Solution of Time-Fractional Navier-Stokes Equation by Using

(1)

ISSN 2219-7184; Copyright ICSRS Publication, 2012c 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

1E-mail: [email protected]

2E-mail: [email protected]

3E-mail: m s [email protected]

4E-mail: mohamed [email protected] (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

(2)

1 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 D_t^αu+ (u· ∇)u=−1

ρ∇P +ν∇²u, (1.1)

Where t is the time, u is the velocity vector,P is the pressure, ν is the kine- matics viscosity andρ 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 functionh(t),t >0, is said to be in the spaceC_µ, µ ∈ R , if there exists a real number p > µ , such that h(t) = t^ph₁(t) , where h₁(t) ∈ C(0,∞) , and it is said to be in the space C_µⁿ if and only if h⁽ⁿ⁾ ∈ C_µ , n ∈ N

Definition 2.2. The Riemann-Liouville fractional integral operator (J^α) of orderα ≥0, of a functionh ∈ C_µ , µ≥ −1 is defined as

J^αh(t) = 1 Γ (α)

Z t

0

(t−τ)^α−1h(τ)dτ] (2.1)

(3)

J⁰h(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^αJ^βh(t) =J^α+βh(t), J^αJ^βh(t) =J^βJ^αh(t), J^αt^γ = Γ (γ+ 1)

Γ (α+γ+ 1) t^γ+α . Definition 2.3. The fractional derivative (D^α) of h(t) in the Caputo’s sense is defined as follows

D^αh(t) = 1 Γ (n−α)

Z t

0

(t−τ)^n−α−1h(τ)dτ (2.2) for

n−1< α < n, n ∈ N , t >0 , h ∈ Cⁿ₋₁

The following are two basic properties of Caputo’s fractional Derivative [?]

Let h ∈ Cⁿ₋₁ , n ∈ N then D^αh , 0 ≤ α ≤ n is well defined and D^αh ∈C−1

Let−1< α < n , n ∈ N and h ∈ Cⁿ_µ , µ≥ −1 then

(J^αD^α)h(t) =h(t)−

n−1

X

k=0

h^(k) 0⁺ t^k

k! (2.3)

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)

WhereN is a nonlinear operator for the problem,r and t denote independent variables andu(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) Whereq is the embedding parameter,q ∈[0, 1], h6= 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 andq = 1, it holds that

∈(r, t,0) =u₀(r, t),∈(r, t,1) = u(r, t). (3.3)

(4)

Liao [10] expanded ∈ (x, t, q) in Taylor series with respect to the embedding parameterq, as follows:

∈(r, t, q) =u₀(r, t) +

∞

X

m=1

u_m(r, t)q^m. (3.4) Where

u_m(r, t) = 1 m!

∂^m ∈(r, t, q)

∂q^m q=0

. (3.5)

Assume that the auxiliary linear operator, the initial guess, the auxiliary pa- rameterh and the auxiliary functionH(x, t) are selected such that the series (3.4) is convergent atq = 1 ,then we have from (3.4)

u(r, t) =u₀(r, t) +

∞

X

m=1

u_m(r, t). (3.6)

Let us define the vector

−

→u_n(t) = {u₀(r, t), u₁(r, t), u₂(r, t), . . . u_n(r, t)}. (3.7) Differentiating (3.2)mtimes with respect to q, then settingq = 0 and dividing bym!, that the mth-order deformation equation

L(um(r, t)−χmum−1(r, t)) = hH(r, t)Rm(−→um−1). (3.8) Where

R_m(−→u_m−1) = 1 (m−1)!

∂^m−1 N[∈(r, t, q)]

∂q^m−1

q=0

. (3.9)

And

χ_m =

0 m≤1

1 m >1 (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 Mathe- matica, 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

D^α_tu=P +ν(∂²u

∂r² +1 r

∂u

∂r). (4.1)

(5)

4.1 Application 1

Firstly, we consider

D^α_tu=P + ∂²u

∂r² +1 r

∂u

∂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)] =D^α_t ∈(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)] =D_t^α∈(r, t, q)− ∂² ∈(r, t, q)

∂r² − 1 r

∂ ∈(r, t, q)

∂r −P. (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 uniquelyH(x, t) = 1, and

R_m(−→um−1) = D_t^αum−1− ∂²

∂r²u

m−1

− 1 r

∂

∂rum−1−P (1−χ_m). (4.6) Now the solution of the mth-order deformation equations (3.8) for m ≥ 1 becomes

u_m(r, t) = χ_mum−1(r, t) +hL⁻¹R_m(−→um−1). (4.7) So, the first few terms of the solution are

u0(r, t) = 1−r² u1(r, t) = −h(−4 +P)t^α Γ[1 +α]

u₂(r, t) = −h(1 +h)(−4 +P)t^α

Γ[1 +α] u₃(r, t) = −h(1 +h)²(−4 +P)t^α Γ[1 +α]

u₄(r, t) = −h(1 +h)³(−4 +P)t^α Γ[1 +α]

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)t^α Γ[1 +α]

1 + (1 +h) + (1 +h)²+ (1 +h)³. . .

. (4.8)

(6)

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

u(r, t) = 1−r²+ (−4 +P)t^α

Γ[1 +α] (4.9)

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 time- fractional derivative.

Fig.1 (α= 1) Fig.2 (α = 0.5)

4.2 Application 2

Consider the equation in the form D_t^αu= ∂²u

∂r² + 1 r

∂u

∂r. (4.10)

Subject to the initial condition

u(r,0) =r. (4.11)

Similarly, choosing

N[∈(r, t, q)] = D_t^α ∈(x, t, q)−∂² ∈(r, t, q)

∂r² − 1 r

∂ ∈(r, t, q)

∂r . (4.12)

(7)

Then,

Rm(−→um−1) =D^α_tum−1 − ∂²

∂r²u

m−1− 1 r

∂

∂rum−1. (4.13) So, first terms of the series are

u₀(r, t) =r

u₁(r, t) =− ht^α rΓ[1 +α]

u₂(r, t) =−h(1 +h)t^α

rΓ[1 +α] + h²t^2α r³Γ[1 + 2α]

u₃(r, t) =−h(1 +h)²t^α

rΓ[1 +α] +h²(1 +h)t^2α

r³Γ [1 + 2α] − 9h³t^3α r⁵Γ[1 + 3α]

u₄(r, t) =−h(1 +h)³t^α

rΓ[1 +α] +3h²(1 +h)²t^2α

r³Γ [1 + 2α] −3h³(1 +h)9t^3α

r⁵Γ[1 + 3α] + 9h⁴25t^4α r⁷Γ[1 + 4α]

Then, we can conclude that

u(r, t) = u₀(r, t) +u₁(r, t) +u₂(r, t) +u₃(r, t) +. . . Or

u(r, t) =r− ht^α rΓ[1 +α]

1 + (1 +h) + (1 +h)² + (1 +h)³. . . + h²t^2α

r³Γ[1 + 2α]

1 + 2 (1 +h) + 3(1 +h)²+. . .

− 9h³t^3α r⁵Γ[1 + 3α]

1 + 3 (1 +h) + 6(1 +h)²+. . . +. . .

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+ t^α

rΓ[1 +α] + t^2α

r³Γ[1 + 2α] + 9t^3α

r⁵Γ[1 + 3α] + 25t^4α r⁷Γ[1 + 4α]. . . Therefore the solution is

u(r, t) =r+

∞

X

n=1

1²×3²× · · · ×(2n−3)² r²ⁿ⁻¹

t^nα Γ(nα+ 1) Which is the same solution given by S. Momani and Z. Odibat [11].

Atα= 1 we have

u(r, t) = r+

∞

X

n=1

1²×3²× · · · ×(2n−3)² r²ⁿ⁻¹

tⁿ n!

(8)

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) whenα = 1 andα= 0.5, respectively.

Fig.1 (α= 1) Fig.2 (α = 0.5)

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

[1] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).

(9)

[2] J. Biazar, E. Babolian, G. Kember, A. Nouri and R. Islam, An alter- nate algorithm for computing Adomian polynomials in special cases,Appl.

Math. Comput., 138(2002), 523-529.

[3] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, (1974).

[4] K.M. Hemida, A new approach to the gas dynamics equation: An application of the homotopy analysis method, Adv. Research in Scientific 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 Applied Math., 2(2010), 39-45.

[7] K.S. Miller and B. Ross, An Introduction to the Fractional and Fractional Differential Equations, John Wiley and Sons, New York, (1993).

[8] M. El-Shahed and A. Salem, On the generalized Navier–Stokes equations, Appl. Math. Comput., 156(2004), 287-293.

[9] S.J. Liao, An explicit, totally analytic approximation of Blasius viscous flow problems,Int. J. Nonlinear Mech., 34(1999), 759-778.

[10] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton: Chapman & Hall, (2003).

[11] S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier- Stokes equation by Adomian decomposition method, Appl Math Comp, 177(2006), 488-494.

[12] Y. Luchko and R. Gorenflo, The Initial-Value Problem for Some Frac- tional Differential Equations with Caputo Derivative (Preprint Series A08-98), Fachbereich Mathematik and Informatic, Freie Universitat, Berlin, (1998).