On Non-Linear Differential Systems with Mixed Boundary Conditions
For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/866 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850239974082871296 |
|---|---|
| author | Miklós Rontó |
| author_facet | Miklós Rontó |
| author_sort | Miklós Rontó |
| collection | DOAJ |
| description | For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions. |
| format | Article |
| id | doaj-art-f4dbb15b1ad74932b054a5e92d7fa2a1 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-f4dbb15b1ad74932b054a5e92d7fa2a12025-08-20T02:01:00ZengMDPI AGAxioms2075-16802024-12-01131286610.3390/axioms13120866On Non-Linear Differential Systems with Mixed Boundary ConditionsMiklós Rontó0Institute of Mathematics, University of Miskolc, H-3515 Miskolc, HungaryFor the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions.https://www.mdpi.com/2075-1680/13/12/866non-linear differential systemsmixed boundary conditionsperiodic and non-linear two-point boundary conditionsparametrization techniquesuccessive approximations |
| spellingShingle | Miklós Rontó On Non-Linear Differential Systems with Mixed Boundary Conditions Axioms non-linear differential systems mixed boundary conditions periodic and non-linear two-point boundary conditions parametrization technique successive approximations |
| title | On Non-Linear Differential Systems with Mixed Boundary Conditions |
| title_full | On Non-Linear Differential Systems with Mixed Boundary Conditions |
| title_fullStr | On Non-Linear Differential Systems with Mixed Boundary Conditions |
| title_full_unstemmed | On Non-Linear Differential Systems with Mixed Boundary Conditions |
| title_short | On Non-Linear Differential Systems with Mixed Boundary Conditions |
| title_sort | on non linear differential systems with mixed boundary conditions |
| topic | non-linear differential systems mixed boundary conditions periodic and non-linear two-point boundary conditions parametrization technique successive approximations |
| url | https://www.mdpi.com/2075-1680/13/12/866 |
| work_keys_str_mv | AT miklosronto onnonlineardifferentialsystemswithmixedboundaryconditions |