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In this paper we study inviscid and viscid Burgers equations with initial conditions in the half plane . First we consider the Burgers equations with initial conditions admitting two and three shocks and use the HOPF-COLE transformation to linearize the problems and explicitly solve them. Next we study the Burgers equation and solve the initial value problem for it. We study the asymptotic behavior of solutions and we show that the exact solution of boundary value problem for viscid Burgers equation as viscosity parameter is sufficiently small approach the shock type solution of boundary value problem for inviscid Burgers equation. We discuss both confluence and interacting shocks. In this article a new approach has been developed to find the exact solutions. The results are formulated in classical mathematics and proved with infinitesimal technique of non standard analysis.

The nonlinear parabolic partial differential equation

was first introduced by J. M. Burgers [

When ε is null, this equation approaches to the Euler’s equations in one dimension who governs the flows of perfect fluids. It’s the viscid equation. it has the form

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 phenomenon of shock waves with important application in physics [

A remarkable feature of viscid Burgers equation is that its solutions with initial conditions of the form

can be explicitly written down. Hopf [

Then Hopf [

with initial condition

Then

and studied the asymptotic behavior of

Explicit solutions of the Burgers equation (1.1) in the quarter plane with integrable initial data and piecewise constant boundary data were constructed by [

The aim of the present article is to study solutions of Inviscid and Viscid Burgers equation if the initial condition admits several singular points, i.e. in the case of a finite number of shocks. A simple formulation is given for the asymptotic behavior based on the evaluation of integrals which is a method of the non standard perturbation theory of differential equations proposed by Imm Van Den Berg [

Historically the subject non standard was developed by Robinson, Reeb, Lutz and Goze [

In Section 2, we treat the boundary value problem for inviscid Burgers equation, solve it and study it. Section 3 is devoted to useful lemmas for our main results. In Section 4, we study viscid Burgers equation, solve exactly the initial value problems for it, and describe the asymptotic behavior of solutions with a non standard form. Some components, such as multi-leveled equations, graphics, and tables are not prescribed, although the various table text styles are provided. The formatter will need to create these components, incorporating the applicable criteria that follow.

We consider the inviscid Burgers equation:

In:

where

This problem not admits the regular solutions but some weak solutions with certain regularity exist. The Burgers equation on the whole line is known to possess traveling wave solutions. The solution of (1.2) and (1.3) may be given in a parametric form and shocks must be fitted in such that:

where

According to Equation (1.2), the solution at time t is obtained from the initial profile

This is the differential equation for the line cord of shock that checks the condition of entropy such as [

When a number of shocks are produced, in general it is possible for one of them to overtake the shock ahead. Then they combine and continue as a single shock. This is also described by our shock solution.

Consider the curve given by f

As time goes, the points

At this stage the characteristics corresponding to

In the plane

In this section we present some lemmas that are important to prove our main result.

Proposition 3.1. Let

i.e.

Proof: When a shock overtakes another shock, they merge into a single shock of increased strength as described in inviscid solution

In the expression of solution for a single shock given in [

Corresponding to the initial conditions:

Then the solutions of the heat equation are given as:

Using Equations (1.4) and (3.1) we obtain the expression (3.2).

And to prove our results, we use the non standard analysis techniques, for that we consider the following lemma.

Lemma 3.2. (The Van. Den. Berg lemma [

fined on ]0,+∞[ such that :

such that:

where a, r are positive standard, m and q are the both positive

To give estimation to the solution, given by (3.2), we state the following lemma:

Lemma 3.3. Let ε be a positive real small enough. And let ϕ and h be two standard functions such that: h, is a C^{2} class function verified the Lemma 3, and admits on the ξ point a unique absolute minimum

δ is an infinitesimal.

Proof: To prove this lemma, we use the “Van Den Berg” method, lemmas: (5.6), (5.7) [

1) Search for the absolute minimum (maximum) of the function under the exponential sign and bring it out.

2) Bring back the minimum (maximum) to the zero.

3) Searching the galaxy as well as the main galaxy where the function in the exponential sign is appreciable.

4) Calculate the integral.

As consequence we have the following lemma.

Lemma 3.4. Let f the initial condition as

(H_{1}): ^{2}(R).

(H_{2}): There exist a, b, c, d and e in R, with a < b < c < d < e, such that

Then for x and t fixed, the functions defined as:

has at most two minima

And the condition:

Proof: Let f the initial condition given as in

This equation is verified at the two minima

But

The condition of the shock is expressed by (3.12), is the same condition of shock given by (2.2) for invicid Burgers equation.

Our general purpose now is to show that the exact solution of (1.1) and (1.3) endorse the ideas regarding shocks in Section 2, we want to confirm that as _{1}), (H_{2}) in the lemma (3.4). Then we proved the following result:

Theorem 4.1. Under the assumptions: (H_{1}), (H_{2}) in lemma (3.4), the problem (1.1) and (1.3) admits a unique solution for

Such a solution is confluence of shocks and for ε sufficiently small, this solution is infinitely close to the solution of the reduced problem given in (2.2).

Proof: 1) From

2) Let

From the Lemma 3.2 we have

where δ > 0 is an infinitesimal. And we will have the following estimate

To conclude we have the following corollary.

Corollary 4.2. Let

And the center of the shock when

Proof: Using lemma (3.4), outside the region of each shock. For

With

Using lemma (3.3) we obtain

And it follows that:

If

If

In this section, we discuss the interacting shocks case; before going further in this case we need the following proposition and lemma.

Now since any

Proposition 4.3. Let

i.e.

Proof: When a shock overtakes another, they merge into a single shock of increased strength as described in inviscid solution

In the solution for a single shock given in [

shock are taken to be zero and

heat equation

And

Corresponding to the initial conditions:

Then the solutions of the heat equation are given as:

and

Using (1.4) and (4.4) we obtain the expression (4.3). Then we have the following.

Theorem 4.4. For

Such a solution is interacting shocks and for ε sufficiently small, it is infinitely close to the solution of the reduced problems (1.2) and (1.3).

Proof. 1) In the interacting shock case we have three shocks, when a shock overtakes another they merge into a single shock of increased strength and the lowest minimum dominating. Then we go back to the single shock case. Using the proposition (4.3), we deduce the uniqueness of solution explicitly given by (3.4). The uniqueness is due to the entropic condition [

2) Let

From the Lemma 3.2 we have:

where δ > 0 is an infinitesimal. And we will have the following estimate

from which the following corollary

Corollary 4.5. Let

Proof: For

where

where δ is an infinitesimal positive real.

As shown in

In:

The symbol “≃” means infinitely close to [

The transition from

Since,

for_{1} and u_{3}, moving with the velocity

on the path determined by

I thank the editor and the referee for their comments.