Elzaki transform homotopy perturbation method for solving gas dynamics equation

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 12 | Dec-2013, Available @ http://www.ijret.org 260
ELZAKI TRANSFORM HOMOTOPY PERTURBATION METHOD FOR
SOLVING GAS DYNAMICS EQUATION
Prem Kiran G. Bhadane1
, V. H. Pradhan2
1
Assistant Professor, Department of Applied Sciences, RCPIT, Maharashtra, India,
2
Associate Professor, Department of Applied Mathematics and Humanities, SVNIT, Gujarat, India
omprem07@gmail.com, pradhan65@yahoo.com
Abstract
In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate
analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant
combination of the new integral transform “ELzaki Transform” and the homotopy perturbation method. The method is really capable
of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed
iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this
technique over the decomposition method is that no calculation of Adomian’s polynomials is needed.
Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas
dynamics equation
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1. INTRODUCTION
In recent years, many research workers have paid attention to
find the solution of nonlinear differential equations by using
various methods. Among these are the Adomian
decomposition method [Hashim, Noorani, Ahmed. Bakar,
Ismail and Zakaria, (2006)], the tanh method, the homotopy
perturbation method [ Sweilam, Khader (2009), Sharma and
Giriraj Methi (2011), Jafari, Aminataei (2010), (2011) ], the
differential transform method [(2008)], homotopy perturbation
transform method and the variational iteration method.
Various ways have been proposed recently to deal with these
nonlinearities; one of these combinations is ELzaki transform
and homotopy perturbation method. ELzaki transform is a
useful technique for solving linear differential equations but
this transform is totally incapable of handling nonlinear
equations [6, 7] because of the difficulties that are caused by
the nonlinear terms. This paper is using homotopy
perturbation method to decompose the nonlinear term, so that
the solution can be obtained by iteration procedure. This
means that we can use both ELzaki transform and homotopy
perturbation methods to solve many nonlinear problems. The
main aim of this paper is to consider the effectiveness of the
Elzaki transform homotopy perturbation method in solving
nonlinear gas dynamics equation. This method provides the
solution in a rapid convergent series which may leads the
solution in a closed form. The fact that the proposed technique
solves nonlinear problems without using so-called Adomian's
polynomials is a clear advantage of this algorithm over the
decomposition method.
Gas dynamics is a science in the branch of fluid dynamics
concerned with studying the motion of gases and its effects on
physical systems, based on the principles of fluid mechanics
and thermodynamics. The science arises from the studies of
gas flows, often around or within physical bodies, some
examples of these studies include but not limited to choked
flows in nozzles and valves, shock waves around jets,
aerodynamic heating on atmospheric reentry vehicles and
flows of gas fuel within a jet engine.
The equations of gas dynamics [4] are mathematical
expressions based on the physical laws of conservation,
namely, the laws of conservation of mass, conservation of
momentum, conservation of energy, and so forth. The
nonlinear equations of ideal gas dynamics are applicable for
three types of nonlinear waves like shock fronts, rare factions,
and contact discontinuities. Different types of gas dynamics
equations in physics have been solved by Evans and Bulut
[2002], Polyanin and Zaitsev [2004], Elizarova, Jafari et
al.[2009], Hossein Aminikhah, Ali Jamalian [2013] and Ames,
by applying various kinds of analytical and numerical
methods.
In this paper, we obtained the solution of gas dynamics
equation by using a new method called as ELzaki transform
homotopy perturbation method. We obtain an analytical
approximation to the solution of the nonlinear gas dynamic
equation using ETHPM. The gas dynamics equation as a
nonlinear model is as follows [4]

__________________________________________________________________________________________
∂u
∂t
+ u
∂u
∂x
− u 1 − u = 0, 0 ≤ x ≤ 1, t > 0, (1)
The present paper discusses the analytical approximate
solution of the nonlinear gas dynamic equation.
2. ELZAKI TRANSFORM HOMOTOPY
PERTURBATION METHOD [6]
Consider a general nonlinear non-homogenous partial
differential equation with initial conditions of the form:
Du x, t + Ru x, t + Nu x, t = g x, t (2)
u x, 0 = h x , ut x, 0 = f(x)
where D is linear differential operator of order two, R is linear
differential operator of less order than D , N is the general
nonlinear differential operator and g (x , t ) is the source term.
Taking ELzaki transform on both sides of equation (2), to get:
E Du x, t + E Ru x, t + E Nu x, t = E g x, t (3)
Using the differentiation property of ELzaki transform and
above initial conditions, we have:
E u x, t = v2
E g x, t + v2
h x + v3
f x
− v2
E Ru x, t + Nu x, t (4)
Applying the inverse ELzaki transform on both sides of
equation (4), to find:
u x, t = G x, t − E−1
v2
E Ru x, t + Nu x, t (5)
where G(x, t) represents the term arising from the source term
and the prescribed initial condition.
Now, we apply the homotopy perturbation method,
u x, t = pn
un(x, t)
∞
n=0
(6)
and the nonlinear term can be decomposed as
N u x, t = pn
Hn(u)
∞
n=0
(7)
where Hn(u) are He’s polynomial and given by
Hn u0, u1, … … . , un =
1
n!
∂n
∂pn
N pi
ui
∞
i=0 p=0
,
n = 0,1,2, … … … (8)
Substituting equations (6) and (7) in equation (5), we get:
pn
un x, t
∞
n=0
= G x, t −
p E−1
v2
E R pn
un(x, t)
∞
n=0
+ pn
Hn(u)
∞
n=0
(9)
This is the coupling of the ELzaki transform and the
homotopy perturbation method. Comparing the coefficient of
like powers of p , the following approximations are obtained.
p0
: u0 x, t = G x, t
p1
: u1 x, t = −E−1
v2
E Ru0 x, t + H0(u)
p2
: u2 x, t = −E−1
v2
E Ru1 x, t + H1(u)
p3
: u3 x, t = −E−1
v2
E Ru2 x, t + H2(u)
Then the solution is
u x, t = lim
p→1
un x, t
= u0 x, t + u1 x, t + u2 x, t + … (10)
3. APPLICATIONS
In this section, the effectiveness and the usefulness of ELzaki
transform homotopy perturbation method (ETHPM) are
demonstrated by finding exact solutions of homogeneous and
nonhomogeneous nonlinear gas dynamics equations.
Example 3.1 Consider the nonlinear gas dynamic equation
∂u
∂t
+ u
∂u
∂x
− u 1 − u = 0, (11)
with the following initial condition
u x, 0 = e−x
. (12)
By applying ELzaki transform on both side of equation (11)
E
∂u
∂t
= E u − u2
− u
∂u
∂x
(13)
Using differential properties of ELzaki transform, Eq. (13) can
be written as
1
v
E u(x, t) − vu x, 0 = E u − u2
− u
∂u
∂x
(14)

__________________________________________________________________________________________
Using initial condition (12), Eq. (14) can be written as
E u(x, t) = v2
e−x
+ vE u − u2
− u
∂u
∂x
(15)
Taking inverse ELzaki transform on both sides of Eq. (15), we
get
u(x, t) = e−x
+ E−1
v E u − u2
− u
∂u
∂x
(16)
Now we apply the homotopy perturbation method,
u x, t = pn
un(x, t)
∞
n=0
(17)
N u x, t = pn
Hn(u)
∞
n=0
(18)
Using Eqs. (17)- (18) into Eq. (16), we get
pn
un x, t
∞
n=0
=
e−x
+ pE−1
v E pn
un(x, t)
∞
n=0
− pn
Hn(u)
∞
n=0
(19)
where Hn(u) are He’s polynomials. The first three
components of He’s polynomials are given by
H0 u = u0
2
+ u0
∂u0
∂x
;
H1 u = 2u0u1 + u0
∂u1
∂x
+ u1
∂u0
∂x
;
H2 u = 2u0u2 + u0
∂u2
∂x
+ u1
∂u1
∂x
+ u2
∂u0
∂x
+ u1
2
;
Comparing the coefficient of various power of p in (19), we
get
p0
: u0 x, t = e−x
,
p1
: u1 x, t = E−1
v E u0 − H0 u
= E−1
v E u0 − u0
2
− u0
∂u0
∂x
= e−x
t ,
p2
: u2 x, t = E−1
v E u1 − H1 u
= E−1
v E u1 − 2u0u1 − u0
∂u1
∂x
− u1
∂u0
∂x
= E−1
v E e−x
t = e−x
t2
2!
,
p3
: u3 x, t = E−1
v E u2 − H2 u
= E−1
v E u2 − 2u0u2 − u0
∂u2
∂x
− u1
∂u1
∂x
− u2
∂u0
∂x
− u1
2
= e−x
t3
3!
,
Proceeding in similar manner we can obtain further values,
p4
: u4 x, t = e−x
t4
4!
,
p5
: u5 x, t = e−x
t5
5!
substituting above values in equation (10), we get solution in
the form of a series
u x, t = u0 x, t + u1 x, t + u2 x, t + …
= e−x
+ e−x
t + e−x
t2
2!
+ e−x
t3
3!
+ e−x
t4
4!
+ e−x
t5
5!
+ ⋯
= e−x
1 + t +
t2
2!
+
t3
3!
+
t4
4!
+
t5
5!
+ ⋯ … … . . = et−x
, (20)
This is the solution of (11) and which is exactly the exact
solution of the problem [2, 3, 4, 5]. Figure 1 shows the graph
of u = et−x
, t > 0 𝑎𝑛𝑑 0 ≤ 𝑥 ≤ 1.
Fig -1: Graph of u = et−x
, t > 0 𝑎𝑛𝑑 0 ≤ 𝑥 ≤ 1.

__________________________________________________________________________________________
Example 3.2 Consider the following nonhomogeneous
nonlinear gas dynamic equation
∂u
∂t
+ u
∂u
∂x
− u 1 − u = −et−x
, (21)
with the following initial condition
u x, 0 = 1 − e−x
. (22)
By applying ELzaki transform on both side of equation (21)
E
∂u
∂t
= E u − u2
− u
∂u
∂x
− E et−x
(23)
Using properties of ELzaki transform, we have
1
v
E u(x, t) − vu x, 0 = E u − u2
− u
∂u
∂x
− e−x
v2
1 − v
(24)
Using initial condition (22), Eq. (24) can be written as
E u x, t =
v2
1 − e−x
+ vE u − u2
− u
∂u
∂x
− ve−x
v2
1 − v
(25)
Taking inverse ELzaki transform on both sides of Eq. (25), we
get
u x, t = 1 − e−x
+ E−1
vE u − u2
− u
∂u
∂x
− e−x
v3
1 − v
Since
E−1
e−x
v3
1 − v
= e−x
E−1
v2
1 − v
− v2
= e−x
et
− 1 ,
Therefore, we get
u x, t =
1 − e−x
+ E−1
vE u − u2
− u
∂u
∂x
− e−x
et
− 1
(26)
Now we apply the homotopy perturbation method,
u x, t = pn
un(x, t)
∞
n=0
(27)
N u x, t = pn
Hn(u)
∞
n=0
(28)
Using Eqs. (27)- (28) into Eq. (26), we get
pn
un x, t
∞
n=0
=
1 − et−x
+ pE−1
v E pn
un x, t
∞
n=0
− pn
Hn u
∞
n=0
(29)
where Hn(u) are He’s polynomials. The first three
components of He’s polynomials are given by
H0 u = u0
2
+ u0
∂u0
∂x
;
H1 u = 2u0u1 + u0
∂u1
∂x
+ u1
∂u0
∂x
;
H2 u = 2u0u2 + u0
∂u2
∂x
+ u1
∂u1
∂x
+ u2
∂u0
∂x
+ u1
2
; … … ..
Comparing the coefficient of various power of p in (19), we
get
p0
: u0 x, t = 1 − et−x
,
p1
: u1 x, t = E−1
v E u0 − H0 u
= E−1
v E u0 − u0
2
− u0
∂u0
∂x
= E−1
v E 1 − et−x
− 1 − et−x 2
− 1 − et−x
et−x
= E−1
v E 0 = 0,
Proceeding in similar manner we can obtain further values,
p2
: u2 x, t = 0 ,
p3
: u3 x, t = 0 , … ….
substituting above values in equation (10), we get solution in
the form of a series
u x, t = u0 x, t + u1 x, t + u2 x, t + …
u x, t = 1 − et−x
+ 0 + 0 + 0 … = 1 − et−x
.
This is the solution of (21) and which is exactly the exact
solution of the problem [2, 3, 4, 5]. Figure 2 shows the graph
of u = 1 − et−x
, t > 0 𝑎𝑛𝑑 0 ≤ 𝑥 ≤ 1.

__________________________________________________________________________________________
Fig -2: Graph of u = 1 − et−x
, t > 0 𝑎𝑛𝑑 0 ≤ 𝑥 ≤ 1.
CONCLUSIONS
The main goal of this paper is to show the applicability of the
mixture of new integral transform “ELzaki transform” with
the homotopy perturbation method to construct an analytical
solution for gas dynamics equation. This combination of two
methods successfully worked to give very reliable and exact
solutions to the equation. This method provides an analytical
approximation in a rapidly convergent sequence with in
exclusive manner computed terms. Its rapid convergence
shows that the method is trustworthy and introduces a
significant improvement in solving nonlinear partial
differential equations over existing methods.
ACKNOWLEDGEMENTS
I am deeply grateful to the management of Shirpur Education
Society, Shirpur (Maharashtra) without whose support my
research work would not have been possible. I would also like
to extend my gratitude to the Prin. Dr. J. B. Patil and Mr. S. P.
Shukla, Head of Department of Applied Sciences, RCPIT for
helping and inspiring me for the research work.
REFERENCES
[1] D.J. Evans, H. Bulut (2002), A new approach to the gas
dynamics equation: An application of the
decomposition method, Appl. Comp. Math., 79, pp817-
822.
[2] H. Jafari, M. Zabihi, M. Saidy (2008), Application of
homotopy perturbation method for solving gas
dynamics equation, Appl. Math. Scie., 2, pp 2393-
2396.
[3] Hossein Jafari, Hassan Hosseinzadehb and Elham
Salehpoor (2008), A New Approach to the Gas
Dynamics Equation : An Application of the Variational
Iteration Method, Applied Mathematical Sciences,
Vol.2,No.48,pp2397-2400.
[4] Hossein Aminikhah, Ali Jamalian (2013), Research
Article: Numerical Approximation for Nonlinear Gas
Dynamic Equation, International Journal of Partial
Differential Equations, Vol.2013, Article ID 846749,
7pages.
[5] M. Matinfar, M.Saeidy, M. Mahdavi and M. Rezael
(2011), Variational iteration method for exact solution
of gas dynamics equation using He’s polynomials,
Bulletin of Mathematical Analysis and Applications,
Vol.3,No.3,pp 50-55.
[6] Tarig M. Elzaki and Eman M. A. Hilal (2012),
Homotopy Perturbation and ELzaki Transform for
solving Nonlinear Partial Differential equations,
Mathematical Theory and Modeling, Vol.2,No.3, pp33-
42.
[7] Tarig M. Elzaki and Salih M. Elzaki (2011),
Applications of New Transform “ELzaki Transform” to
Partial Differential Equations, Global Journal of Pure
and Applied Mathematics, Vol.7, No.1,pp65-70.
[8] Tarig M. Elzaki (2011), The New Integral Transform
“ELzaki Transform”, Global Journal of Pure and
Applied Mathematics, Vol.7, No.1,pp57-64.

Elzaki transform homotopy perturbation method for solving gas dynamics equation

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