Abstract
We study the asymptotic behavior of the first boundary value problem for a secondorder elliptic equation in the case where the small parameter is a factor at only one of the higher derivatives and the limit equation is an ordinary differential equation. Although the limit equation is of the same order as the original one, the problem under consideration is bisingular. We investigate the asymptotic behavior of this problem using the method of matched asymptotic expansions.
Similar content being viewed by others
References
M. I. Vishik and L. A. Lyusternik, “A regular degeneration and a boundary layer for linear differential equations with a small parameter,” Uspekhi Mat. Nauk 12 (5), 3–122 (1957).
V. A. Trenogin, “The development and applications of the asymptotic method of Lyusternik and Vishik,” Russ. Math. Surv. 25 (4), 119–156 (1970).
A. Nayfeh, Perturbation Methods (Wiley, New York, 1973; Mir, Moscow, 1976).
M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic, New York, 1964; Mir, Moscow, 1967).
A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Amer. Math. Soc., Providence, RI, 1992).
E. F. Lelikova, “Asymptotic behavior of the solution to an equation with a small parameter,” Dokl. Math. 80 (3), 852–855 (2009).
E. F. Lelikova, “The asymptotics of the solution of an equation with a small parameter in a domain with angular points,” Sb. Math. 201 (9–10), 1495–1510 (2010).
A. M. Il’in and E. F. Lelikova, “On asymptotic approximations of solutions of an equation with a small parameter,” St. Petersb. Math. J. 22 (6), 927–939 (2011).
E. F. Lelikova, “Asymptotic behavior of the solution of a small-parameter equation in a domain with a conical point on the boundary,” Dokl. Math. 85 (1), 33–39 (2012).
E. F. Lelikova, “On the asymptotic behavior of a solution to an equation with a small parameter in a neighborhood of a boundary inflection point,” Dokl. Math. 86 (3), 756–759 (2012).
E. F. Lelikova, “On the asymptotics of a solution of a second order elliptic equation with small parameter at a higher derivative,” Proc. Steklov Inst. Math., Suppl. 1, S129–S143 (2003).
E. F. Lelikova, “The asymptotics of a solution of a second order elliptic equation with a small parameter multiplying one of the highest order derivatives,” Trans. Moscow Math. Soc., 141–174 (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.F. Lelikova, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 1.
Rights and permissions
About this article
Cite this article
Lelikova, E.F. On the Asymptotics of a Solution to an Elliptic Equation with a Small Parameter in a Neighborhood of an Inflection Point. Proc. Steklov Inst. Math. 299 (Suppl 1), 132–147 (2017). https://doi.org/10.1134/S0081543817090164
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543817090164