Nonlinear phenomena play a crucial role in applied mathematics and physics.
These phenomenons can be described by the means of their nonlinear governing
equations if those are valid, so happens in shock waves. The shock waves
appear in explosion, traffic flow, glacier waves and airplanes breaking
the sound barrier and so on with partial nonlinear differential equation
form. The analytical solution of the nonlinear equations of such nonlinear
phenomena can help the authors know the described process deeply but except
a limited number, most of them do not have precise analytical solutions.
Therefore, these nonlinear equations are to be solved using other methods.
In recent decades, numerical analysis (Burden and Faires, 1993) has remarkably
developed for nonlinear partial equations related to fully developed shock
wave equation. Also there are many standard semi-exact analytical methods
for solving nonlinear partial differential equations; for instance, Backland
transformation method (Rogers and Shadwich, 1982), Lie group method (Olver,
1986), Adomians decomposition method (Adomian, 1988) and variational
iteration method. But as numerical calculation methods were improving,
semi-exact analytical methods were, too. Most scientists believe that
the combination of numerical and semi-exact analytical methods can also
end with remarkable results. Homotopy Analysis Method (HAM) is one of
the well-known and resent methods to solve the nonlinear equations, which
was introduced by Liao (1992, 1995, 1997, 1999, 2003, 2004, 2005). This
method eliminated the classical perturbation method problem, because of
the existence a small parameter in the equation.
Recently, Allan and Al-Khaled, (in press) applied ADM solution for shock
wave equation and achieved better results compared to numerical approaches
(Allan and Al-Khaled, 2007). In this survey, we extend this work by applying
HAM and VIM which are one of the latest analytical methods for solving
linear and nonlinear equations to analysis the shock wave equation.
BASIC OF THE HES VARIATIONAL ITERATION METHOD (VIM)
To clarify the basic ideas of VIM, we consider the following differential
where, L is a linear operator, N a nonlinear operator and g(t) an inhomogeneous
According to VIM, we can write down a correction functional as follows:
where, λ is a general lagrangian multiplier which can be identified
optimally via the variational theory Liao (1999, 2005). The subscript
n indicates the nth approximation and
is considered as a restricted variation (He, 1997, 1998a, b, 1999), i.e.,.
BASIC OF THE LIAOS HOMOTOPY ANALYSIS METHOD (HAM)
Consider the following differential equation (Liao, 1992, 1995, 1997,
1999, 2003, 2004, 2005):
where, N is a nonlinear operator, τ denotes an independent variable,
u(τ) is an unknown function. For simplicity, we ignore all boundary
or initial conditions, which can be treated in the similar way. By means
of generalizing the traditional homotopy method, Liao (2003) constructed
the so-called zero-order deformation equation as:
where, pε[0, 1] is the embedding parameter, h≠0 a nonzero auxiliary
parameter, h(τ)≠0 an auxiliary function, L an auxiliary linear
operator, μ0(τ) an initial guess of μ(τ)
and φ(τ; p) is an unknown function. It is important to have
enough freedom to choose auxiliary unknowns in HAM. Obviously, when p
= 0 and p = 1 it holds:
Thus, as p increases from 0 to 1, the solution φ(τ; p) varies
from the initial guess μ0(τ) to the solution μ(τ).
Expanding φ(τ; p) in Taylor series with respect to p, we have:
If the auxiliary linear operator, the initial guess, the auxiliary parameter
h and the auxiliary function are quite properly chosen, the series of
Eq. 5 converges at p = 1 then we have:
This must be one of the solutions of the original nonlinear equation,
as proved by He (1997). While h = -1 and H(τ) = 1, Eq.
This is mostly used in Homotopy Perturbation Method (HPM); whereas the solution
can be obtained directly without using Taylor series (He, 2006).
According to the Eq. 5, the governing equation can be deduced from the
zero-order deformation Eq. 4. The vector is defined as:
Differentiating Eq. 4 for m times with respect to the embedding parameter
p, then setting p = 0 and finally dividing them by m, we will have the
so-called mth order deformation equation as:
It should be emphasized that um(τ) for m≥1 is governed
by the linear Eq. 8 with the linear boundary conditions
coming from the original problem, which can be easily solved using symbolic
In the following, we apply HAM and VIM to solve shock wave equation.
The shock wave equation which describes the flow of most gases is given
by Chow (1979), Al-Khled (1996) and Smaller (1983):
where, c0, γ are constant and γ is the specific
heat. Study the case under c0 = 2 and γ = 3/2 which corresponds
to the flow of air, results to the following shock wave equation:
And the initial condition is assumed to be:
Implementation of VIM: In order to solve Eq.
12 by means of VIM, we construct a correction functional, as follows:
The Lagrange multiplier can therefore be simply identified as λ
= 1 and the following iteration formula can be obtained:
Now we start with an arbitrary initial approximation that satisfies the
initial conditions, as follows:
By the above iteration formula (15) we can obtain the other components
In the same manner the rest of components of the iteration formula Eq.
15 can be obtained. The excellent agreement between the solution of
Eq. 12 which is obtained from VIM after three iterations
and the Runge-Kuttas algorithm will be shown in results and discussion.
Implementation of HAM: Now we apply homotopy analysis method Eq.
4 to 12. We construct a homotopy in the following form:
Consider Eq. 17 and let us solve them through HAM
with proper assignment of H(τ) = 1 subject to the initial condition
and assuming m = 2.
By two times derivation of Eq. 17 with respect to
the p, two equations will be obtained. Consider the following equation:
Substitution Eq. 18 into both of them, we have following
Subject to the boundary condition (13) gives us:
Further from Eq. 24 it will be shown that HAM requires only two steps
to achieve accurate results.
RESULTS AND DISCUSSION
In this study, the shock wave equation is solved using HAM and VIM. The
result shown in Fig. 1-4 indicates that both methods
experience a high accuracy. In addition, in comparison with most other
analytical methods, a considerable reduction of the volume of the calculation
can be seen in both propose method especially for the HAM. It can be approved
that HPM and VIM are powerful and efficient techniques in finding analytical
solutions for a wide classes of nonlinear problems.
The behavior of the solution of Eq. 12 obtained from
the Runge-Kuttas algorithm is shown in Fig. 1. Also
the result of the HAM and VIM are depicted in Fig. 2a
and b, respectively. These figures show obviously the excellent agreement
between proposed methods and the Runge-Kuttas method.
For further verification, the surface error of the HAM and VIM solutions
(Fig. 2) with respect to the results obtained from Runge-Kuttas
algorithm (Fig. 1) are shown in Fig 3a
and b. Also the surface error of HAM for t = 2s and
h = -0.6 is showed in Fig 3c.
Here we studied and compared the diagram of the results obtained from
HAM for h = -0.5, h = -1, h = -2 in comparison with the results obtained
from Runge-Kuttas algorithm and VIM. Also the results obtained from HAM
for different value of h, VIM and the Runge-Kuttas method is shown in
Fig. 4a-c in 2D space for t = ls. For better understanding
of the efficiency of the proposed methods, the results from ADM solution
(Allan and Al-Khaled, 2007) is shown and compared to the previous ones
in Fig. 4d. As we can see; the results obtained from
HAM for h = -1 are closer to the results obtained from Runge-Kuttas algorithm
than other values of h (Fig. 4a-d) also HAM experiences
more accurate results.
||Behavior of solution obtained from Runge-Kuttas algorithm
||Behavior of solutions obtained by: HAM (a), VIM (b)
||Surface error of solutions obtained by: HAM (a, c), VIM (b) Error
||Comparison of the results with auxiliary parameter variation
In this study, we have successfully developed HAM and VIM for shock wave
equation. The results obtained from the proposed method are in good agreed
well with those obtained from the Runge-Kuttas algorithm and ADM. It
is apparently seen that HAM and VIM are very powerful and efficient techniques
in finding analytical solutions for wide classes of nonlinear problems.
It is worth pointing out that these two methods present a rapid convergence
for the solutions.
In conclusion, both proposed methods are convenient and efficient to
approach such problems and they also do not require large computer memory
and discrimination of the variables t and x, while applying HAM to the
discussed problems has got more advantages than VIM and most other methods;
it overcomes the difficulties arising in the calculation of other methods
and the auxiliary parameter help us to obtain the solution for fewer approximations.
Also HAM does not require small parameters in the equation so that the
limitations of the traditional perturbation methods can be eliminated
and thereby the calculations are simple and straightforward.