < n , n ∈ N
and h ∈ C nµ , µ ≥ −1 then
(J α Dα ) h (t) = h (t) −
n−1
X
k=0
3
h(k) 0+
tk
k!
(2.3)
The Homotopy Analysis Method (HAM)
The HAM [10] is applied to the nonlinear homogeneous fractional equation
with a general form
N [u (r, t)] = 0.
(3.1)
Where N is a nonlinear operator for the problem, r and t denote independent
variables and u(r; t) is an unknown function. By means of the HAM, one first
constructs the zero-order deformation equation
(1 − q) L (∈ (r, t, q) − u0 (r, t)) = qhH (r, t) N [∈ (r, t)] .
(3.2)
Where q is the embedding parameter, q ∈ [0, 1] , h 6= 0 is an auxiliary parameter, H(r; t) 6= 0 is an auxiliary function , L is an auxiliary linear operator,
u0 (r, t) is an initial guess.
Obviously, when q = 0 and q = 1, it holds that
∈ (r, t, 0) = u0 (r, t) , ∈ (r, t, 1) = u (r, t) .
(3.3)
16
A.A.Ragab et al.
Liao [10] expanded ∈ (x, t, q) in Taylor series with respect to the embedding
parameter q, as follows:
∈ (r, t, q) = u0 (r, t) +
∞
X
um (r, t)q m .
(3.4)
m=1
Where
1 ∂ m ∈ (r, t, q) um (r, t) =
.
m!
∂q m
q=0
(3.5)
Assume that the auxiliary linear operator, the initial guess, the auxiliary parameter h and the auxiliary function H(x, t) are selected such that the series
(3.4) is convergent at q = 1 , then we have from (3.4)
u (r, t) = u0 (r, t) +
∞
X
um (r, t).
(3.6)
m=1
Let us define the vector
−
→ (t) = {u (r, t) , u (r, t) , u (r, t) , . . . . . . . . . u (r, t)} .
u
n
0
1
2
n
(3.7)
Differentiating (3.2) m times with respect to q, then setting q = 0 and dividing
by m!, that the mth-order deformation equation
−
L (um (r, t) − χm um−1 (r, t)) = hH (r, t) Rm (→
u m−1 ) .
Where
−
Rm (→
u m−1 ) =
(3.8)
∂ m−1 N [∈ (r, t, q)] 1
.
(m − 1)!
∂q m−1
q=0
And
χm =
0 m≤1
1 m>1
(3.9)
(3.10)
The mth-order deformation Eq. (3.8) becomes linear and it can be easily
solved, especially by means of symbolic computation software such as Mathematica, Maple, Matlab.
4
Time Fractional Navier-Stokes Equation
The Navier-Stokes equation (1.1) in cylindrical coordinates for unsteady one
dimensional motion of a viscous fluid is given by
Dtα u = P + ν(
∂ 2 u 1 ∂u
+
).
∂r2
r ∂r
(4.1)
17
Solution of Time-Fractional...
4.1
Application 1
Firstly, we consider
Dtα u = P +
∂ 2 u 1 ∂u
.
+
∂r2
r ∂r
(4.2)
With initial condition
u (r, 0) = 1 − r2 .
(4.3)
According to the (HAM), and apply (HAM) as [4], [5] and [6] we choose the
auxiliary operator as
L [∈ (r, t; q)] = Dtα ∈ (r, t, q) .
(4.4)
With property L [c] = 0 where c is a constant.
We define a nonlinear operator as
N [∈ (r, t, q)] = Dtα ∈ (r, t, q) −
∂ 2 ∈ (r, t, q) 1 ∂ ∈ (r, t, q)
−
− P.
∂r2
r
∂r
(4.5)
In order to obey the rule of solution expression and the rule of the coefficient
periodicity [9], the auxiliary function can be determined uniquely H(x, t) = 1,
and
1 ∂
∂2
−
−
um−1 − P (1 − χm ) .
Rm (→
u m−1 ) = Dtα um−1 − 2 u
∂r m−1 r ∂r
(4.6)
Now the solution of the mth-order deformation equations (3.8) for m ≥ 1
becomes
−
um (r, t) = χm um−1 (r, t) + hL−1 Rm (→
u m−1 ) .
(4.7)
So, the first few terms of the solution are
u0 (r, t) = 1 − r2
h(1 + h)(−4 + P )tα
Γ[1 + α]
h(1 + h)3 (−4 + P )tα
u4 (r, t) = −
Γ[1 + α]
u2 (r, t) = −
h(−4 + P )tα
Γ[1 + α]
h(1 + h)2 (−4 + P )tα
u3 (r, t) = −
Γ[1 + α]
u1 (r, t) = −
Then, we can conclude that
u (r, t) = u0 (r, t) + u1 (r, t) + u2 (r, t) + u3 (r, t) + u4 (r, t) . . .
Then
u (r, t) = 1 − r2 −
h(−4 + P )tα 1 + (1 + h) + (1 + h)2 + (1 + h)3 . . . . (4.8)
Γ[1 + α]
18
A.A.Ragab et al.
Using the Geometric series as in [4], when the series tends to infinity and h
must be less than 0, the solution becomes independent of h and takes the
following form
(−4 + P )tα
(4.9)
u (r, t) = 1 − r2 +
Γ[1 + α]
Which represent the exact solution of equation (4.2), also the solution is the
same as S. Momani and Z. Odibat [11], and Fig. 1 and 2 shows the evolution
results for the time-fractional Eq. (4.2) when α = 1 and α = 0.5 respectively.
It is easy to conclude that the solution continuously depends on the timefractional derivative.
Fig.1 (α = 1)
4.2
Fig.2 (α = 0.5)
Application 2
Consider the equation in the form
Dtα u =
∂ 2 u 1 ∂u
+
.
∂r2
r ∂r
(4.10)
Subject to the initial condition
u (r, 0) = r.
(4.11)
Similarly, choosing
N [∈ (r, t, q)] = Dtα ∈ (x, t, q) −
∂ 2 ∈ (r, t, q) 1 ∂ ∈ (r, t, q)
−
.
∂r2
r
∂r
(4.12)
19
Solution of Time-Fractional...
Then,
∂2
1 ∂
−
Rm (→
u m−1 ) = Dtα um−1 − 2 u
um−1 .
−
∂r m−1 r ∂r
So, first terms of the series are
(4.13)
u0 (r, t) = r
htα
rΓ[1 + α]
h(1 + h)tα
h2 t2α
u2 (r, t) = −
+ 3
rΓ[1 + α]
r Γ[1 + 2α]
u1 (r, t) = −
α
h(1 + h)2 t
h2 (1 + h)t2α
9h3 t3α
u3 (r, t) = −
+ 3
− 5
rΓ[1 + α]
r Γ [1 + 2α]
r Γ[1 + 3α]
α
3h2 (1 + h)2 t2α 3h3 (1 + h)9t3α
9h4 25t4α
h(1 + h)3 t
+ 3
− 5
+ 7
u4 (r, t) = −
rΓ[1 + α]
r Γ [1 + 2α]
r Γ[1 + 3α]
r Γ[1 + 4α]
Then, we can conclude that
u (r, t) = u0 (r, t) + u1 (r, t) + u2 (r, t) + u3 (r, t) + . . .
Or
htα
1 + (1 + h) + (1 + h)2 + (1 + h)3 . . .
rΓ[1 + α]
h2 t2α
1 + 2 (1 + h) + 3(1 + h)2 + . . .
+ 3
r Γ[1 + 2α]
9h3 t3α
− 5
1 + 3 (1 + h) + 6(1 + h)2 + . . . + . . .
r Γ[1 + 3α]
u (r, t) =r −
As the series tends to infinity (using Geometric series as [4] where h must be
less than 0), the solution becomes independent of h and takes the following
form
u (r, t) = r +
tα
t2α
9t3α
25t4α
+ 3
+ 5
+ 7
...
rΓ[1 + α] r Γ[1 + 2α] r Γ[1 + 3α]
r Γ[1 + 4α]
Therefore the solution is
u (r, t) = r +
∞
X
12 × 32 × · · · × (2n − 3)2
r2n−1
n=1
tnα
Γ(nα + 1)
Which is the same solution given by S. Momani and Z. Odibat [11].
At α = 1 we have
u (r, t) = r +
∞
X
12 × 32 × · · · × (2n − 3)2 tn
n=1
r2n−1
n!
20
A.A.Ragab et al.
Which is the same solution given by Biazar et al. [2].
Fig. 3 and 4 shows the evolution results for the time-fractional Eq. (4.10)
when α = 1 and α = 0.5, respectively.
Fig.1 (α = 1)
5
Fig.2 (α = 0.5)
Conclusion
In this paper, the homotopy analysis method (HAM) is applied to obtain the
solution of time-fractional Navier–Stokes equation in cylindrical coordinates.
The results show that (HAM) is powerful and efficient techniques in finding
exact and approximate solutions for nonlinear fractional partial differential
equations.
The (HAM) provides us with a convenient way to control the convergence of
approximation series which is a fundamental qualitative difference in analysis
between (HAM) and other method. Thus the auxiliary parameter h plays an
important role within the frame of the (HAM). Mathematica has been used
for computations in this paper.
References
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Solution of Time-Fractional...
21
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Math. Comput., 138(2002), 523-529.
[3] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press,
New York, (1974).
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Computing, 3(2011), 1-7.
[5] K.M. Hemida, K.A. Gepreel and M.S. Mohamed, Analytic approximate
solution to the time-space nonlinear partial fractional differential equation, Int. J. of Pure and Applied Math., 78(2) (2012), 233-243.
[6] K.M. Hemida and M.S. Mohamed, Application of the homotopy analysis method to fractional order gas dynamics equation, Adv. Research in
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