On the necessary and sufficient conditions for the convergence of the difference schemes for the general boundary value problem for the linear systems of ordinary differential equations

Malkhaz Ashordia

Mathematica Bohemica (2021)

  • Volume: 146, Issue: 3, page 333-362
  • ISSN: 0862-7959

Abstract

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We consider the numerical solvability of the general linear boundary value problem for the systems of linear ordinary differential equations. Along with the continuous boundary value problem we consider the sequence of the general discrete boundary value problems, i.e. the corresponding general difference schemes. We establish the effective necessary and sufficient (and effective sufficient) conditions for the convergence of the schemes. Moreover, we consider the stability of the solutions of general discrete linear boundary value problems, in other words, the continuous dependence of solutions on the small perturbation of the initial dates. In the direction, there are obtained the necessary and sufficient condition, as well. The proofs of the results are based on the concept that both the continuous and discrete boundary value problems can be considered as so called generalized ordinary differential equation in the sense of Kurzweil. Thus, our results follow from the corresponding well-posedness results for the linear boundary value problems for generalized differential equations.

How to cite

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Ashordia, Malkhaz. "On the necessary and sufficient conditions for the convergence of the difference schemes for the general boundary value problem for the linear systems of ordinary differential equations." Mathematica Bohemica 146.3 (2021): 333-362. <http://eudml.org/doc/298038>.

@article{Ashordia2021,
abstract = {We consider the numerical solvability of the general linear boundary value problem for the systems of linear ordinary differential equations. Along with the continuous boundary value problem we consider the sequence of the general discrete boundary value problems, i.e. the corresponding general difference schemes. We establish the effective necessary and sufficient (and effective sufficient) conditions for the convergence of the schemes. Moreover, we consider the stability of the solutions of general discrete linear boundary value problems, in other words, the continuous dependence of solutions on the small perturbation of the initial dates. In the direction, there are obtained the necessary and sufficient condition, as well. The proofs of the results are based on the concept that both the continuous and discrete boundary value problems can be considered as so called generalized ordinary differential equation in the sense of Kurzweil. Thus, our results follow from the corresponding well-posedness results for the linear boundary value problems for generalized differential equations.},
author = {Ashordia, Malkhaz},
journal = {Mathematica Bohemica},
keywords = {general linear boundary value problem; linear ordinary differential systems; numerical solvability; convergence of difference schemes; effective necessary and sufficient conditions; generalized ordinary differential equations in the Kurzweil sense},
language = {eng},
number = {3},
pages = {333-362},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the necessary and sufficient conditions for the convergence of the difference schemes for the general boundary value problem for the linear systems of ordinary differential equations},
url = {http://eudml.org/doc/298038},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Ashordia, Malkhaz
TI - On the necessary and sufficient conditions for the convergence of the difference schemes for the general boundary value problem for the linear systems of ordinary differential equations
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 3
SP - 333
EP - 362
AB - We consider the numerical solvability of the general linear boundary value problem for the systems of linear ordinary differential equations. Along with the continuous boundary value problem we consider the sequence of the general discrete boundary value problems, i.e. the corresponding general difference schemes. We establish the effective necessary and sufficient (and effective sufficient) conditions for the convergence of the schemes. Moreover, we consider the stability of the solutions of general discrete linear boundary value problems, in other words, the continuous dependence of solutions on the small perturbation of the initial dates. In the direction, there are obtained the necessary and sufficient condition, as well. The proofs of the results are based on the concept that both the continuous and discrete boundary value problems can be considered as so called generalized ordinary differential equation in the sense of Kurzweil. Thus, our results follow from the corresponding well-posedness results for the linear boundary value problems for generalized differential equations.
LA - eng
KW - general linear boundary value problem; linear ordinary differential systems; numerical solvability; convergence of difference schemes; effective necessary and sufficient conditions; generalized ordinary differential equations in the Kurzweil sense
UR - http://eudml.org/doc/298038
ER -

References

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