N Wave and Periodic Wave Solutions for Burgers Equations

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Zahia Nouri
Saida Bendaas
Houssem Eddine Kadem

Abstract

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.

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References

  1. Alexy, Samokhin, Gradient catastrophes and saw tooth solution for a generalized Burgers equation on an interval, J. Geom. Phys. 85 (2014), 177-184.
  2. S. Bendaas, The Asymptotic Behavior of Viscid Burgers solution in the N wave shock case. A new Approach, Asian J. Math. Computer Res. 12 (3) (2016), 221-232.
  3. S. Bendaas, Confluence of Shocks in Burgers Equation. A new Approach, Int. J. Differ. Equ. Appl. 14 (2015), 369-382.
  4. S. Bendaas, Boundary value problems for Burgers equations through Nonstandard Analysis, Appl. Math. 6 (2015), 1086- 1098.
  5. S. Bendaas, L'equation de Burgers avec un terme dissipatif. Une approche non standard, Analele universitatii Oradea, Fasc. Math. 15 (2008), 239-252.
  6. S. Bendaas, Quelques applications de l'A.N.S aux E.D.P, Th`ese de Doctorat. Universit ´e de Haute Alsace, France, (1994).
  7. J. M. Burgers, The non linear diffusion equation asymptotic solution and satatistical propblems, Reidel, 1974.
  8. D. Euvrard, Resolution numerique des equations aux derivees partielles, Differences finies, elements finis/Masson. Paris, Milan, Barcelone, Mexico (1989).
  9. E. Hopf, The partial differential equation ut + uux = νuxx, Commun. Pure Appl. Mech. 3 (1950), 201-230.
  10. S. Kida, Asymptotic properties of Burgers turbulence, Journal of Fluid mechanics, 93(1979), 337-377.
  11. R. Lutz and M. Goze, Non Standard Analysis. A. Practical Guide with application Lectures, Notes in Math. N 861. Springer Verlag, Berlin (1981).
  12. M.O. Olayiwola, A.W. Gbolagade and F.O. Akinpelu, Numerical solution of generalized Burger's-Huxley equation by modified variational iteration method, J. Nigerian Assoc. Math. Phys. 17 (2010), 433-438.
  13. A. V. Samokhin, Evolution of initial data for Burgers equation with fixed boundary values, Sci. Herald MSTUCA, 194 (2013), 63-70 (In Russian).
  14. K. Shah and T. Singh, A solution of the Burger's equation arising in the longitudinal dispersion phenomena in fluid flow through porous media by mixture of new integral transform and homotopy perturbation method, J. Geosci. Environ. Protect. 3 (2015), 24-30.
  15. Z. S. She, E. Aurell and U. Frisch, The Inviscid Burgers equation with initial data of brownien type, Commun. Math. Phys. 148 (1992), 623-641.
  16. Ya. G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation, Commun. Math. Phys. 148 (1992), 601-621.
  17. I. Van Den Berg, Non Standard Asymptotic Analysis Lectures, Notes in Math. Vol. 1249. Springer Verlag.
  18. A. M. Wazwaz, Travelling wave solution of generalized forms of Burgers, Burgers-KdV and Burger's-Huxley equations, App. Math. Comput. 169 (2005), 639-656.