N WAVE AND PERIODIC WAVE SOLUTIONS FOR BURGERS EQUATIONS

This article concerns the initial boundary value problem for the non linear dissipative Burgers equation. Our general purpose is to describe the asymptotic behavior of the solution in the Cauchy problem with a small parameter ε for this equation and to discuss in particular the cases of the N wave shock and periodic wave shock. we show that the solution of Cauchy problem of viscid equation approach the shock type solution for the Cauchy problem of the inviscid equation for each case. The results are formulated in classical mathematics and proved with infinitesimal techniques of Non Standard Analysis.


Introduction
Burgers equation is the scalar partial differential equation where x ∈ X ⊆ R, t 0, and u : X × R + → R. The parameter ε is typically referred to as the viscosity due to the connection between this equation and the stydy of fluid dynamics. When ε > 0 it is often reffered to the viscous Burgers equation, and when ε = 0, it is often reffered to the inviscid Burgers equation.  [2], [3]. If the viscous term is null, the remaining equation is hyperbolic this is the inviscid Burgers equation If the viscous term is dropped from the Burgers equation, discontinuities may appear in finite time, even if the initial condition is smooth they give rise to the phenomen of shock waves with important application in physics [10]. This properties make Burgers equation a proper model for testing numerical algorithms in flows where severe gradients or shocks are anticipated [1], [16], [18]. Recently, Kunjan and Twinkle [14]  solutions of the reduiced problem to deduce existence and asymptotic behaviour of the solutions as ε tends to 0, the passage of the limit is very complicated, but in general the limit exist and it's a solution for the reduced problem (when ε = 0). For ε small the solution u(x, t) is approximated by this limit [10]. Other methods are based on a weak formulation of burgers equation seen as a conservation law satisfied on each of the computational domain called cell or finite volume. Stochastic particle method is so used for with different initial conditions. In particular we will take the same initial conditions that we took for the inviscid.
This paper completes recent works on the study of boundary value problems of Burgers equations for different initial conditions [2]- [4]. In the presented work, our general purpose is to describe the asymptotic behavior of solutions in boundary value problems with a small parameter ε and to discuss in particular the cases of N wave and periodic wave shocks with new techniques infinitesimal of Non-Standard Analysis. We can conclude that the solution of the Cauchy problem of inviscid equation in each case is infinitely close to the solution of he Cauchy problem of viscid equation as ε is a parameter positif sufficiently small. We introduce the infinitesimal techniques to give a simple formulation for the asymptotic behaviour. It is worth noting that our contribution is an elegant combination of infinitesimal techniques of Non-Standard Analysis and the Van Den Berg method [2], [3], [17].
Historically the subject non standard was developed by Robinson, Reeb, Lutz and Gose [11]. The Nonstandard perturbation theory of differential equations, which is today a well-established tool in asymptotic theory, has its roots in the seventies, when the Reebian school (see [11], [17]) introduced the use of Non-Standard Analysis into the field of perturbed differential equations. Our goal in this paper is to generalize these techniques on PDE.
The paper is organised as follows: Section 2 concerns the boundary value problems of inviscid Burgers equation, we start with the Fitting discontinuous shock then we describe the asymptotic behaviour of solutions for this problem in the N wave and periodic wave cases. Section 3 concerns the boundary value problems of viscid Burgers equation, it contains basic preliminaries results and deals with our main results about N wave and periodic wave shock cases and its proof, we present it in a non standard form.

Inviscid Burgers Equation
We will focus first on equation (1.2). Specifically, we will deal with the initial value problem As it as has been suggested previously, although (2.1) seems to be a very innocent problem a priori it hides many unexpected phenomena.This problem does not admit the regular solutions but some weak solutions with certain regularity exist.The Burgers equation on the whole line is known to possess traveling waves solutions. Using the characteristic method, the solution of the problem (2.1) may be given in a parametric and shocks must be fitted in such that Where f : R → R. is a standard continuous function. ξ 1 and ξ 2 are the values of ξ on the two sides of the shock [6].
According to (2.2), the solution at time t is obtained from the initial profile u = f (ξ) by translating each point a distance f (ξ)t to the right. The shock cuts out the part corresponding to ξ 2 ≥ ξ ≥ ξ 1 . If the discontinuity line it is a straight line chord betewen the points ξ = ξ 1 and ξ = ξ 2 on the curve f (ξ).
Moreover since areas are preserved under the mapping, the equal area property still holds. The chord on the f curve cuts off lobes of equal area. The shock determination can then be describe entirely on the fixe f (ξ) curve by drawing all the chords with the equal area propperty can be written analytically as This is the differential equation for the shock line chord wich verifies the entropic condition such as [6].
Since the left hand side is the area under the chord and the right hand side is the area under the f curve.
If the shock is at x = s(t) at time t, we also have To describe the solutions of the problem (2.1), we assume that the initial condition f verify the following assumptions : The theorem bellow gives the assymptotic behavior to the solution behind the shock and at the shock. with 0 < ξ < ξ 2 and the asymptotic form is Proof. We consider the problem (2.1) and we suppose that (H 1 ) and (H 2 ) are satisfied. Equation (2.4) may be written as As time goes on ξ 1 increase and eventually exceed L. At this stage f (ξ 1 ) = u 0 and the shock is moving into the constant region u = u 0 . The function ξ 1 (t) can then be eliminated for we have.
There for where ξ 2 satisfies (2.11). As t is infinitely large we have ξ 2 infinitesimal and f (ξ 2 ) approach u 0 , hence the equation for ξ 2 (t) takes the limiting form is the area of the hump above the undisturbed value u 0 .We have ξ 2 infinitesimal and Therefore the asymptotic formulas for s(t) and u in (2.12) are at the shock. The shock curve is asymptotically parabolic. The solution behind the shock is given by (2.2) with 0 < ξ < ξ 2 . Since ξ 2 is small enouhg as t is small enouhg, all the relevant values of ξ also small enouhg and the asymptotic form is The asymptotic solution and the corresponding (x, t) diagram are shown in Figure 1.  for the front shock approach (0, ∞),where as for the rear shock (ξ 2 , ξ 1 ) approach (−∞, 0). Asymptotically the front shock is and the jump of u is where A is the area of the f curve above u = u 0 .The rear shock has where B is the area below u = u 0 .The solution between the shocks is again asymptotically The asymptotic form and the (x, t) diagramm are shown in Figure 3 and Figure 4.  In this case, the shock equations (2.4) simplify considerably for all times t. Consider one period 0 < ξ < λ as in Figure 5. Relations (2.4) becomes From the difference and sum of (2.5) and (2.6) we have Respectively. The discontinuity in u at the shock is The shock has constant velocity u 0 and this result could have been deduced in advance from the symmetry of the problem. The shock starts with zero strength corresponding to θ = 0 at time t = λ/2πa. It reaches a maximum strength of 2a/u 0 for θ = π/2, t = λ/4a and decays ultimately with θ approach π, when t is infinitely large It is interesting that the final decay formula does not even depend explicitly on the amplitude a, however the condition for its application is t >> λ/a. For any periodic sinusoidal f (ξ) or not ξ 1 − ξ 2 → λ as t infinitely large; thence from (2.7) Between successive shocks, the solution for u is linear in x with slope 1/t as before, and the asymptotic form of the entire profile is the Sawtooth shown in Figure 6.

Viscid Burgers Equation
In this section we shall present and prove our main results, we discuss the N wave and periodic wave cases in the boundary value problem of viscid Burgers equation Before going further in this cases we need the following proposition and lemma : For the Heat equation, the solution for ϕ is The contribution from the neighborhood of a stationary point η = ξ in an integral is given with the lemma 3.2.
δ is a positif real small enough.
Proof. Suppose first that there is only one stationary point ξ(x, t) wich satisfies (3.8) then and in (3.6) we have where ξ(x, t) is defined by (3.8). This asymptotic solution may be rewritten as It is exactly the solution of (2.1) witch was discussed in section 2. The stationary point ξ(x, t) becomes the characteristic variable.
3.3. The Main Results. Since ϕ has a δ function behavior as t is infinitesimal, this is a little hard to interpret as an initial value problem on u. However, for any t > 0,it has the form shown in Figure 2 with a positive and a negative phase and we may take the profile at any t = t 0 to be the initial profile. It should typical of all N wave solutions.
Theorem 3.1. (I). Assume that the initial data f has a profile shown in Figure 2, the problem (1.3) admit an unique solution for t > 0 given by Proof. To obtain an N wave for u, we choose the source solution of the Heat equation for ϕ given by (3.13).
Then the corresponding solution for u is Since ϕ has a δ function behavior as t is infinitely small, this is a little hard to interpret as an initial value problem on u. However for any t > 0 it has the form shown in Fig.6 with a positive and negative phase and we may take the profile at any t = t 0 to be the initial profile. It should be typical of all N Wave solution.
The area under the positive phase of the profile is The positive phase is infinitely small when t is infinitely large. If the value of (3.14) at the initial time t 0 is denoted by A we may introduce a Reynolds number But as time goes on the effective, Reynolds number will be (3.17) and this is infinitely small as t is infinitely large.
If R 0 >> 1, We may expect the "Inviscid Theory" of (2.1) and (2.2) to be a good approximation for some time but as t is infinitely large , R(t) is eventually become dominant. In terms of R 0 and t 0 , a = t 0 (exp(R 0 ) − 1). Hence (3.14) may be written.
and For R 0 >> 1 ( corresponding to t 0 << 0),we have Or u in (3.17) may be approximated by For x and t . Now for fixed t limited and R 0 infinitely large we have This is exactly the inviscid solution.
However, for any fixed a and ε we see directely from (3.15) Theorem 3.2. (I). Assume that the initial data f has a profile shown in Fig 5.When λ 2 /4εt >> 1, the problem (3.1) admit an unique solution for t > 0 given by . Such solution is the periodic wave chock, and for ε small enough, this solution is infinitely close to the solution of the inviscid problem (2.1) given in section 2.
Proof. To obtain a periodic wave for u, we choose for ϕ a distribution of Heat sources spaced a distance λ given by (3.14). Then the corresponding solution for u is Proof. To study the Final decay (λ 2 /4εt) << 1, we may use an alternative form of the solution. The expression (3.14) is periodic in x, and in the interval: −λ/2 < x < λ/2, ϕ → δ(x), as t is infinitely small.
The initial condition can be expanded in a Fourier series as

Conclusion
This paper completes recent works on the study of boundary value problems of Burgers equations for different initial conditions [2]- [4]. Our general purpose is to describe the asymptotic behavior of solutions in boundary value problem with a small parameter ε and to discuss in particular the N wave shocks and Periodic wave shocks cases. The originality of this work consists in introducing new infinitesimal techniques of Non-Standard Analysis. We can conclude that the solution of the Cauchy problem of inviscid equation in each case is infinitely close to the solution of he Cauchy problem of viscid equation as ε is a parameter positif sufficiently small. We introduce the infinitesimal techniques to give a simple formulation for the asymptotic behavior. It is worth noting that our contribution is an elegant combination of infinitesimal techniques of non standard analysis and the Van Den Berg method [2], [3], [17].