Asymptotic integration of singularly perturbed differential algebraic equations with turning points. Part I
Abstract
UDC 517.928
This paper deals with the problem of finding asymptotic solutions for singular perturbed linear differential algebraic equations with simple turning point.
Technique of constructing the asymptotic solutions is developed.
References
R. E. Langer, The asymptotic solutions of ordinary linear differential equations of the second order with special reference to a turning point, Trans. Amer. Math. Soc., 67, 461 – 490 (1949), https://doi.org/10.2307/1990486 DOI: https://doi.org/10.2307/1990486
T. M. Cherry, Uniform asymptotic formulae for functions with transition points, Trans. Amer. Math. Soc., 68, 224 – 257 (1950), https://doi.org/10.2307/1990443 DOI: https://doi.org/10.2307/1990443
A. A. Dorodniczy`n, Asimptoticheskie zakony` raspredeleniya sobstvenny`kh znachenij dlya nekotory`kh osoby`kh vidov differenczial`ny`kh uravnenij vtorogo poryadka, Uspekhi mat. nauk, 7, вып. 6(52), 3 – 96 (1952).
W. Wasow, Asymptotic expansions for ordinary differential equations, Interscience Publishes, New York (1965).
A. M. Samojlenko, Ob asimptoticheskom integrirovanii odnoj sistemy` linejny`kh differenczial`ny`kh uravnenij s maly`m parametrom pri chasti proizvodny`kh, Ukr. mat. zhurn., 54, № 11, 1505 – 1517 (2002).
W. Wasow, The central connection problem at turning points of linear differential equations, Comment. Math. Helv., 46, № 1, 65 – 86 (1971), https://doi.org/10.1007/BF02566828 DOI: https://doi.org/10.1007/BF02566828
Y. Sibuya, Simplification of a system of linear ordinary differential equations about a singular point, Funkcial. Ekvac., 4, 29 – 56 (1962).
M. Iwano, Asymptotic solutions of a system of linear ordinary differential equations containing a small parameter, I, Funkcial. Ekvac., 5, 71 – 134 (1963).
M. Iwano, Asymptotic solutions of a system of linear ordinary differential equations containing a small parameter, II, Funkcial. Ekvac., 6, 89 – 141 (1964).
D. L. Russell, Y. Sibuya, The problem of singular perturbations of linear ordinary differential equations at regular singular points, I, Funkcial. Ekvac., 9, 207 – 218 (1966).
D. L. Russell, Y. Sibuya, The problem of singular perturbations of linear ordinary differential equations at regular singular points, II, Funkcial. Ekvac., 11, 175 – 184 (1968).
S. A. Lomov, Vvedenie v obshhuyu teoriyu singulyarny`kh vozmushhenij, Nauka, Moskva (1981).
A. N. Tikhonov, Sistemy` differenczial`ny`kh uravnenij, soderzhashhikh maly`e parametry` pri proizvodny`kh, Mat. sb., 31 (73), № 3, 575 – 586 (1952).
A. B. Vasil’eva, V. F. Butuzov, L. V. Kalachev, The boundary function method for singular perturbation problems, Soc. Industrial and Appl. Math., Philadelphia (1995), https://doi.org/10.1137/1.9781611970784 DOI: https://doi.org/10.1137/1.9781611970784
S. L. Campbell, Singular systems of differential equations II., Pitman, San-Francisco (1982).
A. M. Samojlenko, M. I. Shkil`, V. P. Yakovecz`, Linijni sistemi diferenczial`nikh rivnyan` z virodzhennyami, Vishha shkola, Kiyiv (2000).
V. F. Chistyakov, A. A. Shcheglova, Selected chapters in the theory of algebro-differential systems, Nauka, Novosibirsk (2003).
P. Kunkel, V. Mehrmann, Differential-algebraic equations. Analysis and numerical solution, Eur. Math. Soc., Zurich (2006), https://doi.org/10.4171/017 DOI: https://doi.org/10.4171/017
R. Riaza, Differential-algebraic systems. Analytical aspects and circuit applications, World Sci. (2008), https://doi.org/10.1142/6746 DOI: https://doi.org/10.1142/6746
E. Hairer, G. Wanner, Solving ordinary differential equations. II. Stiff and differential-algebraic problems, Springer- Verlag, Berlin, (2010), https://doi.org/10.1007/978-3-642-05221-7 DOI: https://doi.org/10.1007/978-3-642-05221-7
C. Tischendorf, Coupled systems of differential algebraic and partial differential equations in circuit and device simulation, Model. and Numer. Anal. (2003).
J. D. Murray, Mathematical biology: biomathematics, Vol. 19, Springer-Verlag (1989), https://doi.org/10.1007/978-3-662-08539-4 DOI: https://doi.org/10.1007/978-3-662-08539-4
R. E. Beardmore, The singularity-induced bifurcation and its Kronecker normal form, SIAM J. Matrix Anal. and Appl., 23, № 1, 126 – 137 (2001), https://doi.org/10.1137/S089547989936457X DOI: https://doi.org/10.1137/S089547989936457X
S. L. Campbell, Singular systems of differential equations, Pitman, San-Francisco (1980). DOI: https://doi.org/10.1080/00036818008839326
G. D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc., 9, № 2, 219 – 231 (1908), https://doi.org/10.2307/1988652 DOI: https://doi.org/10.2307/1988652
J. Tamarkin, Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions, Math. Z., 27, № 1, 1 – 54 (1928), https://doi.org/10.1007/BF01171084 DOI: https://doi.org/10.1007/BF01171084
S. F. Feshhenko, N. I. Shkil`, L. D. Nikolenko, Asimptoticheskie metody` v teorii linejny`kh differenczial`ny`kh uravnenij, Nauk. dumka, Kiyiv (1966).
V. F. Butuzov, Ob osobennostyakh pogranichnogo sloya v singulyarno vozmushhenny`kh zadachakh s kratny`m kornem vy`rozhdennogo uravneniya, Mat. zametki, 94, № 1, 68 – 80 (2013).
V. F. Butuzov, N. N. Nefedov, L. Recke, K. R. Schneider, On a singularly perturbed initial value problem in the case of a double root of the degenerate equation, Nonlinear Anal. Theory, Methods and Appl., 83, 1 – 11 (2013), https://doi.org/10.1016/j.na.2013.01.013 DOI: https://doi.org/10.1016/j.na.2013.01.013
P. F. Samusenko, Asimptotichne integruvannya singulyarno zburenikh sistem diferenczial`no-funkczional`nikh rivnyan` z virodzhennyami, Vid-vo Nacz. ped. un-tu im. M. P. Dragomanova, Kiyiv (2011).
W. Wasow, Linear turning point theory, Springer-Verlag, New York (1985), https://doi.org/10.1007/978-1-4612-1090-0 DOI: https://doi.org/10.1007/978-1-4612-1090-0
A. Ostrowski, Solution of equations in Euclidean and Banach spaces, Acad. Press, 1 New York (1973).
Copyright (c) 2020 П. Ф. Самусенко, А. М. Самойленко
This work is licensed under a Creative Commons Attribution 4.0 International License.
