Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations

1982 ◽  
Vol 42 (3) ◽  
pp. 502-531 ◽  
Author(s):  
Charles G. Lange ◽  
Robert M. Miura
Keyword(s):  
Singular Perturbation ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Perturbation Analysis ◽  
Boundary Value ◽  
Singular Perturbation Analysis

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

2019 ◽  
Vol 33 ◽  
pp. 204-217
Author(s):  
Sergei Chuiko ◽  
Yaroslav Kalinichenko ◽  
Nikita Popov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Oscillations ◽  
Bounded Operator ◽  
Boundary Value ◽  
Homogeneous Part ◽  
Solvability Conditions ◽  
Engineering Control ◽  
Generalized Green Operator

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

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