Advertisement

Asymptotic periodic solutions of some generalized Burgers equations

  • Smriti Nath
  • Ch. Srinivasa RaoEmail author
Article
  • 9 Downloads

Abstract

In this paper, we construct asymptotic periodic solutions of some generalized Burgers equations using a perturbative approach. These large time asymptotics (constructed) are compared with relevant numerical solutions obtained by a finite difference scheme.

Keywords

periodic solution large time asymptotics generalized Burgers equations 

MR Subject Classification

39A23 35B40 35K55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M Abramowitz, I A Stegun (eds.). Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover, New York, 1972.zbMATHGoogle Scholar
  2. [2]
    L C Andrews. Special functions for engineers and applied mathematicians, Macmillan Publishing Company, New York, 1985.Google Scholar
  3. [3]
    C M Bender, S A Orszag. Advanced mathematical methods for scientists and engineers, McGraw-Hill Book Company, Singapore, 1978.zbMATHGoogle Scholar
  4. [4]
    T F Chen, H A Levine, P E Sacks. Analysis of a convective reaction–diffusion equation, Nonlinear Anal TMA, 1988, 12(12): 1349–1370.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C N Dawson. Godunov-mixed methods for advective flow problems in one space dimension, SIAM J Numer Anal, 1991, 28(5): 1282–1309.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Y Duan, L Kong, R Zhang, A lattice Boltzmann model for the generalized Burgers–Huxley equation, Physica A, 2012, 391: 625–632.CrossRefGoogle Scholar
  7. [7]
    Y Duan, R Liu, Y Jiang. Lattice Boltzmann model for the modified Burgers’ equation, Appl Math Comput, 2008, 202: 489–497.MathSciNetzbMATHGoogle Scholar
  8. [8]
    R E Grundy, P L Sachdev, C N Dawson. Large time solution of an initial value problem for a generalized Burgers equation, In: P L Sachdev and R E Grundy (eds.), Nonlinear Diffusion Phenomenon, Narosa Publishing House, New Delhi, 1994, 68–83.Google Scholar
  9. [9]
    M Kato. Large time behavior of solutions to the generalized Burgers equations, Osaka J Math, 2007, 44: 923–943.MathSciNetzbMATHGoogle Scholar
  10. [10]
    H A Levine, L E Payne, P E Sacks, B Straughan. Analysis of a convective reaction–diffusion equation II, SIAM J Math Anal, 1989, 20(1): 133–147.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A Mishra, R Kumar. Exact solutions of variable coefficient nonlinear diffusion–reaction equations with a nonlinear convective term, Phys Lett A, 2010, 374: 2921–2924.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    H Moritz. The strange behavior of asymptotic series in mathematics, celestial mechanics and physical geodesy, In: E W Grafarend, F W Krumm and V S Schwarze (eds.), Geodesy -The challenge of the 3rd millennium, Springer-Verlag Berlin Heidelberg, 2003, 371–377.CrossRefGoogle Scholar
  13. [13]
    J J C Nimmo, D G Crighton. Bäcklund transformations for nonlinear parabolic equations: the general results, Proc R Soc Lond A, 1982, 384: 381–401.CrossRefzbMATHGoogle Scholar
  14. [14]
    Ö Oruç, F Bulut, A Esen. A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation, J Math Chem, 2015, 53: 1592–1607.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M A Pinsky. Partial differential equations and boundary-value problems with applications, American Mathematical Society, Providence, Rhode Island, 2011.zbMATHGoogle Scholar
  16. [16]
    O A Pocheketa, R O Popovych, O O Vaneeva. Group classification and exact solutions of variable-coefficient generalized Burgers equations with linear damping, Appl Math Comput, 2014, 243: 232–244.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Ch S Rao, E Satyanarayana. Asymptotic N-wave solutions of the nonplanar Burgers equation, Stud Appl Math, 2008, 121: 199–221.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    P L Sachdev. A generalised Cole–Hopf transformation for nonlinear parabolic and hyperbolic equations, J Appl Math Phys, 1978, 29(6): 963–970.MathSciNetzbMATHGoogle Scholar
  19. [19]
    P L Sachdev. Self-similarity and beyond. Exact solutions of nonlinear problems, Chapman & Hall/CRC, Boca Raton, Florida, 2000.CrossRefzbMATHGoogle Scholar
  20. [20]
    P L Sachdev, B O Enflo, Ch S Rao, B M Vaganan, P Goyal. Large-time asymptotics for periodic solutions of some generalized Burgers equations, Stud Appl Math, 2003, 110: 181–204.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    P L Sachdev, Ch S Rao, B O Enflo. Large-time asymptotics for periodic solutions of the modified Burgers equation, Stud Appl Math, 2005, 114: 307–323.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Smriti Nath. Large time asymptotics to solutions of some generalized Burgers equations, Ph. D thesis, IIT Madras, India, 2016.Google Scholar
  23. [23]
    A S Tersenov. On the generalized Burgers equation, Nonlinear Differ Equ Appl, 2010, 17: 437–452.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A S Tersenov. On the first boundary value problem for quasilinear parabolic equations with two independent variables, Arch Rational Mech Anal, 2000, 152: 81–92.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Ar S Tersenov. On solvability of some boundary value problems for a class of quasilinear parabolic equations, Sib Math J, 1999, 40(5): 972–980.MathSciNetCrossRefGoogle Scholar
  26. [26]
    B M Vaganan, S Padmasekaran. Large-time asymptotics for periodic solutions of nonplanar Burgers equation with linear damping, Int J Pure Appl Math, 2007, 41(3): 301–316.MathSciNetzbMATHGoogle Scholar
  27. [27]
    B M Vaganan, S Padmasekaran. Large time asymptotic behaviors for periodic solutions of generalized Burgers equations with spherical symmetry or linear damping, Stud Appl Math, 2009, 124: 1–18.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    B M Vaganan, S Padmasekaran. Large-time asymptotics for periodic solutions of modified nonplanar and modified nonplanar damped Burgers equations, In: B M Vaganan (ed.), Nonlinear Waves and Diffusion Processes, Narosa Publishing House, New Delhi, 2006, 50–59.Google Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

Personalised recommendations