Boundary Value Problems for Burgers Equations, through Nonstandard Analysis

Saida Bendaas

doi:10.4236/am.2015.66099

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Applied Mathematics > Vol.6 No.6, June 2015

Boundary Value Problems for Burgers Equations, through Nonstandard Analysis ()

Saida Bendaas

Department of Mathematics, Faculty of Sciences, University of Setif 1, Elbez, Algeria.

DOI: 10.4236/am.2015.66099 PDF HTML XML 3,613 Downloads 4,207 Views Citations

Abstract

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.

Keywords

Non Standard Analysis, Boundary Value Problem, Viscid Burgers Equation, Inviscid Burgers Equation, Heat Equation

Share and Cite:

Bendaas, S. (2015) Boundary Value Problems for Burgers Equations, through Nonstandard Analysis. Applied Mathematics, 6, 1086-1098. doi: 10.4236/am.2015.66099.

Conflicts of Interest

The authors declare no conflicts of interest.

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