INTRODUCTION
The one-dimensional non-linear partial equation
is known as Burgers equation. Burgers model of turbulence is a very important
fluid dynamics model and the study of this model and the theory of shock
waves have been considered by many authors both for conceptual understanding
of a class of physical flows and for testing various numerical methods.
The distinctive feature of Eq. 1 is that it is the simplest
mathematical formulation of the competition between non-linear advection
and the viscous diffusion. It contains the simplest form of non-linear
advection term u ux and dissipation term εuxx
where ε = 1/Re (ε: kinematics viscosity and Re: Reynolds number)
for simulating the physical phenomena of wave motion and thus deteermines
the behavior of the solution. The mathematical properties of Eq.
1 have been studied by Cole (1951). Particularly, the detailed relationship
between Eq. 1 and both turbulence theory and shock wave
theory were described by Cole. He also gave an exact solution of Burgers
equation. Benton and Platzman (1972) have demonstrated about 35 distinct
exact solutions of Burgers-like equations and their classifications. It
is well known that the exact solution of Burgers equation can only be
computed for restricted values of ε which represent the kinematics
viscosity of the fluid motion. Because of this fact, various numerical
methods were employed to obtain the solution of Burgers’ equation with
small ε values.
Many numerical solutions for Eq. 1 have been adopted
over the years. Finite element techniques have been employed frequently.
For example, Varoglu and Finn (1980) presented an isoparametric spaceBtime
finite-element approach for solving Burgers equation, utilizing the hyperbolic
differential equation associated with Burgers equation. Another approach
which has been used by Caldwell et al. (1981) is the finite-element
method such that by altering the size of the element at each stage using
information from the previous steps. Caldwell et al. (1980) give
an indication of how complementary variational principles can be applied
to Burgers equation. Later, Saunders et al. (1984) have demonstrated
how a variational-iterative scheme based on complementary variational
principles can be applied to non-linear partial differential equations
and the test problem chosen is the steady-state version of Burgers equation
Özis and Özdes (1996) applied a direct variational method to
generate limited form of the solution of Burgers equation. Özis et al. (2003) applied a simple finite-element approach
with linear elements to Burgers equation reduced by Hopf-Cole transformation.
Aksan and Özdes (2004) have reduced Burgers equation to the system
of non-linear ordinary differential equations by discretization in time
and solved each non-linear ordinary differential equation by Galerkin
method in each time step. As they claimed, for moderately small kinematics
viscosity, their approach can provide high accuracy while using a small
number of grid points (i.e., N = 5) and this makes the approach very economical
computational wise. In the case where the kinematics viscosity is small
enough i.e., ε = 0.0001, the exact solution is not available and
a discrepancy exists in the literature, their results clarify the behavior
of the solution for small times, i.e., T = tmax≤0.15. Also
it is demonstrated that the parabolic structure of the equation decayed
for tmax = 0.5. And finally, Aksan et al. (2006) applied
least squares method to solution this equation.
In this study, again, the reduced Burgers equation is solved by He’s
homotopy perturbation method and variational iteration method. The Variational
Iteration Method (VIM) has been previously employed by Abu and Soliman
(2005) to obtain a solution to Burgers equation in the form of an infinite
power series. It is well-known that the HPM and VIM converge very fast
to the results. Moreover, contrary to the conventional methods which require
the initial and boundary conditions, the HPM and VIM provide an analytical
solution by using only the initial conditions. The boundary conditions
can be used only to justify the obtained result. In the present study,
it is aimed to establish the existence of the solution first using the
Homotopy Perturbation Method (HPM) (He, 1999b, 2006; Zhang and He, 2006)
and then by the variational iteration method (VIM) (He, 1999a, 2000; Momani
and Abuasad, 2006; Ganji and Sadighi, 2007; Ganji et al., 2007;
Sweilam and Khader, 2007; Bildik and Konuralp, 2006). A comparison will
be made between the two methods to show that both methods are equally
able to arrive at exact solutions of Burgers’ equation. Numerical examples
are also presented for moderate values of ε since the exact solution
is not available for lower values.
BASIC IDEA OF HE’S HOMOTOPY-PERTURBATION METHOD
To illustrate the basic ideas of HPM, we consider the following nonlinear
differential equation :
with the boundary conditions of
Where, A, B, f(r) and Γ are a general differential operator, a boundary
operator, a known analytical function and the boundary of the domain Ω,
respectively.
Generally speaking the operator A can be divided into a linear part L
and a nonlinear part N(u). Eq. 2 can therefore, be rewritten
as:
By the Homotopy technique, we construct a homotopy v (r,p): Ωx[0,1]→R, which satisfies:
or
Where, pε[0, 1] is an embedding parameter, while u0 is
an initial approximation of Eq. 2, which satisfies the
boundary conditions. Obviously, from Eq. 5 and 6
we will have:
The changing process of p from zero to unity is just that of v(r, p)
from u0(r) to u(r). In topology, this is called deformation,
while L(v)-L(u0) and A(v)-f(r) are called homotopy.
According to the HPM, we can first use the embedding parameter p as a
small parameter and assume that the solution of Eq. 5
and 6 can be written as a power series in p:
Setting p = 1 yields in the approximate solution of Eq.
2 to:
The combination of the perturbation method and the homotopy method is
called the HPM, which eliminates the drawbacks of the traditional perturbation
methods while keeping all its advantage.
The series (10) is convergent for most cases. However, the convergent
rate depends on the nonlinear operator A(v). Moreover, He (1999a) made
the following suggestions:
| • |
The second derivative of N(v) with respect to v must
be small because the parameter may be relatively large, i.e., p→1. |
| • |
The norm of 
must be smaller than one so that the series converges. |
BASIC IDEA OF VARIATIONAL ITERATION METHOD
To clarify the basic ideas of VIM (He, 1999a, 2000; Momani and Abuasad,
2006; Ganji and Sadighi, 2006; Sweilam and Khader, 2007; Bildik and Konuralp,
2006) we consider the following differential equation:
Where, L is a linear operator, N is a nonlinear operator and g(t) is
an inhomogeneous term.
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. The subscript n indicates the nth approximation
and un is considered as a restricted variation, i.e., δ ũ
n = 0.
HPM APPLICATIONS IN SOLVING OF BURGERS EQUATION
Let us consider Burgers Eq. 1 with the following initial
and boundary conditions:
Where, Ω, is the interval (0, 1).
The exact solution of Eq. 1 with conditions 13 and 14
was given by Cole (1951) as:
Where:
and
Upon separating the linear and nonlinear parts of the Eq.
1, we apply homotopy-perturbation to Eq. 5. A homotopy
can be constructed as follows:
Substituting the value of v from Eq. 9 into 15
and rearranging based on powers of p-terms yields:
With the following conditions
The solutions of Eq. 16-19 by using
the conditions (20), may be re-written as follows:
Similarly, the other components were obtained using the maple software
package.
Substituting Eq. 21-24 into 10, then
re-written as follows:
It is noteworthly that this exact solution (Cole, 1951) is obtained by
using only the initial conditions. Moreover, the solution can be used
to justify the given boundry conditions.
APPLICATION OF VARIATIONAL ITERATION METHOD
To solve the Eq. 1 by means of VIM, one can construct
the following correction functional:
Its stationary conditions can be obtained as follows:
We obtain the lagrangian multiplier:
As a result, we obtain the following iteration formula:
Now we start with an arbitrary initial approximation that satisfies the
initial condition:
Using the above variational formula (29), we have:
Substituting Eq. 30 into 31 and after
simplifications, we have:
In the same way, we obtain u2 (x,t) as follows:
and so on. In the same way the rest of the components of the iteration
formula can be obtained.
COMPARISON OF HPM, VIM AND EXACT SOLUTIONS
In order to demonstrate the adoptability and accuracy of the present
approaches, we have applied it to the problem given by Eq.
1 which exact solution exists and is given by Cole (1951) in terms
of infinite series. To emphasize the accuracy of the method for moderate
size viscosity values, we have given the comparisons with analytical solutions
obtained from the infinite series of Cole (1951) for ε = 1 and 0.05.
Both Table 1-2 show that solutions
are in good agreement with analytical solutions. In the case ε is
smaller than 0.01, the exact solution is not available and a discrepancy
exists in the literature Also, it is not practical to evaluate the analytical
solution at these values due to slow convergence of the infinite series
and thus the exact solution in this regime is unknown.
| Table 1: |
Comparison of the HPM solutions obtained for ε
= 1 at different times with the exact solutions |
 |
| Table 2: |
Comparison of the HPM solutions obtained for ε
= 0.05 at different times with the exact solutions |
 |
CONCLUSION
In this study, the homotopy perturbation method and variational iteration
method have been successfully applied to the Burgers equation with specified
initial conditions. The results showed that these method are powerful
mathematical tools for solving Burgers equation and very effective, convenient
and quite accurate to systems of partial differential equations. They
provide more realistic series solutions that converge very rapidly in
real physical problems. The obtained results reinforce the conclusions
made by many researchers about the efficiency of the HPM and VIM. Therefore
these methods can be widely applied to engineering problems.
