We establish asymptotic estimates for the solutions of singularly perturbed boundary-value problems with initial jumps.
Similar content being viewed by others
References
A. N. Tikhonov, “On the dependence of solutions of differential equations on a small parameter,” Mat. Sb., 22 (64), No. 2, 193–204 (1948).
M. I. Vishik and L. A. Lyusternik, “Regular degeneration and the boundary layer for linear differential equations with small parameter,” Usp. Mat. Nauk, 12, No. 5, 3–122 (1957).
A. B. Vasil’eva, “Asymptotics of solutions of some boundary-value problems for quasilinear equations with small parameter at the higher derivative,” Dokl. Akad. Nauk SSSR, 123, No. 4, 583–586 (1958).
N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).
E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameter and Relaxation Oscillations [in Russian], Nauka, Moscow (1979).
M. I. Imanaliev, “Asymptotic methods in the theory of singularly perturbed integrodifferential systems,” in: Investigations of Integrodifferential Equations [in Russian], Ilim, Frunze, Vol. 2 (1962), pp. 21–39.
S. A. Lomov, Introduction to the General Theory of Singular Perturbations [in Russian], Nauka, Moscow (1981).
V. F. Butuzov, “Angular boundary layer in mixed singularly perturbed problems for hyperbolic equations of the second order,” Dokl. Akad. Nauk SSSR, 235, No. 5, 997–1000 (1977).
M. I. Vishik and L. A. Lyusternik, “On the initial jump for nonlinear differential equations containing a small parameter,” Dokl. Akad. Nauk SSSR, 132, No. 6, 1242–1245 (1960).
K. A. Kasymov and D. N. Nurgabyl, “Asymptotic behavior of solutions of linear singularly perturbed general separated boundaryvalue problems with initial jump,” Ukr. Mat. Zh., 55, No. 11, 1496–1508 (2003); English translation: Ukr. Math. J., 55, No. 11, 1777–1792 (2003).
K. A. Kasymov and D. N. Nurgabyl, “Asymptotic estimates for the solutions of singularly perturbed boundary-value problems with initial jump for linear differential equations,” Differents. Uravn., 40, No. 4, 597–607 (2004).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 5, pp. 629–641, May, 2013.
Rights and permissions
About this article
Cite this article
Kasymov, K.A., Nurgabyl, D.N. & Uaisov, A.B. Asymptotic estimates for the solutions of boundary-value problems with initial jump for linear differential equations with small parameter in the coefficients of derivatives. Ukr Math J 65, 694–708 (2013). https://doi.org/10.1007/s11253-013-0807-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0807-5