
On the regularization of the Cauchy problem for a system of linear difference equations
Publication year - 2018
Publication title -
vìsnik harkìvsʹkogo nacìonalʹnogo unìversitetu ìmenì v.n. karazìna. serìâ: matematika, prikladna matematika ì mehanìka
Language(s) - English
Resource type - Journals
eISSN - 2523-4641
pISSN - 2221-5646
DOI - 10.26565/2221-5646-2018-88-03
Subject(s) - mathematics , regularization (linguistics) , cauchy problem , cauchy distribution , bounded function , boundary value problem , cauchy's convergence test , mathematical analysis , coefficient matrix , cauchy matrix , linear system , initial value problem , cauchy boundary condition , physics , mixed boundary condition , computer science , eigenvalues and eigenvectors , quantum mechanics , artificial intelligence
The article proposes unusual regularization conditions as well as a scheme for finding solutions of the linear Cauchy problem for a system of difference equations in the critical case, significantly using the Moore-Penrose matrix pseudo-inversion technology. The problem posed in the article continues the study of the regularization conditions for linear Noetherian boundary value problems in the critical case given in the monographs by S.G. Krein, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. The general case is studied in which a linear bounded operator corresponding to a homogeneous part of a linear Cauchy problem has no inverse. In the article, a generalized Green operator is constructed and the type of a linear perturbation of a regularized linear Cauchy problem for a system of difference equations in the critical case is found. The proposed regularization conditions, as well as the scheme for finding solutions to linear Cauchy problems for a system of difference equations in the critical case, are illustrated in details with examples. In contrast to the earlier articles of the authors, the regularization problem for a linear Cauchy problem for a system of difference equations in the critical case has been resolved constructively, and sufficient conditions has been obtained for the existence of a solution to the regularization problem.