Асимптотичне інтегрування сингулярно збурених диференціально-алгебраїчних рівнянь з точками повороту. І
Анотація
УДК 517.928
Розроблено алгоритм знаходження асимптотичних розв'язків сингулярно збуреної диференціально-алгебраїчної системи з простою точкою повороту.
Посилання
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).
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