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

[1]   Burgers, J.M. (1948) A Mathematical Model Illustrating the Theory of Turbulence. In: Von Mises, R. and Von Karman, T., Eds., Advances in Applied Mechanics, Vol. 1, Academic Press, New York, 171-199.
http://www.sciencedirect.com/science/article/pii/S0065215608701005
http://dx.doi.org/10.1016/s0065-2156(08)70100-5

[2]   Burgers, J.M. (1975) The Non Linear Diffusion Equation Asymptotic Solution and Statistical Problems.
http://www.amazon.it/The-Non-Linear-Diffusion-Equation-Statistical/dp/9027704945

[3]   Kida, S. (1979) Asymptotic Properties of Burgers Turbulence. Journal of Fluid Mechanics, 93, 337-377.
http://journals.cambridge.org/action/displayAbstract;jsessionid=0174C93AF976CBF18771
FEB41E0FEFA9.journals?fromPage=online&aid=388434
http://dx.doi.org/10.1017/S0022112079001932

[4]   Samokhin, A. (2014) Gradient Catastrophes and Saw Tooth Solution for a Generalized Burgers Equation on an Interval. Journal of Geometry and Physics, 85, 177-184.
http://www.sciencedirect.com/science/article/pii/
S0393044014000965
http://dx.doi.org/10.1016/j.geomphys.2014.05.007

[5]   Samokhin, A.V. (2013) Evolution of Initial Data for Burgers Equation with Fixed Boundary Values. Sci Herald of MSTUCA, 194, 63-70.
http://www.mstuca.ru/scientific_work/scientific_work/files/194.pdf

[6]   Euvrard, D. (1992) Résolution Numérique des Equations aux Dérivées Partielles. Différences finies, Eléments finis. Masson, Paris.

[7]   Sinai, G. (1992) Statistics of Shocks in Solutions of Inviscid Burgers Equation. Communications in Mathematical Physics, 148, 601-621.
http://citeseerx.ist.psu.edu/showciting?cid=205153
http://dx.doi.org/10.1007/bf02096550

[8]   She, Z.S., Aurell, E. and Frich, U. (1992) The Inviscid Burgers Equation with Initial Data of Brownian Type. Communications in Mathematical Physics, 148, 623-641.
http://www.researchgate.net/publication/38331252_The_inviscid_Burgers_equation_
with_initial_data_of_Brownian_type
http://dx.doi.org/10.1007/BF02096551

[9]   Hopf, E. (1950) The Partial Differential Equation: ut + uux= εuxx. Communications on Pure and Applied Mathematics, 3, 201-230.
http://www.researchgate.net/publication/259149172_The_partial_
differential_equation_ut__uux__xx
http://dx.doi.org/10.1002/cpa.3160030302

[10]   Cole, J.D. (1951) On a Quasilinear Parabolic Equation Occurring in Aerodynamics. Quarterly of Applied Mathematics, 9, 225-236.
http://www.researchgate.net/publication/238286127_On_a_quasilinear_parabolic_
equation_occurring_in_aerodynamics

[11]   Joseph, K.T. (1988) Burgers Equation in the Quarter Plane, a Formula for the Weak Limit. Communications on Pure and Applied Mathematics, 41, 133-149.
http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160410202/abstract
http://dx.doi.org/10.1002/cpa.3160410202

[12]   Kevorkian, J. and Cole, J.D. (1981) Perturbation Methods in Applied Mathematics. Springer Verlag, New York.
http://www.amazon.com/Perturbation-Methods-Mathematics-
Mathematical-Sciences/dp/0387905073
http://dx.doi.org/10.1007/978-1-4757-4213-8

[13]   Van Den Berg, I. (1987) Non Standard Asymptotic Analysis. Lecture Notes in Mathematics, 1249.
http://www.researchgate.net/publication/44553479_Nonstandard_asymptotic_
analysis__Imme_van_den_Berg

[14]   Lutz, R. and Goze, M. (1981) Non Standard Analysis. A Practical Guide with Application. Lecture Notes in Mathematics, 861.

[15]   Lutz, R. and Sari, T. (1982) Applications of Nonstandard Analysis in Boundary Value Problems in Singular Perturbatio Theory; Theory and Application of Singular Perturbation (Oberwolfach 1981). Lecture Notes in Mathematics, 942, 113-135.
http://www.researchgate.net/publication/225547873_Applications_of_
nonstandard_analysis_to_boundary_value_problems_in_singular_perturbation_theory

[16]   Bendaas, S. (1994) Quelques applications de l’A.N.S aux E.D.P. Ph.D. Thesis, Haute Alsace University, France.

[17]   Bendaas, S. (2008) L’équation de Burgers avec un Terme Dissipatif. Une approche non standard. Analele Universitatii din Oradea. Fascicola Matematica, 15, 239-252.