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Computer Physics Communications 185 (2014) 1947–1954
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Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc
New analytical method for gas dynamics equation arising in shock fronts Sunil Kumar c , Mohammad Mehdi Rashidi a,b,∗ a
Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran
b
University of Michigan-Shanghai Jiao Tong University Joint Institute, Jiao Tong University, Shanghai, People’s Republic of China
c
Department of Mathematics, National Institute of Technology, Jamshedpur, 801014, Jharkhand, India
article
info
Article history: Received 16 July 2013 Received in revised form 6 January 2014 Accepted 24 March 2014 Available online 3 April 2014 Keywords: Laplace transform method Fractional gas dynamics equation Fractional derivatives Analytic solution Mittag-Leffler function Fractional homotopy analysis transform method (FHATM)
abstract This work suggests a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving nonlinear homogeneous and nonhomogeneous time-fractional gas dynamics equations. The FHATM is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. In this paper, it can be observed that the auxiliary parameter }, which controls the convergence of the HATM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more qualitative difference in analysis between HATM and other methods. The solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. The proposed method is illustrated by solving some numerical examples. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The fractional calculus has a long history, starting from 30 September 1695 when the derivative of order α = 1/2 was described by Leibniz. The theory of derivatives and integrals of noninteger order goes back to Leibniz, Liouville, Grunewald, Letnikov, and Riemann. Many important phenomena are well described by fractional differential equations in electromagnetics, acoustics, viscoelasticity, electrochemistry, and materials science. Fractional order ordinary differential equations, as generalizations of classical integer-order ordinary differential equations, are increasingly used to model problems in fluid flow, mechanics, viscoelasticity, biology, physics and engineering, and other applications. Fractional derivatives provide an excellent medium for the description of memory and hereditary properties of various materials and processes. Half-order derivatives and integrals proved to be
∗ Corresponding author at: Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran. Tel.: +98 811 8257409; fax: +98 811 8257400. E-mail addresses: [email protected], [email protected] (S. Kumar), [email protected], [email protected] (M.M. Rashidi). http://dx.doi.org/10.1016/j.cpc.2014.03.025 0010-4655/© 2014 Elsevier B.V. All rights reserved.
more useful for the formulation of certain electrochemical problems than the classical models [1–6]. Gas dynamics are the mathematical expressions of conservation laws that exist in engineering practices such as conservation of mass, conservation of momentum, conservation of energy, etc. Few different types of gas dynamics equations in physics have been solved by Jafari et al. [7], Jawad et al. [8], Elizarova [9], Evans and Bulut [10], and Steger and Warming [11] by applying various kinds of analytical and numerical methods. In 1985, Aziz and Anderson [12] used a pocket computer to solve some problems arising in gas dynamics. In 2003, Rasulov and Karaguler [13] applied the difference scheme for solving nonlinear system of equations of gas dynamics problems for a class of discontinuous functions. In 1990, Liu [14] studied nonlinear hyperbolic–parabolic partial differential equations related to gas dynamics and mechanics. Recently, Biazar and Eslami [15], Das and Kumar [16], and Kumar et al. [17] have used differential transform and homotopy perturbation transform method to obtain the solutions of the homogeneous and nonhomogeneous time-fractional gas dynamics equation. In this paper, the homotopy analysis transform method (HATM) basically illustrates how the Laplace transform can be used to approximate the solutions of linear and nonlinear fractional
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differential equations by manipulating the homotopy analysis method (HAM). The proposed method involves coupling of the HAM and Laplace transform method. The main advantage of this proposed method is its capability of combining two powerful methods for obtaining rapid convergent series for fractional partial differential equations arising in sciences and engineering. This method provides us with a convenient way to control the convergence of the series solution and allows the adjustment of the convergence region wherever it is needed. The HATM solution generally agrees with the exact solution at large domains as compared with HPM and HAM solutions. The HAM was first introduced and applied by Liao [18–24] in 1992. Cheng et al. [25] derived an explicit series approximation solution for an American put option by means of the HAM with the help of the Laplace transformation. The Black–Scholes formula is used as a model for valuing European or American call and put options on a non-dividend paying stock. The Black–Scholes option pricing equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear subproblems in a fixed domain through the deformation equations. Cheng et al. [26] investigated a train of periodic deepwater waves propagating on a steady shear current with a vertical distribution of vorticity via HAM. They found that the HAM can be applied to solve many types of nonlinear partial differential equations with variable coefficients in science, finance, and also engineering. In 2010, Liao [27] used an optimal homotopy analysis approach with three convergences to solve the nonlinear Blasius equation, which was applied successfully by Wang for solving the Kawahara equation [28]. In 2009, Liang and Jeffrey [29] compared homotopy analysis and homotopy perturbation method through an evolution equation. The HAM has been successfully applied by many researchers for solving linear and nonlinear partial differential equations [30–40]. In recent years, many authors have paid attention to studying the solutions of linear and nonlinear partial differential equations by using various methods combined with the Laplace transform. These include the Laplace decomposition methods [41,42] and the homotopy perturbation transform method [43–45]. Recently, Khan et al. [46], Kumar et al. [47,48], and Arife et al. [49] have coupled homotopy analysis with Laplace transform method to obtain the solutions of the nonlinear partial differential and Volterra integral equation. The main aim of this article is to present approximate analytical solutions of time-fractional homogeneous and nonhomogeneous gas dynamics equations by using HATM. We discuss how to solve time-fractional gas dynamics equations by using HATM. Our concern in this work is to consider the numerical solution of the nonlinear gas dynamics equations with time-fractional derivatives of the form:
∂u ∂αu +u − u (1 − u) = 0, 0 < α ≤ 1, (1.1) ∂tα ∂x with initial condition u0 (x, t ) = h (x) and α being the parameter describing the order of the time-fractional derivatives. In case of α = 1, Eq. (1.1) reduces to the classical gas dynamics equation [10]. Definition 1.1. The Laplace transform L [f (t )] of the Riemann– Liouville fractional integral is defined as [4]
L Itα f (t ) = s−α F (s) .
(1.2)
Definition 1.2. The Laplace transform L [f (t )] of the Caputo fractional derivative is defined as [4]
Definition 1.3. The Mittag-Leffler function Eα (z ) with α > 0 is defined by the following series representation, valid in the whole complex plane [50]: Eα (z ) =
∞
zn
n =0
Γ (α n + 1)
.
(1.4)
2. Basic idea of the new Fractional Homotopy Analysis Transform Method (FHATM) To illustrate the basic idea of the HATM for the fractional partial differential equation, we consider the following fractional partial differential equation as Dnt α u (x, t ) + R [x] u (x, t ) + N [x] u (x, t ) = g (x, t ) , t > 0, x ∈ R, , n − 1 < nα ≤ n,
(2.1)
∂ nα , ∂ t nα
Dnt α
where = R [x] is the linear operator in x, N [x] is the general nonlinear operator in x, and g (x, t ) are continuous functions. For simplicity, we ignore all initial and boundary conditions, which can be treated in similar way. On applying Laplace transform on both sides of Eq. (2.1), we get L Dnt α u (x, t ) + L [R [x] u (x, t ) + N [x] u (x, t )]
= L [g (x, t )] .
(2.2)
Now, using the differentiation property of the Laplace transform, we have L [u (x, t )] −
n −1 1 (nα−k−1) k s u (x, 0) snα k=0
1 L (R [x] u (x, t ) + N [x] u (x, t ) − g (x, t )) = 0. snα We define the nonlinear operator:
+
(2.3)
n −1 1 (nα−k−1) k s u (x, 0) snα k=0
N [φ (r , t ; q)] = L [φ (x, t ; q)] −
1 L (R [x] u (x, t ) + N [x] u (x, t ) − g (x, t )) , (2.4) snα where q ∈ [0, 1] is an embedding parameter and φ (x, t ; q) is the real function of x, t, and q. By means of generalizing the traditional homotopy methods, Liao [18–24] constructed the zeroorder deformation equation:
+
(1 − q) L [φ (x, t ; q) − u0 (x, t )] = } q H (x, t ) N [φ (x, t ; q)] ,
(2.5)
where } is a nonzero auxiliary parameter, H (x, t ) ̸= 0 is an auxiliary function, u0 (x, t ) is an initial guess of u (x, t ), and φ (x, t ; q) is an unknown function. It is important that one has great freedom to choose the auxiliary parameter in HATM. Obviously, when q = 0 and q = 1, it holds
φ (x, t ; 0) = u0 (x, t ) ,
φ (x, t ; 1) = u (x, t ) ,
(2.6)
respectively. Thus, as q increases from 0 to 1, the solution varies from the initial guess u0 (x, t ) to the solution u (x, t ). Expanding φ (x, t ; q) in Taylor’s series with respect to q, we have
φ (x, t ; q) = u0 (x, t ) +
∞
qm um (x, t ) ,
(2.7)
m=1
L Dαt f (t ) = sα F (s) −
n −1
where
s(α−k−1) f (k) (0) ,
k=0
n − 1 < α ≤ n.
(1.3)
um ( x , t ) =
1 ∂ m φ (x, t ; q) m!
∂ qm
q =0
.
(2.8)
S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
The convergence of series solution (2.7) is controlled by }. If the auxiliary linear operator R, the initial guess u0 (x, t ), the auxiliary parameter }, and the auxiliary function H are properly chosen, and the series (2.7) converges at q = 1, then we have u (x, t ) = u0 (x, t ) +
∞
um ( x , t ) ,
(2.9)
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(for more details, see Refs. [25,27,51,52]). For the current approximation, the optimal value of } is given by the minimum value of the Eκ corresponding to the nonlinear algebraic equation: dEκ d}
= 0.
(2.18)
m=1
which must be one of the solutions of original nonlinear equations. The above expression provides us with a relationship between the initial guess u0 (x, t ) and the exact solution u (x, t ) by means of the terms um (x, t ) (m = 1, 2, 3, . . .), which are still to be determined. We define the vector as
⃗n = {u0 (x, t ) , u1 (x, t ) , u2 (x, t ) , . . . , un (x, t )} . u
(2.10)
Differentiating the zero-order deformation equation (2.5) m times with respect to embedding parameter q and then setting q = 0 and finally dividing them by m!, we obtain the mth-order deformation equation:
⃗m−1 , x, t . (2.11) L [um (x, t ) − χm um−1 (x, t )] = } q H (x, t ) Rm u
Operating the inverse Laplace transform on both sides, we get um (x, t ) = χm um−1 (x, t )
+ } q L−1 H (x, t ) Rm u⃗m−1 , x, t ,
⃗m−1 , x, t = Rm u
1 ∂ m φ (x, t ; q)
∂ qm
m!
,
∂u ∂αu +u − u (1 − u) = 0, ∂tα ∂x
0 < α ≤ 1,
Applying the Laplace transform on both sides in Eq. (3.1) and after using the differentiation property of Laplace transform, we get
q =0
sα L [u (x, t )] − sα−1 u (x, 0) + L u ux − u + u2 = 0.
0, 1,
m ≤ 1, m > 1.
um (x, t ) = χm um−1 (x, t ) − χm
+ h H (x, t ) L
−1
⃗ m , x, t Rm u
.
(2.14)
um (x, t ) ,
(2.15)
when M → ∞, we get an accurate approximation of the original equation (2.1). The solution of problem (2.1) is obtained by putting these um (x, t )’s in (2.9) and choosing a suitable value of } for the convergence of the series. The residual error formula for the HATM solution is defined as follows: dt α
= 0.
(3.3)
(3.4)
with property £ [c] = 0, where c is constant. We now define a nonlinear operator as N [ϕ (x, t ; q)] = L [ϕ (x, t ; q)] − s−1 e−x
+ s−α L ϕ (x, t ; q) ϕx (x, t ; q) − ϕ (x, t ; q) + ϕ 2 (x, t ; q) . (3.5)
m=0
dα u (x, t )
u (x, t ) − e−x + L−1 s−α L u ux − u + u2 £ [φ (x, t ; q)] = L [φ (x, t ; q)] ,
It is easy to obtain um (x, t ) for m ≥ 1, and at Mth-order, we have
Res =
(3.2)
We choose the linear operator as
s(nα−k−1) uk (x, 0)
k=0
u (x, t ) =
On simplifying and operating the inverse Laplace transform on both sides, we get
n −1
M
(3.1)
(2.13)
Operating the inverse Laplace transform on both sides of Eq. (2.11), we have
+ u (x, t )
du (x, t )
1
(Q + 1) (W + 1)
i=0 j=0
(1 − q) £ [ϕ (x, t ; q) − u0 (x, t )] = q } N [ϕ (x, t ; q)] .
(2.16)
Res
κ
u (i∆x, j∆t )
, (2.17)
z =0
with Q = W = 1, ∆x = 0.05/Q , and ∆t = 0.05/W . It should be stated that the selection of optimal value of the auxiliary parameter in the HATM and HAM has approximately similar procedures
(3.6)
Obviously, when q = 0 and q = 1,
ϕ (x, t ; 0) = u0 (x, t ) ,
ϕ (x, t ; 1) = u (x, t ) .
(3.7)
Thus, we obtain the mth-order deformation equation:
In order to select the optimal value of the auxiliary parameter (}), the averaged residual error is introduced as Q W
Using the above definition, with assumption H (x, t ) = 1, we construct the zeroth-order deformation equation:
⃗m−1 , x, t . £ [um (x, t ) − χm um−1 (x, t )] = } Rm u
dx
− u (x, t ) (1 − u (x, t )) .
Eκ =
Example 1. We consider the following homogeneous nonlinear time-fractional gas dynamics equation [16,17] as
with initial condition u (x, 0) = e−x , and the solution u (x, t ) = e−x+t is an exact solution of standard gas dynamics, that is, for α = 1.
and
χm =
In this section, we discuss the implementation of our new numerical method and investigate its accuracy and stability by applying it to numerical examples with known n analytical solutions. Three numerical examples of nonlinear fractional order homogeneous and nonhomogeneous time-fractional gas dynamics equations are solved to demonstrate the performance and efficiency of the HAM coupled with Laplace transform method.
(2.12)
where
3. Illustrative examples
(3.8)
Operating the inverse Laplace transform on both sides in Eq. (3.9), we get
⃗m−1 , x, t um (x, t ) = χm um−1 (x, t ) + } q L−1 Rm u
,
(3.9)
where
⃗m−1 , x, t = L [um−1 (x, t )] − (1 − χm ) e−x Rm u
+s
−α
L
m−1
m−1
k=0
um−1−k (uk )x − um−1 +
k=0
um−1−k uk .
(3.10)
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S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
u4 (x, t ) = −
−
} (1 + })3 e−x t α
+
Γ (α + 1) 3 }3 (1 + }) e−x t 3α Γ (3α + 1)
3 }2 (1 + })2 e−x t 2α
+
Γ (2α + 1) }4 e−x t 4α Γ (4α + 1)
,....
In a similar manner, the rest of the components un (x, t ) for n ≥ 5 can be completely obtained and the series solutions are thus entirely determined. Hence, the solution of Eq. (3.1) is given as u ( x , t ) = u0 ( x , t ) +
∞
um (x, t ) .
(3.12)
m=0
Fig. 1. Plot of u (x, t ) versus time t for different values of α at x = 1 and } = −1.
However, mostly, the results given by the Laplace decomposition method and homotopy perturbation transform method converge to the corresponding numerical solutions in a rather small region. However, different from those two methods, the HATM provides us with a simple way to adjust and control the convergence region of solution series by choosing a proper value for the auxiliary parameter }. It is easy to observe that the so-called Laplace decomposition method [41,42] and homotopy perturbation transform [43–45] are only special cases of the new proposed method (HATM), when } = −1. It can be observed that the proposed method in this work generalizes these two methods. If set } = −1, then u ( x, t )
=e
−x
= e −x
Fig. 2. Comparison between the exact and approximate solutions at x = 1, a = 1, and } = −1.
Now the solution of mth-order deformation equation (3.8) is given as um (x, t ) = (χm + }) um−1 − } (1 − χm ) e−x
+ }L
−1
s
−α
L
m−1
m−1
um−1−k (uk )x − um−1 +
k=0
um−1−k uk
.
k=0
(3.11) Using the initial approximation u0 (x, t ) = u (x, 0) = e−x and the iterative scheme (3.11), we obtain the various iterates: u1 (x, t ) = −
} e −x t α
, Γ (α + 1) } (1 + }) e−x t α }2 e−x t 2α u2 (x, t ) = − + , Γ (α + 1) Γ (2α + 1) u3 (x, t ) = −
−
} (1 + })2 e−x t α
Γ (α + 1)
}3 e−x t 3α
Γ (3α + 1)
,
+
2 }2 (1 + }) e−x t 2α
Γ (2α + 1)
1+
tα
Γ (α + 1)
∞
tk α
k=0
Γ (kα + 1)
+
t 2α
Γ (2α + 1)
+
t 3α
Γ (3α + 1)
+ ···
= e −x E α ( t α ) .
(3.13)
As α = 1, this series has the closed form et −x , which is an exact solution of the classical gas dynamics. The above result is in complete agreement with that of Das and Kumar [16]. The solution u (x, t ) = et −x grows exponentially with time t. However, higher degree of accuracy may be obtained by using higher degree polynomials approximations. Fig. 1 shows the behavior of the approximate solution u (x, t ) for different fractional Brownian motion α = 0.7, 0.8, 0.9 and for standard motion, that is, at α = 1 for Example 1. It is seen from Fig. 1 that the solution obtained by HATM increases very rapidly with increases in t at the value of x = 1 and } = −1. Fig. 2 shows the comparison between the well-known exact solution and approximate solution obtained by the proposed method in less order of approximations. Result indicates the performance of the method for nonlinear gas dynamics in accurately approximating the solution of less computational cost. Fig. 3 shows the } curve obtained from the fourth-order HATM approximation solution of time-fractional gas dynamics equation (3.1). It is obvious from Fig. 3 that the range for the admissible values of } is −1.98 ≤ } < 0. We still have freedom to choose the auxiliary parameter according to } curve. From this figure, the valid regions of convergence correspond to the line segments nearly parallel to the horizontal axis. According to the optimal auxiliary parameter finding method that was displayed in Eqs. (2.16)–(2.18), Fig. 4 illustrates the residual error of Eq. (2.16) for the HATM solution and a different value of the auxiliary parameter; especially for the optimal value of the auxiliary parameter (} = −1). In this example, during numerical computations, only four iterations are considered. It is evident that the accuracy of the results can be improved by introducing more terms and the error converges to zero. Example 2. In this example, we consider the following homogeneous nonlinear time-fractional gas dynamic equation [17]:
∂αu ∂u +u − u (1 − u) log a = 0, ∂tα ∂x
a > 0, 0 < α ≤ 1,
(3.14)
S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
1951
Fig. 3. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.01.
Fig. 4. Plot of residual error of Eq. (2.16) using HATM approximation at α = 1, x = 0.5.
with initial condition u (x, 0) = a−x , and the solution u (x, t ) = at −x is the exact solution for α = 1.
u3 (x, t ) = −
Now, applying the aforesaid technique as Example 1, we define a nonlinear operator as N [ϕ (x, t ; q)] = L [ϕ (x, t ; q)] − s−1 a−x
+ s−α L ϕ (x, t ; q) ϕx (x, t ; q) − ϕ (x, t ; q) log a + ϕ 2 (x, t ; q) log a . (3.15) Thus, we obtain the mth-order deformation equation:
⃗m−1 , x, t um (x, t ) = χm um−1 (x, t ) + } q L−1 Rm u
,
(3.17)
where
⃗m−1 , x, t = L [um−1 (x, t )] − s−1 (1 − χm ) a−x Rm u
m−1
+ s−α L
um−1−k (uk )x − um−1 log a
k=0
m−1
+ log a
um−1−k uk .
(3.18)
k=0
Now the solution of mth-order deformation Eq. (3.16) is given as um (x, t ) = (χm + }) um−1 − } (1 − χm ) a−x
+ }L
−1
s
−α
L
m−1
um−1−k (uk )x
k=0
−
m−1
um−1 −
um−1−k uk
log a
.
(3.19)
k=0
Using the initial approximation u0 (x, t ) = u (x, 0) = a−x and the iterative scheme (3.19), we obtain the various iterates: u1 ( x , t ) = − u2 (x, t ) = −
} a−x (t α log a)
Γ (α + 1)
,
} (1 + }) a−x (t α log a)
Γ (α + 1)
}2 a−x (t α log a)
2
+
Γ (2α + 1)
,
−x
Proceeding in this manner, the rest of the components un (x, t ) for n ≥ 5 can be completely obtained and the series solutions are thus entirely determined. So the solution u (x, t ) of Eq. (3.13), when } = −1, is given as
(3.16)
Operating the inverse Laplace transform on both sides in Eq. (3.16), we get
3 (t α log a)2 }3 a−x (t α log a) − ,.... Γ (2α + 1) Γ (3α + 1)
u (x, t ) = u0 +
⃗m−1 , x, t . £ [um (x, t ) − χm um−1 (x, t )] = } Rm u
Γ (α + 1)
2 } (1 + }) a 2
+
} (1 + })2 a−x (t α log a)
∞
um = a
−x
1+
m=0
+
(t α log a)3 Γ (3α + 1)
= a− x
t α log a
Γ (α + 1)
+
(t α log a)2 Γ (2α + 1)
+ ···
∞ (t α log a)k = a−x Eα (t α log a) . Γ k α + 1 ( ) k=0
(3.20)
Now for the standard case, that is, for α = 1, the series has the form at −x , which is an exact solution of the classical gas dynamics equation (3.14). The above result is in complete agreement with the results of Kumar et al. [16] Fig. 5 shows the behavior of the approximate solution u (x, t ) for different fractional Brownian motion α = 0.7, 0.8, 0.9 and for standard motion, that is, at α = 1 for Example 2. It is seen from Fig. 5 that the solution obtained by HATM increases very rapidly with the increases in t at the value of x = 1, a = 2, and } = −1. Fig. 6 shows the comparison between the well-known exact solution and the approximate solution obtained by the proposed method in less order of approximations. Figs. 7 and 8 show the } curve obtained from the fourth-order HATM approximation solution of time-fractional gas dynamics equation (3.14). From these figures, the valid regions of convergence correspond to the line segments nearly parallel to the horizontal axis. Example 3. In this example, we consider the following inhomogeneous fractional gas dynamics equation [15] as follows:
∂u ∂αu +u + (1 + t )2 u2 = x2 , 0 < α ≤ 1, (3.21) ∂tα ∂x x with initial condition u (x, 0) = x, and the solution u (x, t ) = (1+ t) is the exact solution for α = 1.
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S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
Fig. 5. Plot of u (x, t ) versus time t for different values of α at x = 1, a = 2, and } = −1.
Fig. 8. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.01, a = 1.5.
We now define a nonlinear operator as N [ϕ (x, t ; q)] = L [ϕ (x, t ; q)] − s−1 x
+ s−α L ϕ (x, t ; q) ϕx (x, t ; q) + (1 + t )2 ϕ 2 (x, t ; q) − x2 . (3.22) Thus, we obtain the mth-order deformation equation:
⃗m−1 , x, t . £ [um (x, t ) − χm um−1 (x, t )] = } Rm u
(3.23)
Operating the inverse Laplace transform on both sides in Eq. (3.23), we get
⃗m−1 , x, t um (x, t ) = χm um−1 (x, t ) + } q L−1 Rm u
,
(3.24)
where
⃗m−1 , x, t = L [um−1 ] − } (1 − χm ) Rm u
m−1
+s
−α
L
x s
+
x2
um−1−k (uk )x + (1 + t )
sα+1
m−1 2
k=0
Fig. 6. Comparison between the exact and approximate solutions at x = 1, α = 1, a = 2, and } = −1.
um−1−k uk .
(3.25)
k=0
Now the solution of mth-order deformation equations (3.24) is given as x2 t α
um (x, t ) = (χm + }) um−1 − } (1 − χm ) x +
m−1
+ L−1 s−α L
Γ (α + 1)
m−1
um−1−k (uk )x + (1 + t )2
k=0
um−1−k uk
.
k=0
(3.26) Using the initial approximation u0 (x, t ) = u (x, 0) = x and the iterative scheme (3.25), we obtain the various iterates: u1 (x, t ) =
Fig. 7. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.01, a = 2.
}x t α
2}x2 t α+1
2}x2 t α+2
, Γ (α + 1) Γ (α + 2) Γ (α + 3) } (} + 1) x t α 2} (1 + }) x2 t α+1 u2 (x, t ) = + Γ (α + 1) Γ (α + 2) 2 α+2 2 2} (1 + }) x t 2 } x (x + 1) t 2α + + Γ (α + 3) Γ (2α + 1) 4 }2 x2 (α + 2) t 2α+1 2 }2 x2 (2x + 3) t 2α+1 + + Γ (2α + 2) Γ (2α + 3) 2 }2 x2 (α + 3) (α + 2) t 2α+2 2 }2 x2 (2x + 3) t 2α+2 + + Γ (2α + 3) Γ (2α + 3) +
+
S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
1953
method, and studied its validity in a wide range with three examples of nonlinear time-fractional gas dynamics equations. The method gives more realistic series solutions that converge very rapidly in nonlinear time-fractional gas dynamics equations. It is worth mentioning that the method is capable of reducing the volume of the computational work as compared to the classical methods with high accuracy of the numerical result and will considerably benefit mathematicians and scientists working in the field of fractional calculus. It may be concluded that the HATM is very powerful and efficient in finding the approximate solutions as well as analytical solutions of many physical problems arising in sciences and engineering. Acknowledgments The authors are very grateful to the anonymous referees for carefully reading the paper and for their constructive comments and suggestions that have improved the paper. Fig. 9. Plot of u (x, t ) versus time t for different values of α at x = 1 and } = −1.
Fig. 10. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.001.
+
8 }2 x3 (α + 3) t 2α+2
+
8 }2 x3 (α + 4) t 2α+3
Γ (2α + 3) Γ (2α + 4)
+
4 }2 x3 (α + 4) (α + 3) t 2α+3
+
4 }2 x3 (α + 4) (α + 5) t 2α+4
Γ (2α + 4) Γ (2α + 5)
,....
Continuing in this manner, the rest of the components un (x, t ) for n ≥ 3 can be completely obtained and the series solutions are thus entirely determined. Fig. 9 shows the behavior of the approximate solution u (x, t ) for different fractional Brownian motion α = 0.7, 0.8, 0.9 and for standard motion, that is, at α = 1 for Example 3. It is seen from Fig. 9 that the solution obtained by HATM increases very rapidly with the increases in t at the values of x = 1 and } = −1. Fig. 10 shows the } curve obtained from the fourth-order HATM approximation solution of time-fractional gas dynamics equation (3.21). It is obvious from Fig. 10 that the range for the admissible values of } is −2.13 ≤ } < 0. 4. Concluding remarks This paper develops an effective and new modification of HAM, which is coupled with homotopy analysis and Laplace transform
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Contents lists available at ScienceDirect
Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc
New analytical method for gas dynamics equation arising in shock fronts Sunil Kumar c , Mohammad Mehdi Rashidi a,b,∗ a
Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran
b
University of Michigan-Shanghai Jiao Tong University Joint Institute, Jiao Tong University, Shanghai, People’s Republic of China
c
Department of Mathematics, National Institute of Technology, Jamshedpur, 801014, Jharkhand, India
article
info
Article history: Received 16 July 2013 Received in revised form 6 January 2014 Accepted 24 March 2014 Available online 3 April 2014 Keywords: Laplace transform method Fractional gas dynamics equation Fractional derivatives Analytic solution Mittag-Leffler function Fractional homotopy analysis transform method (FHATM)
abstract This work suggests a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving nonlinear homogeneous and nonhomogeneous time-fractional gas dynamics equations. The FHATM is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. In this paper, it can be observed that the auxiliary parameter }, which controls the convergence of the HATM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more qualitative difference in analysis between HATM and other methods. The solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. The proposed method is illustrated by solving some numerical examples. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The fractional calculus has a long history, starting from 30 September 1695 when the derivative of order α = 1/2 was described by Leibniz. The theory of derivatives and integrals of noninteger order goes back to Leibniz, Liouville, Grunewald, Letnikov, and Riemann. Many important phenomena are well described by fractional differential equations in electromagnetics, acoustics, viscoelasticity, electrochemistry, and materials science. Fractional order ordinary differential equations, as generalizations of classical integer-order ordinary differential equations, are increasingly used to model problems in fluid flow, mechanics, viscoelasticity, biology, physics and engineering, and other applications. Fractional derivatives provide an excellent medium for the description of memory and hereditary properties of various materials and processes. Half-order derivatives and integrals proved to be
∗ Corresponding author at: Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran. Tel.: +98 811 8257409; fax: +98 811 8257400. E-mail addresses: [email protected], [email protected] (S. Kumar), [email protected], [email protected] (M.M. Rashidi). http://dx.doi.org/10.1016/j.cpc.2014.03.025 0010-4655/© 2014 Elsevier B.V. All rights reserved.
more useful for the formulation of certain electrochemical problems than the classical models [1–6]. Gas dynamics are the mathematical expressions of conservation laws that exist in engineering practices such as conservation of mass, conservation of momentum, conservation of energy, etc. Few different types of gas dynamics equations in physics have been solved by Jafari et al. [7], Jawad et al. [8], Elizarova [9], Evans and Bulut [10], and Steger and Warming [11] by applying various kinds of analytical and numerical methods. In 1985, Aziz and Anderson [12] used a pocket computer to solve some problems arising in gas dynamics. In 2003, Rasulov and Karaguler [13] applied the difference scheme for solving nonlinear system of equations of gas dynamics problems for a class of discontinuous functions. In 1990, Liu [14] studied nonlinear hyperbolic–parabolic partial differential equations related to gas dynamics and mechanics. Recently, Biazar and Eslami [15], Das and Kumar [16], and Kumar et al. [17] have used differential transform and homotopy perturbation transform method to obtain the solutions of the homogeneous and nonhomogeneous time-fractional gas dynamics equation. In this paper, the homotopy analysis transform method (HATM) basically illustrates how the Laplace transform can be used to approximate the solutions of linear and nonlinear fractional
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S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
differential equations by manipulating the homotopy analysis method (HAM). The proposed method involves coupling of the HAM and Laplace transform method. The main advantage of this proposed method is its capability of combining two powerful methods for obtaining rapid convergent series for fractional partial differential equations arising in sciences and engineering. This method provides us with a convenient way to control the convergence of the series solution and allows the adjustment of the convergence region wherever it is needed. The HATM solution generally agrees with the exact solution at large domains as compared with HPM and HAM solutions. The HAM was first introduced and applied by Liao [18–24] in 1992. Cheng et al. [25] derived an explicit series approximation solution for an American put option by means of the HAM with the help of the Laplace transformation. The Black–Scholes formula is used as a model for valuing European or American call and put options on a non-dividend paying stock. The Black–Scholes option pricing equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear subproblems in a fixed domain through the deformation equations. Cheng et al. [26] investigated a train of periodic deepwater waves propagating on a steady shear current with a vertical distribution of vorticity via HAM. They found that the HAM can be applied to solve many types of nonlinear partial differential equations with variable coefficients in science, finance, and also engineering. In 2010, Liao [27] used an optimal homotopy analysis approach with three convergences to solve the nonlinear Blasius equation, which was applied successfully by Wang for solving the Kawahara equation [28]. In 2009, Liang and Jeffrey [29] compared homotopy analysis and homotopy perturbation method through an evolution equation. The HAM has been successfully applied by many researchers for solving linear and nonlinear partial differential equations [30–40]. In recent years, many authors have paid attention to studying the solutions of linear and nonlinear partial differential equations by using various methods combined with the Laplace transform. These include the Laplace decomposition methods [41,42] and the homotopy perturbation transform method [43–45]. Recently, Khan et al. [46], Kumar et al. [47,48], and Arife et al. [49] have coupled homotopy analysis with Laplace transform method to obtain the solutions of the nonlinear partial differential and Volterra integral equation. The main aim of this article is to present approximate analytical solutions of time-fractional homogeneous and nonhomogeneous gas dynamics equations by using HATM. We discuss how to solve time-fractional gas dynamics equations by using HATM. Our concern in this work is to consider the numerical solution of the nonlinear gas dynamics equations with time-fractional derivatives of the form:
∂u ∂αu +u − u (1 − u) = 0, 0 < α ≤ 1, (1.1) ∂tα ∂x with initial condition u0 (x, t ) = h (x) and α being the parameter describing the order of the time-fractional derivatives. In case of α = 1, Eq. (1.1) reduces to the classical gas dynamics equation [10]. Definition 1.1. The Laplace transform L [f (t )] of the Riemann– Liouville fractional integral is defined as [4]
L Itα f (t ) = s−α F (s) .
(1.2)
Definition 1.2. The Laplace transform L [f (t )] of the Caputo fractional derivative is defined as [4]
Definition 1.3. The Mittag-Leffler function Eα (z ) with α > 0 is defined by the following series representation, valid in the whole complex plane [50]: Eα (z ) =
∞
zn
n =0
Γ (α n + 1)
.
(1.4)
2. Basic idea of the new Fractional Homotopy Analysis Transform Method (FHATM) To illustrate the basic idea of the HATM for the fractional partial differential equation, we consider the following fractional partial differential equation as Dnt α u (x, t ) + R [x] u (x, t ) + N [x] u (x, t ) = g (x, t ) , t > 0, x ∈ R, , n − 1 < nα ≤ n,
(2.1)
∂ nα , ∂ t nα
Dnt α
where = R [x] is the linear operator in x, N [x] is the general nonlinear operator in x, and g (x, t ) are continuous functions. For simplicity, we ignore all initial and boundary conditions, which can be treated in similar way. On applying Laplace transform on both sides of Eq. (2.1), we get L Dnt α u (x, t ) + L [R [x] u (x, t ) + N [x] u (x, t )]
= L [g (x, t )] .
(2.2)
Now, using the differentiation property of the Laplace transform, we have L [u (x, t )] −
n −1 1 (nα−k−1) k s u (x, 0) snα k=0
1 L (R [x] u (x, t ) + N [x] u (x, t ) − g (x, t )) = 0. snα We define the nonlinear operator:
+
(2.3)
n −1 1 (nα−k−1) k s u (x, 0) snα k=0
N [φ (r , t ; q)] = L [φ (x, t ; q)] −
1 L (R [x] u (x, t ) + N [x] u (x, t ) − g (x, t )) , (2.4) snα where q ∈ [0, 1] is an embedding parameter and φ (x, t ; q) is the real function of x, t, and q. By means of generalizing the traditional homotopy methods, Liao [18–24] constructed the zeroorder deformation equation:
+
(1 − q) L [φ (x, t ; q) − u0 (x, t )] = } q H (x, t ) N [φ (x, t ; q)] ,
(2.5)
where } is a nonzero auxiliary parameter, H (x, t ) ̸= 0 is an auxiliary function, u0 (x, t ) is an initial guess of u (x, t ), and φ (x, t ; q) is an unknown function. It is important that one has great freedom to choose the auxiliary parameter in HATM. Obviously, when q = 0 and q = 1, it holds
φ (x, t ; 0) = u0 (x, t ) ,
φ (x, t ; 1) = u (x, t ) ,
(2.6)
respectively. Thus, as q increases from 0 to 1, the solution varies from the initial guess u0 (x, t ) to the solution u (x, t ). Expanding φ (x, t ; q) in Taylor’s series with respect to q, we have
φ (x, t ; q) = u0 (x, t ) +
∞
qm um (x, t ) ,
(2.7)
m=1
L Dαt f (t ) = sα F (s) −
n −1
where
s(α−k−1) f (k) (0) ,
k=0
n − 1 < α ≤ n.
(1.3)
um ( x , t ) =
1 ∂ m φ (x, t ; q) m!
∂ qm
q =0
.
(2.8)
S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
The convergence of series solution (2.7) is controlled by }. If the auxiliary linear operator R, the initial guess u0 (x, t ), the auxiliary parameter }, and the auxiliary function H are properly chosen, and the series (2.7) converges at q = 1, then we have u (x, t ) = u0 (x, t ) +
∞
um ( x , t ) ,
(2.9)
1949
(for more details, see Refs. [25,27,51,52]). For the current approximation, the optimal value of } is given by the minimum value of the Eκ corresponding to the nonlinear algebraic equation: dEκ d}
= 0.
(2.18)
m=1
which must be one of the solutions of original nonlinear equations. The above expression provides us with a relationship between the initial guess u0 (x, t ) and the exact solution u (x, t ) by means of the terms um (x, t ) (m = 1, 2, 3, . . .), which are still to be determined. We define the vector as
⃗n = {u0 (x, t ) , u1 (x, t ) , u2 (x, t ) , . . . , un (x, t )} . u
(2.10)
Differentiating the zero-order deformation equation (2.5) m times with respect to embedding parameter q and then setting q = 0 and finally dividing them by m!, we obtain the mth-order deformation equation:
⃗m−1 , x, t . (2.11) L [um (x, t ) − χm um−1 (x, t )] = } q H (x, t ) Rm u
Operating the inverse Laplace transform on both sides, we get um (x, t ) = χm um−1 (x, t )
+ } q L−1 H (x, t ) Rm u⃗m−1 , x, t ,
⃗m−1 , x, t = Rm u
1 ∂ m φ (x, t ; q)
∂ qm
m!
,
∂u ∂αu +u − u (1 − u) = 0, ∂tα ∂x
0 < α ≤ 1,
Applying the Laplace transform on both sides in Eq. (3.1) and after using the differentiation property of Laplace transform, we get
q =0
sα L [u (x, t )] − sα−1 u (x, 0) + L u ux − u + u2 = 0.
0, 1,
m ≤ 1, m > 1.
um (x, t ) = χm um−1 (x, t ) − χm
+ h H (x, t ) L
−1
⃗ m , x, t Rm u
.
(2.14)
um (x, t ) ,
(2.15)
when M → ∞, we get an accurate approximation of the original equation (2.1). The solution of problem (2.1) is obtained by putting these um (x, t )’s in (2.9) and choosing a suitable value of } for the convergence of the series. The residual error formula for the HATM solution is defined as follows: dt α
= 0.
(3.3)
(3.4)
with property £ [c] = 0, where c is constant. We now define a nonlinear operator as N [ϕ (x, t ; q)] = L [ϕ (x, t ; q)] − s−1 e−x
+ s−α L ϕ (x, t ; q) ϕx (x, t ; q) − ϕ (x, t ; q) + ϕ 2 (x, t ; q) . (3.5)
m=0
dα u (x, t )
u (x, t ) − e−x + L−1 s−α L u ux − u + u2 £ [φ (x, t ; q)] = L [φ (x, t ; q)] ,
It is easy to obtain um (x, t ) for m ≥ 1, and at Mth-order, we have
Res =
(3.2)
We choose the linear operator as
s(nα−k−1) uk (x, 0)
k=0
u (x, t ) =
On simplifying and operating the inverse Laplace transform on both sides, we get
n −1
M
(3.1)
(2.13)
Operating the inverse Laplace transform on both sides of Eq. (2.11), we have
+ u (x, t )
du (x, t )
1
(Q + 1) (W + 1)
i=0 j=0
(1 − q) £ [ϕ (x, t ; q) − u0 (x, t )] = q } N [ϕ (x, t ; q)] .
(2.16)
Res
κ
u (i∆x, j∆t )
, (2.17)
z =0
with Q = W = 1, ∆x = 0.05/Q , and ∆t = 0.05/W . It should be stated that the selection of optimal value of the auxiliary parameter in the HATM and HAM has approximately similar procedures
(3.6)
Obviously, when q = 0 and q = 1,
ϕ (x, t ; 0) = u0 (x, t ) ,
ϕ (x, t ; 1) = u (x, t ) .
(3.7)
Thus, we obtain the mth-order deformation equation:
In order to select the optimal value of the auxiliary parameter (}), the averaged residual error is introduced as Q W
Using the above definition, with assumption H (x, t ) = 1, we construct the zeroth-order deformation equation:
⃗m−1 , x, t . £ [um (x, t ) − χm um−1 (x, t )] = } Rm u
dx
− u (x, t ) (1 − u (x, t )) .
Eκ =
Example 1. We consider the following homogeneous nonlinear time-fractional gas dynamics equation [16,17] as
with initial condition u (x, 0) = e−x , and the solution u (x, t ) = e−x+t is an exact solution of standard gas dynamics, that is, for α = 1.
and
χm =
In this section, we discuss the implementation of our new numerical method and investigate its accuracy and stability by applying it to numerical examples with known n analytical solutions. Three numerical examples of nonlinear fractional order homogeneous and nonhomogeneous time-fractional gas dynamics equations are solved to demonstrate the performance and efficiency of the HAM coupled with Laplace transform method.
(2.12)
where
3. Illustrative examples
(3.8)
Operating the inverse Laplace transform on both sides in Eq. (3.9), we get
⃗m−1 , x, t um (x, t ) = χm um−1 (x, t ) + } q L−1 Rm u
,
(3.9)
where
⃗m−1 , x, t = L [um−1 (x, t )] − (1 − χm ) e−x Rm u
+s
−α
L
m−1
m−1
k=0
um−1−k (uk )x − um−1 +
k=0
um−1−k uk .
(3.10)
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S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
u4 (x, t ) = −
−
} (1 + })3 e−x t α
+
Γ (α + 1) 3 }3 (1 + }) e−x t 3α Γ (3α + 1)
3 }2 (1 + })2 e−x t 2α
+
Γ (2α + 1) }4 e−x t 4α Γ (4α + 1)
,....
In a similar manner, the rest of the components un (x, t ) for n ≥ 5 can be completely obtained and the series solutions are thus entirely determined. Hence, the solution of Eq. (3.1) is given as u ( x , t ) = u0 ( x , t ) +
∞
um (x, t ) .
(3.12)
m=0
Fig. 1. Plot of u (x, t ) versus time t for different values of α at x = 1 and } = −1.
However, mostly, the results given by the Laplace decomposition method and homotopy perturbation transform method converge to the corresponding numerical solutions in a rather small region. However, different from those two methods, the HATM provides us with a simple way to adjust and control the convergence region of solution series by choosing a proper value for the auxiliary parameter }. It is easy to observe that the so-called Laplace decomposition method [41,42] and homotopy perturbation transform [43–45] are only special cases of the new proposed method (HATM), when } = −1. It can be observed that the proposed method in this work generalizes these two methods. If set } = −1, then u ( x, t )
=e
−x
= e −x
Fig. 2. Comparison between the exact and approximate solutions at x = 1, a = 1, and } = −1.
Now the solution of mth-order deformation equation (3.8) is given as um (x, t ) = (χm + }) um−1 − } (1 − χm ) e−x
+ }L
−1
s
−α
L
m−1
m−1
um−1−k (uk )x − um−1 +
k=0
um−1−k uk
.
k=0
(3.11) Using the initial approximation u0 (x, t ) = u (x, 0) = e−x and the iterative scheme (3.11), we obtain the various iterates: u1 (x, t ) = −
} e −x t α
, Γ (α + 1) } (1 + }) e−x t α }2 e−x t 2α u2 (x, t ) = − + , Γ (α + 1) Γ (2α + 1) u3 (x, t ) = −
−
} (1 + })2 e−x t α
Γ (α + 1)
}3 e−x t 3α
Γ (3α + 1)
,
+
2 }2 (1 + }) e−x t 2α
Γ (2α + 1)
1+
tα
Γ (α + 1)
∞
tk α
k=0
Γ (kα + 1)
+
t 2α
Γ (2α + 1)
+
t 3α
Γ (3α + 1)
+ ···
= e −x E α ( t α ) .
(3.13)
As α = 1, this series has the closed form et −x , which is an exact solution of the classical gas dynamics. The above result is in complete agreement with that of Das and Kumar [16]. The solution u (x, t ) = et −x grows exponentially with time t. However, higher degree of accuracy may be obtained by using higher degree polynomials approximations. Fig. 1 shows the behavior of the approximate solution u (x, t ) for different fractional Brownian motion α = 0.7, 0.8, 0.9 and for standard motion, that is, at α = 1 for Example 1. It is seen from Fig. 1 that the solution obtained by HATM increases very rapidly with increases in t at the value of x = 1 and } = −1. Fig. 2 shows the comparison between the well-known exact solution and approximate solution obtained by the proposed method in less order of approximations. Result indicates the performance of the method for nonlinear gas dynamics in accurately approximating the solution of less computational cost. Fig. 3 shows the } curve obtained from the fourth-order HATM approximation solution of time-fractional gas dynamics equation (3.1). It is obvious from Fig. 3 that the range for the admissible values of } is −1.98 ≤ } < 0. We still have freedom to choose the auxiliary parameter according to } curve. From this figure, the valid regions of convergence correspond to the line segments nearly parallel to the horizontal axis. According to the optimal auxiliary parameter finding method that was displayed in Eqs. (2.16)–(2.18), Fig. 4 illustrates the residual error of Eq. (2.16) for the HATM solution and a different value of the auxiliary parameter; especially for the optimal value of the auxiliary parameter (} = −1). In this example, during numerical computations, only four iterations are considered. It is evident that the accuracy of the results can be improved by introducing more terms and the error converges to zero. Example 2. In this example, we consider the following homogeneous nonlinear time-fractional gas dynamic equation [17]:
∂αu ∂u +u − u (1 − u) log a = 0, ∂tα ∂x
a > 0, 0 < α ≤ 1,
(3.14)
S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
1951
Fig. 3. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.01.
Fig. 4. Plot of residual error of Eq. (2.16) using HATM approximation at α = 1, x = 0.5.
with initial condition u (x, 0) = a−x , and the solution u (x, t ) = at −x is the exact solution for α = 1.
u3 (x, t ) = −
Now, applying the aforesaid technique as Example 1, we define a nonlinear operator as N [ϕ (x, t ; q)] = L [ϕ (x, t ; q)] − s−1 a−x
+ s−α L ϕ (x, t ; q) ϕx (x, t ; q) − ϕ (x, t ; q) log a + ϕ 2 (x, t ; q) log a . (3.15) Thus, we obtain the mth-order deformation equation:
⃗m−1 , x, t um (x, t ) = χm um−1 (x, t ) + } q L−1 Rm u
,
(3.17)
where
⃗m−1 , x, t = L [um−1 (x, t )] − s−1 (1 − χm ) a−x Rm u
m−1
+ s−α L
um−1−k (uk )x − um−1 log a
k=0
m−1
+ log a
um−1−k uk .
(3.18)
k=0
Now the solution of mth-order deformation Eq. (3.16) is given as um (x, t ) = (χm + }) um−1 − } (1 − χm ) a−x
+ }L
−1
s
−α
L
m−1
um−1−k (uk )x
k=0
−
m−1
um−1 −
um−1−k uk
log a
.
(3.19)
k=0
Using the initial approximation u0 (x, t ) = u (x, 0) = a−x and the iterative scheme (3.19), we obtain the various iterates: u1 ( x , t ) = − u2 (x, t ) = −
} a−x (t α log a)
Γ (α + 1)
,
} (1 + }) a−x (t α log a)
Γ (α + 1)
}2 a−x (t α log a)
2
+
Γ (2α + 1)
,
−x
Proceeding in this manner, the rest of the components un (x, t ) for n ≥ 5 can be completely obtained and the series solutions are thus entirely determined. So the solution u (x, t ) of Eq. (3.13), when } = −1, is given as
(3.16)
Operating the inverse Laplace transform on both sides in Eq. (3.16), we get
3 (t α log a)2 }3 a−x (t α log a) − ,.... Γ (2α + 1) Γ (3α + 1)
u (x, t ) = u0 +
⃗m−1 , x, t . £ [um (x, t ) − χm um−1 (x, t )] = } Rm u
Γ (α + 1)
2 } (1 + }) a 2
+
} (1 + })2 a−x (t α log a)
∞
um = a
−x
1+
m=0
+
(t α log a)3 Γ (3α + 1)
= a− x
t α log a
Γ (α + 1)
+
(t α log a)2 Γ (2α + 1)
+ ···
∞ (t α log a)k = a−x Eα (t α log a) . Γ k α + 1 ( ) k=0
(3.20)
Now for the standard case, that is, for α = 1, the series has the form at −x , which is an exact solution of the classical gas dynamics equation (3.14). The above result is in complete agreement with the results of Kumar et al. [16] Fig. 5 shows the behavior of the approximate solution u (x, t ) for different fractional Brownian motion α = 0.7, 0.8, 0.9 and for standard motion, that is, at α = 1 for Example 2. It is seen from Fig. 5 that the solution obtained by HATM increases very rapidly with the increases in t at the value of x = 1, a = 2, and } = −1. Fig. 6 shows the comparison between the well-known exact solution and the approximate solution obtained by the proposed method in less order of approximations. Figs. 7 and 8 show the } curve obtained from the fourth-order HATM approximation solution of time-fractional gas dynamics equation (3.14). From these figures, the valid regions of convergence correspond to the line segments nearly parallel to the horizontal axis. Example 3. In this example, we consider the following inhomogeneous fractional gas dynamics equation [15] as follows:
∂u ∂αu +u + (1 + t )2 u2 = x2 , 0 < α ≤ 1, (3.21) ∂tα ∂x x with initial condition u (x, 0) = x, and the solution u (x, t ) = (1+ t) is the exact solution for α = 1.
1952
S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
Fig. 5. Plot of u (x, t ) versus time t for different values of α at x = 1, a = 2, and } = −1.
Fig. 8. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.01, a = 1.5.
We now define a nonlinear operator as N [ϕ (x, t ; q)] = L [ϕ (x, t ; q)] − s−1 x
+ s−α L ϕ (x, t ; q) ϕx (x, t ; q) + (1 + t )2 ϕ 2 (x, t ; q) − x2 . (3.22) Thus, we obtain the mth-order deformation equation:
⃗m−1 , x, t . £ [um (x, t ) − χm um−1 (x, t )] = } Rm u
(3.23)
Operating the inverse Laplace transform on both sides in Eq. (3.23), we get
⃗m−1 , x, t um (x, t ) = χm um−1 (x, t ) + } q L−1 Rm u
,
(3.24)
where
⃗m−1 , x, t = L [um−1 ] − } (1 − χm ) Rm u
m−1
+s
−α
L
x s
+
x2
um−1−k (uk )x + (1 + t )
sα+1
m−1 2
k=0
Fig. 6. Comparison between the exact and approximate solutions at x = 1, α = 1, a = 2, and } = −1.
um−1−k uk .
(3.25)
k=0
Now the solution of mth-order deformation equations (3.24) is given as x2 t α
um (x, t ) = (χm + }) um−1 − } (1 − χm ) x +
m−1
+ L−1 s−α L
Γ (α + 1)
m−1
um−1−k (uk )x + (1 + t )2
k=0
um−1−k uk
.
k=0
(3.26) Using the initial approximation u0 (x, t ) = u (x, 0) = x and the iterative scheme (3.25), we obtain the various iterates: u1 (x, t ) =
Fig. 7. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.01, a = 2.
}x t α
2}x2 t α+1
2}x2 t α+2
, Γ (α + 1) Γ (α + 2) Γ (α + 3) } (} + 1) x t α 2} (1 + }) x2 t α+1 u2 (x, t ) = + Γ (α + 1) Γ (α + 2) 2 α+2 2 2} (1 + }) x t 2 } x (x + 1) t 2α + + Γ (α + 3) Γ (2α + 1) 4 }2 x2 (α + 2) t 2α+1 2 }2 x2 (2x + 3) t 2α+1 + + Γ (2α + 2) Γ (2α + 3) 2 }2 x2 (α + 3) (α + 2) t 2α+2 2 }2 x2 (2x + 3) t 2α+2 + + Γ (2α + 3) Γ (2α + 3) +
+
S. Kumar, M.M. Rashidi / Computer Physics Communications 185 (2014) 1947–1954
1953
method, and studied its validity in a wide range with three examples of nonlinear time-fractional gas dynamics equations. The method gives more realistic series solutions that converge very rapidly in nonlinear time-fractional gas dynamics equations. It is worth mentioning that the method is capable of reducing the volume of the computational work as compared to the classical methods with high accuracy of the numerical result and will considerably benefit mathematicians and scientists working in the field of fractional calculus. It may be concluded that the HATM is very powerful and efficient in finding the approximate solutions as well as analytical solutions of many physical problems arising in sciences and engineering. Acknowledgments The authors are very grateful to the anonymous referees for carefully reading the paper and for their constructive comments and suggestions that have improved the paper. Fig. 9. Plot of u (x, t ) versus time t for different values of α at x = 1 and } = −1.
Fig. 10. Plot of } curves for different values of α = 0.7, 0.8, 0.9, 1 at x = 0.5, t = 0.001.
+
8 }2 x3 (α + 3) t 2α+2
+
8 }2 x3 (α + 4) t 2α+3
Γ (2α + 3) Γ (2α + 4)
+
4 }2 x3 (α + 4) (α + 3) t 2α+3
+
4 }2 x3 (α + 4) (α + 5) t 2α+4
Γ (2α + 4) Γ (2α + 5)
,....
Continuing in this manner, the rest of the components un (x, t ) for n ≥ 3 can be completely obtained and the series solutions are thus entirely determined. Fig. 9 shows the behavior of the approximate solution u (x, t ) for different fractional Brownian motion α = 0.7, 0.8, 0.9 and for standard motion, that is, at α = 1 for Example 3. It is seen from Fig. 9 that the solution obtained by HATM increases very rapidly with the increases in t at the values of x = 1 and } = −1. Fig. 10 shows the } curve obtained from the fourth-order HATM approximation solution of time-fractional gas dynamics equation (3.21). It is obvious from Fig. 10 that the range for the admissible values of } is −2.13 ≤ } < 0. 4. Concluding remarks This paper develops an effective and new modification of HAM, which is coupled with homotopy analysis and Laplace transform
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