Stability of Differential Equations with Random Impulses and Caputo-Type Fractional Derivatives
In this paper, we study nonlinear differential equations with Caputo fractional derivatives with respect to other functions and impulses. The main characteristic of the impulses is that the time between two consecutive impulsive moments is defined by random variables. These random variables are inde...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/855 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850041714373296128 |
|---|---|
| author | Snezhana Hristova Billur Kaymakçalan Radoslava Terzieva |
| author_facet | Snezhana Hristova Billur Kaymakçalan Radoslava Terzieva |
| author_sort | Snezhana Hristova |
| collection | DOAJ |
| description | In this paper, we study nonlinear differential equations with Caputo fractional derivatives with respect to other functions and impulses. The main characteristic of the impulses is that the time between two consecutive impulsive moments is defined by random variables. These random variables are independent. As the distribution of these random variables is very important, we consider the Erlang distribution. It generalizes the exponential distribution, which is very appropriate for describing the time between the appearance of two consecutive events. We describe a detailed solution to the studied problem, which is a stochastic process. We define the p-exponential stability of the solutions and obtain sufficient conditions. The study is based on the application of appropriate Lyapunov functions. The obtained sufficient conditions depend not only on the nonlinear function and impulsive functions, but also on the function used in the fractional derivative. The obtained results are illustrated using some examples. |
| format | Article |
| id | doaj-art-fe9d40cdb4e848d1925d6b911e064a85 |
| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-fe9d40cdb4e848d1925d6b911e064a852025-08-20T02:55:42ZengMDPI AGAxioms2075-16802024-12-01131285510.3390/axioms13120855Stability of Differential Equations with Random Impulses and Caputo-Type Fractional DerivativesSnezhana Hristova0Billur Kaymakçalan1Radoslava Terzieva2Faculty of Mathematics and Informatics, Plovdiv University, 4000 Plovdiv, BulgariaFaculty of Engineering, University of Turkish Aeronautical Association, Ankara 06790, TurkeyDepartment of Mathematics, Physics and Chemistry, Technical University of Sofia-Branch Plovdiv, 4000 Plovdiv, BulgariaIn this paper, we study nonlinear differential equations with Caputo fractional derivatives with respect to other functions and impulses. The main characteristic of the impulses is that the time between two consecutive impulsive moments is defined by random variables. These random variables are independent. As the distribution of these random variables is very important, we consider the Erlang distribution. It generalizes the exponential distribution, which is very appropriate for describing the time between the appearance of two consecutive events. We describe a detailed solution to the studied problem, which is a stochastic process. We define the p-exponential stability of the solutions and obtain sufficient conditions. The study is based on the application of appropriate Lyapunov functions. The obtained sufficient conditions depend not only on the nonlinear function and impulsive functions, but also on the function used in the fractional derivative. The obtained results are illustrated using some examples.https://www.mdpi.com/2075-1680/13/12/855fractional differential equationsimpulsesrandom moments of impulsesErlang distributionp-moment exponential stability |
| spellingShingle | Snezhana Hristova Billur Kaymakçalan Radoslava Terzieva Stability of Differential Equations with Random Impulses and Caputo-Type Fractional Derivatives Axioms fractional differential equations impulses random moments of impulses Erlang distribution p-moment exponential stability |
| title | Stability of Differential Equations with Random Impulses and Caputo-Type Fractional Derivatives |
| title_full | Stability of Differential Equations with Random Impulses and Caputo-Type Fractional Derivatives |
| title_fullStr | Stability of Differential Equations with Random Impulses and Caputo-Type Fractional Derivatives |
| title_full_unstemmed | Stability of Differential Equations with Random Impulses and Caputo-Type Fractional Derivatives |
| title_short | Stability of Differential Equations with Random Impulses and Caputo-Type Fractional Derivatives |
| title_sort | stability of differential equations with random impulses and caputo type fractional derivatives |
| topic | fractional differential equations impulses random moments of impulses Erlang distribution p-moment exponential stability |
| url | https://www.mdpi.com/2075-1680/13/12/855 |
| work_keys_str_mv | AT snezhanahristova stabilityofdifferentialequationswithrandomimpulsesandcaputotypefractionalderivatives AT billurkaymakcalan stabilityofdifferentialequationswithrandomimpulsesandcaputotypefractionalderivatives AT radoslavaterzieva stabilityofdifferentialequationswithrandomimpulsesandcaputotypefractionalderivatives |