Periodic wave shock solutions of Burgers equations

Abstract: In this paper, we present an new approach for the study of Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the cauchy problem for viscid equation with small parametr and to discuss in particular the case of periodic wave shock. We show that the solution of this problem approaches the shock-type solution for the cauchy problem of the inviscid burgers equation. The results are formulated in classical mathematics and proved with infinitesimal techniques of nonstandard analysis.


Introduction
One of the most important PDEs in the theory of nonlinear consevation laws is Burgers equation, she combining both nonlinear propagation effects and diffusive effect. This equation is the approximation for the one-dimensional propagation of weak shock waves in a fluid. It can also be used in the description of the variation in vehicle density in highway traffic. The equation is one of the fundamental model equations in fluid mechanics. Burgers introduced the equation to describe the behavior of shock waves, traffic flow and acoustic transmission. It is well known that Burgers' equation plays a relevant role in many different areas of the mathematical physics, specially in Fluid Mechanics. Moreover, the simplicity of its formulation, in contrast with the Navier-Stokes system, makes of the Burgers' equation a suitable model equation to test different numerical algorithms and results of a varied nature (Samokhin, 2013(Samokhin, , 2014. This equation is parabolic when the viscous term is included. It has the form (Hopf, 1950): (1.1)

PUBLIC INTEREST STATEMENT
Many phenomena in the world have been described by nonlinear partial differential equations that can be solved by several methods. Burgers equation is one of the famous nonlinear equations. In the presented work, our general purpose is to describe the asymptotic behavior of solutions in boundary value problem with a small parameter and to discuss in particular the case of periodic wave shocks with new techniques infinitesimal of nonstandard analysis. Also, the author compared the present work via another published methods and concluded that the present results are more accurate and efficient than Standard methods studied.

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 (Kida, 1979). This properties make Burgers equation a proper model for testing numerical algorithms in flows where severe gradients or shocks are anticipated (Samokhin, 2014;Sinal, 1992;Wazwaz, 2005). Recently, Shah and Singh (2015) used mixture of new integral transform and Homotopy Perturbation Method to find the solution of Bugers' equation arising in the longitudinal dispersion phenomenon in fluid flow through porous media. Discretization methods are wellknown techniques for solving Burgers equation. Ascher and Mclachlan established many methods as multisymplectic box scheme. Olayiwola, Gbolagade and Akinpelu (2010), also presented the modified variational iteration method for the numerical solution of generalized Burger's-Huxley equation. Pocheketa and Popovych (2017) used enhanced classification techniques for the numerical solution of generalized Burgers equation. To establish numerical solutions for the modified Burgers equation, Ucar, Yagmurlu and Tasbozan (2017) were used in finite difference methods. For the boundary value problem, Sinal (1992) is intersted to the initial condition case: null on R −, and Brownian on R + . She, Aurell and Frisch (1992) with a numerical calculus particularly examine the initial conditions of Brownian fractional type.
This paper completes recent works on the study of boundary value problems of Burgers equations for different initial conditions (Bendaas, 2015a(Bendaas, , 2015b(Bendaas, , 2016. In the presented work, our general purpose is to describe the asymptotic behavior of solutions in boundary value problem with a small parameter and to discuss in particular the case of periodic wave shocks with new techniques infinitesimal of nonstandard analysis. We can conclude that the solution of the cauchy problem of inviscid equation is infinitely close to the solution of the cauchy problem of viscid equation as is a parameter positif sufficiently small. We introduce the infintesimal techniques to give a simple and direct formulation for the asymptotic behavior, unlike classical methods are based on search of solutions of the reduiced problem to deduce existence and asymptotic behavior of the solutions as tends to 0, the passasge of the limit is very complicated, but in general the limit exist and it's a solution for the reduced problem (when = 0). For small the solution u(x, t) is approximated by this limit (Kida, 1979). 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. It is worth noting that our contribution is an elegant combination of infinitesimal techniques of nonstandard analysis and the Van Den Berg method (Bendaas, 2015a(Bendaas, , 2016Van Den Berg, 2006).
Historically, the subject nonstandard was developed by Lutz and Goze (1981). The idea to use nonstandard analysis in perturbation theory of differential equations goes back to the 1970s with the Reebian school. Relative to this use, among many works we refer the interested reader, for instance, to (Benoit, 1997;Diener & Diener, 1995;Lobry & Sari, 2008;Lutz & Sari, 1982;Nelson, 1977;Robinson, 1974) and the references therein. It has become today a well-established tool in asymptotic theory of differential equations. Among the famous discoveries of the nonstandard asymptotic theory of differential equations we can cite the canards which appear in slow-fast vector fields and are closely related to the stability loss delay phenomenon in dynamical bifurcations The paper is organised as follows: Section 2 concerns the boundary value problem of inviscid burgers equation, we start with the Fitting discontinuous shock then we describe the asymptotic behavior of solutions for this problem in the periodic wave case. Section 3 concerns the boundary value problem of viscid burgers equation, we give first the main result about periodic wave shock case, (1.2) u t + uu x = 0 then we give somme basic preliminaries results and deals with our main result proof, we present it in a nonstandard 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 form: and shocks must be fitted in such that: where f : R → R. is a standard continuous function. 1 and 2 are the values of on the two sides of the shock (Bendaas, 1994).
According to (2.2), the solution at time t is obtained from the initial profile u = f ( ) by translating each point a distance f ( )t to the right. The shock cuts out the part corresponding to 2 ≥ ≥ 1 , if the discontinuity line is a straight line chord betewen the points = 1 and = 2 on the curve f ( ).
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 fixed f ( ) curve by drawing all the chords with the equal area property can be written analytically as This is the differential equation for the shock line chord wich verifies the entropic condition such as Bendaas (1994). 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 From (2.5) and (2.6), we have

Single hump
To describe the solutions of the problem (2.1), we assume that the initial condition f verify the following assumptions: The theorem below gives the assymptotic behavior to the solution behind the shock and at the shock.
Theorem 1 Suppose that (H 1 ) and (H 2 ) are satisfied, the solution of the problem (2.1) is given by (2.2) with 0 < < 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 1 increase and eventually exceed L. At this stage f ( 1 ) = u 0 and the shock is moving into the constant region u = u 0 . The function 1 (t) can then be eliminated for we have.
There for At this stage, the shock position and the value of u just behind the shock are given by : where 2 satisfie (2.11). As t is infinitely large we have 2 infinitesimal and f ( 2 ) approach u 0 , hence the equation for 2 (t) takes the limiting form where is the area of the hump above the undisturbed value u 0 . We have 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 < < 2 . Since 2 is small enough as t is small enough, all the relevant values of also small enough and the asymptotic form is: The asymptotic solution and the corresponding (x, t) diagram are shown in Figure 1.
Another interesting problem is that of an initial distribution In this case, the shock equations (2.4) simplify considerably for all times t. Consider one period 0 < < as in Figure 2.

Relations (2.4) becomes and the relevant choice is the trivial one
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 = 0 at time t = ∕2 a. It reaches a maximum strength of 2a∕u 0 for = ∕2, t = ∕4a and decays ultimately with approach , when t is infinitly 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 >> ∕a. For any periodic sinusoidal f ( ) or not 1 − 2 → 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 3.

Viscid Burgers equation
In this section, we shall present and prove our main result, we discuss the periodic wave case in the boundary value problem of viscid Burgers equation:

Periodic wave. The main result
A periodic solution may be obtained by taking for a distribution of heat sources spaced a distance apart. Then The main result of this work is: Theorem 2. 1. Assume that the initial data f have a profile shown in Figure 2 When 2 ∕4 t >> 1, the problem (3.1) admit an unique solution for t > 0 given by

. Such solution is the periodic wave chock, and for small enough, this solution is infinitely close to the solution of the inviscid problem (2.1) given in Section 2.
Before going further in the proof, we need the following proposition and lemma

The Cole-Hopf transformation
Cole and Hopf noted the remarkable result (Hopf, 1950)  The nonlinear transformation just eliminates the nonlinear term. The general solution of the heat equation (3.3) is well known and can be handled by a variety of methods. The basic problem considered in Section 2 is the initial value problem is a positif real small enough.
Proof Suppose first that there is only one stationary point (x, t) wich satisfies (3.9) then and in (3.7) we have where (x, t) is difinied by (3.9).It is exactly the solution of (2.1) which was discussed in section 2. The stationary point (x, t) becomes the characteristic variable Proof To obtain a periodic wave for u, we choose for a distribution of heat sources spaced a distance given by (3.14). Then the corresponding solution for u is For 2 ∕4 t >>, 1 this implies that √ t << ∕2, and ∑ ∞ n=−∞ = 2 ∑ ∞ n=0 , then for n = m we have which gives �x − m � << √ . and the exponential with the minimum value of (x − n ) 2 ∕4 t will dominate over all the others.Therefore, the term whitch will dominate for and (3.15) is approximately This is a sawtooth wave with a periodic set of shocks a distance apart, and u jumps from − ∕2t to ∕2t at each shock. The result agrees with the inviscid solution given by (2.20).

Conclusions
This paper completes recent works on the study of boundary value problems of Burgers equations for different initial conditions (Bendaas, 2015a(Bendaas, , b, 2016. Our general purpose is to describe the asymptotic behavior of solutions in boundary value problem with a small parameter and to discuss in particular the case of periodic wave shocks. The originality of this work consists in introducing new infinitesimal techniques of nonstandard analysis. We can conclude that the solution of the cauchy problem of inviscid equation is infinitely close to the solution of the cauchy problem of viscid equation as is a parameter positif sufficiently small. .e −G∕2 (1 + ) These techniques gives us a simple and direct formulation for the asymptotic behavior. It is worth noting that our contribution is an elegant combination of infinitesimal techniques of nonstandard analysis and the Van Den Berg method (Bendaas, 2015a(Bendaas, , 2016Van Den Berg, 2006).