Stability Study of Stationary Solutions of the Viscous Burgers Equation

Stability study of stationary solutions of the viscous Burgers equation using Fourier mode stability analysis for the stationary solutions where D is constant and , in two cases is analyzed. Firstly when the wave amplitude A is constant and secondly when the wave amplitude A is variable. In the case of constant amplitude, the results found to be: The solution is always stable while the solution is conditionally stable. In the case of variable amplitude, it has been found that the solutions D 1 and are conditionally stable.

are conditionally stable.

Introduction
Consider a system of any nature whatsoever that exists in a state S. We say that S is stable, in one sense or another, if small perturbations or changes in the system do not drastically affect the state S. For example, the solar system currently exists in a time-dependent state in which the planets move about the sun in an orderly fashion. It is known that if a small additional celestial body is introduced into the system, then the original state is not disturbed to any significant degree. We say that the original state is stable to small perturbations. Similar questions of stability arise in every physical problem [19]. The notorious Burgers equation was the subject of interest study in different fields such as analytical solutions, numerical solutions, mathematical modeling, fluid mechanics, stability and bifurcation. Roy and Baker [27] presented and derived the numerical results using a nonlinear subgrid embedded (SGM) finite element basis for D 1 , D 2 and D 3 verification/benchmark linear and nonlinear convectiondiffusion problems such as Burgers equation in steady state.
Burns et al [8] considered the numerical stationary solutions for a viscous Burgers equation on the interval ( ) 1 , 0 with Neumann boundary conditions. Roy and Fleming [28] developed a nonlinear subgrid embedded (SGM) finite element basis for generating multidimensional solutions for convection-dominated computational fluid dynamics (CFD) applications and they applied them to a stationary Burgers equation. Balogh and Krstic [4] considered the viscous Burgers equation under recently proposed nonlinear boundary conditions and they showed that it guarantees global asymptotic stabilization and semi global exponential stabilization in 1 H sense. Balogh et al [5] studied the stationary solutions of a one-parameter family of boundary control problems for a forced viscous Burgers equation. They assumed that the forcing term possesses a special symmetry. Allen et al [2] studied numerically the equilibrium solutions of Burgers equation. Moller [23]  In this paper, the stability of stationary solutions of viscous Burgers equation using Fourier mode stability analysis is investigated.

The Mathematical Model
One of the famous nonlinear diffusion equations is the generalized Burgers-Huxley (gBH) equation [30]: where ∈ is the diffusion coefficient and in fluid flow problems it represents the viscosity and is the reciprocal of the Reynolds number.
which describes nerve pulse propagation in nerve fibers and wall motion in liquid crystals. The parameter a arises in genetics and other fields, the case is what the geneticists refer to as the heterozygote inferiority case [15]. Manaa and Moheemmeed studied the stability [20] and the numerical solution [21] of this case.
The standard real Newell-Whitehead (rNW) equation is a special case of Eq.  Newell and Whitehead examined this equation in 1969 [24]. When 0 = β , Eq. (1) is reduced to the generalized Burgers equation The well known viscous Burgers equation is a special case of Eq.(1) and (5) ,respectively [18].
Burgers equation provides remarkable system that has been studied for some time by Bateman in 1915 and was extensively developed by Burgers in 1940 and 1948 as a simplified fluid flow model which, nonetheless, exhibits some of the important aspects of turbulence. It was later derived by Lighthill in 1956 as a second-order approximation to the one-dimensional unsteady Navier-Stokes equation [5]. The Burgers equation can be seen as a reduction of the Navier-Stokes equation to the case of a single space dimension. In this equation, α controls the nonlinearity and ∈ stands for viscosity. It is perhaps the simplest nonlinear differential second order equations, and it has been considered to describe different physical problems such as sound waves in viscous media, the far field of wave propagation in nonlinear dissipative systems, shock waves, magnetohydrodynamic waves in media with finite electrical conductivity, nonlinear heat diffusion and viscous effects in gas dynamics [6]. The study of the viscous Burgers equation is naturally related to that of the in viscid Burgers equation [11]: The heat equation corresponds to the linearized Burgers equation It is known that nonlinear diffusion equations (3) and (6) (9) shows a prototype model [30] for describing the interaction between reaction mechanisms, convection effects and diffusion transport. Also, Burgers equation is a particular case of following convection-reaction-diffusion equation.

( )
The equation (10) (10) is the Burgers-Fisher equation [30]: There is another Burgers type equation named the generalized Burgers-Korteweg-de Vries equation [31]: are positive real numbers. It reduces to the generalized Burgers and Burgers equations for , respectively. It also reduces to the generalized Korteweg-de Vries (gKdV) and standard Korteweg-de Vries (KdV) equations for 0 ∈= and , The Burgers-Korteweg-de Vries or Burgers-KdV equation [18] is special case of Eq. (14) when which reduces to the Burgers and KdV equations when 0 = γ and 0 ∈= , respectively.

Introduction to the Burgers Type Equations
Burgers type equations are famous nonlinear equations which, appear in different scientific fields and play significant role in the study of the nonlinear evolution equations in applied mathematics. Satsuma-Burgers-Huxley (SBH) equation [9], [10] considers another type of the Burgers type equation with reaction term: The case 0 ∈= , corresponds to the first order equation One of the important models related to both shallow water waves and to turbulence is the b-equation [14]: .The equation (21) contains a family of equations. For , Let us consider the generalized Burgers equation (5), this equation is named generalized since it contains the quantity δ u in the convection term We can get another generalized Burgers equations by changing the properties of the nonlinear term .
x u u δ α The generalizations of Burgers and Evolution n Viscosity y

Stretching Convection n
Burgers-Huxley equations, for which only relaxation of the assumption of weak nonlinearity is made. This means that no change in the original equations is made to introduce other effects, like including a new term to describe dispersion for instance, but just changing the nonlinear properties of the original system, for the generalized Burgers equation, for example, the consideration of the dynamics of diffusion in media where nonlinearity is not just restricted to the simplest case. If we replace the nonlinear term in (5), we get another generalized Burgers equation [6]: is a smooth function of u . The Burgers equation (6) is obtained with the linear function ( ) u Like the Burgers equation (6), the generalized Burgers equation (24) also combines nonlinearity and diffusion, but now nonlinearity is controlled by ( ) u g and may vary according to the model one considers, note that the Burgers equation is defined with the simplest nontrivial function ( ) This equation is named the modified Burgers equation, since it contains nothing but the change 2 3u u → in its nonlinear term. Equation (24) can be written in the form: This form is interesting since it allows a natural extension to systems where two or more configurations interact with each other. The equation (27) can be extended to the system of two coupled Burgers type equations , we can write (28) as: The generalized Burgers equation (24) can be further extended to the following form are smooth functions. Equation (30) represents another generalized Burgers-Huxley or generalized Burgers-Fisher equations, which differ from the equations (1) and (10)  ( )  ) , we get the standard (KdVB) equation (17). The (KdV) and Burgers equations were first added [7] to describe properties of waves in liquid-filled elastic tubes.
The nonlinear differential equations in the generic form [25]:

The Non-dimensional Transformations
For non-dimensional form, we introduce the following nondimensional quantities: By substituting these dimensionless quantities in (6), we get: represents the Reynolds number if we set ∈ = / Re L α and omit the primes in the equation in above, we get: The equation (43) with the boundary conditions represents the nondimensional Burgers equation in x and t .

Fourier Mode Stability Analysis
Let the solution of equation (43) has the following form [19]: is a constant independent of ∈ . The solution will approach the steady state approximately as t e 1 µ , hence for small values of ∈, this will become an extremely slow process. The equation (47)

Stability Analysis in the Case of Constant Amplitude
We assume that the perturbation has the following form [19]: where Ais the wave amplitude , k is the wave, number c is the wave velocity . If 0 2 < c the disturbance will decay as ∞ → t and the solution is stable, but if 0 2 > c the disturbance will grow as ∞ → t and the solution is unstable. The case 0 2 = c , gives the neutral stability curve, which separates between the stable and unstable regions, 2 c is called the stability indicator [22]. Substitute (52) in (48), and after some mathematical manipulation, we get: Equating the real and imaginary parts , we have : Here, in above we neglect the error term since it is small [23]. ( )

Stability Analysis in the Case of Variable Amplitude
We assume the disturbance to have the following form [19], [1]: Equation (58) can be written in the following form: The characteristic equation of Eq. (59) is: ) which has the following solutions: According to the sign of λ Eq. (59) has the following three analytical solutions: then (61) and (62) become: The general solution of Eq. (59) in this case is: ( ) By using the boundary conditions, we obtain:

>
, then the solution as in the case (i).
(2) When, ( )   For simplicity and to determine the value of 2 c , we take 1 , 1 = = a C and after some mathematical manipulation, we get:

Conclusions
The main conclusions from this study in the case of constant amplitude are: