Advertisement

Indian Journal of Pure and Applied Mathematics

, Volume 49, Issue 4, pp 601–620 | Cite as

Asymptotic Behavior of Solutions to the Diffusion Equation

  • Satyanarayana EnguEmail author
  • Ahmed Mohd
  • Manas Ranjan Sahoo
Article
  • 84 Downloads

Abstract

We study asymptotic behavior of solutions to an initial value problem posed for heat equation. For which, we construct an approximate solution to the initial value problem in terms of derivatives of Gaussian by incorporating the moments of initial function. Spatial shifts are introduced into the leading order term as well as penultimate term of the approximation. This paper is continuation to the work of Yanagisawa [14].

Key words

Asymptotic behavior Cole-Hopf transformation heat equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. L. Chern and T. P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Commun. Math. Phys., 110 (1987), 503–517.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693–698.MathSciNetzbMATHGoogle Scholar
  3. 3.
    S. Engu, M. R. Sahoo, and M. Manasa, Higher order asymptotic for Burgers equation and Adhesion model, Commun. Pure Appl. Anal., 16 (2017), 253–272.MathSciNetzbMATHGoogle Scholar
  4. 4.
    E. Hopf, The partial differential equation u t + uu x = vu xx, Comm. Pure Appl. Math., 3 (1950), 201–230.MathSciNetCrossRefGoogle Scholar
  5. 5.
    R. S. Irving, Integers, polynomials, and rings, A course in algebra, Springer-Verlag, New York, 2004.zbMATHGoogle Scholar
  6. 6.
    Y. J. Kim and W. M. Ni, Higher order approximations in the heat equation and the truncated moment problem, SIAM J. Math. Anal., 40 (2009), 2241–2261.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Y. J. Kim, A generalization of the moment problem to a complex measure space and an approximation technique using backward moments, Discrete Contin. Dyn. Syst., 30 (2011), 187–207.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    R. C. Kloosterziel, On the large time asymptotics of the diffusion equation on infinite domains, J. Engineering Mathematics, 24 (2003), 213–236.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. C. Miller and A. J. Bernoff, Rates of convergence to self-similar solutions of Burgers equation, Stud. Appl. Math., 111 (2003), 29–40.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E. L. Rees, Graphical discussion of the roots of a quartic equation, Amer. Math. Monthly, 29 (1922), 51–55.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    I. Rubinstein and L. Rubinstein, Partial differential equations in classical mathematical physics, Cambridge University Press, 1998.zbMATHGoogle Scholar
  12. 12.
    Ch. Srinivasa Rao and S. Engu, Solutions of Burgers equation, Int. J. Nonlinear Sci., 9 (2010), 290–295.MathSciNetzbMATHGoogle Scholar
  13. 13.
    T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153–193.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation, Osaka J. Math., 44 (2007), 99–119.MathSciNetzbMATHGoogle Scholar

Copyright information

© The Indian National Science Academy 2018

Authors and Affiliations

  • Satyanarayana Engu
    • 1
    Email author
  • Ahmed Mohd
    • 1
  • Manas Ranjan Sahoo
    • 2
    • 3
  1. 1.Department of Mathematical and Computational SciencesNational Institute of Technology KarnatakaSurathkalIndia
  2. 2.School of Mathematical SciencesNational Institute of Science Education and ResearchBhubaneswarIndia
  3. 3.Homi Bhaba National Institute (HBNI), Training School Complex, Anushakti NagarMumbaiIndia

Personalised recommendations