Existence and Mittag–Leffler Stability for the Solution of a Fuzzy Fractional System with Application of Laplace Transforms to Solve Fractional Differential Systems
Abstract
:1. Introduction
1.1. Definition of Fuzzy Numbers
- Normality: There exists such that .
- Convexity: For all and ,
- Upper Semi-Continuity: The function is upper semi-continuous on .
- Compact Support: The support of ,
- is a bounded, non-decreasing, left-continuous function;
- is a bounded, non-increasing, left-continuous function;
- for all .
- (1)
- are two fuzzy differentiable functions.
- (2)
- The functions are in .
- (3)
- and are disjoint sets and their union is , and are disjoint sets and their union is .
- (4)
- (5)
- for and
1.2. Mittag–Leffler Function, Laplace Transform, and Adomian Decomposition Method
1.3. Achievements of the Article
2. Preliminaries
- (1)
- and the H-differences exist.
- (2)
- (3)
- (4)
- (1)
- The subset C of is compact-supported if there is a compact set such that U consists of all sets , where .
- (2)
- The subset C of is level-equicontinuous in if, for all , we have
- (3)
- The continuous function is compact if, for every and (C is bounded), the set is relatively compact in .
- (1)
- Let , and . The Riemann–Liouville fuzzy fractional integral is defined as
- (2)
- Let and . The formula for the Caputo fractional gH-derivative of order l is
3. The Non-Local Problem for Fractional Differential Systems
- (1)
- is equicontinuous;
- (2)
- is level-equicontinuous on .
4. Stability Result for the Fractional System
5. Application of Laplace Transforms to Solve Fractional Differential Systems
6. Adomian Decomposition Method for Solving Fuzzy Fractional Equations
7. Numerical Methods to Solve Fuzzy Fractional Equations
8. Conclusions and Future Works
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Finite Difference Method | ||
---|---|---|
Estimate for | Estimate for | |
10 | 1.206168310 | 0.4436465165 + r |
50 | 1.016466996 | 0.02520916477 + r |
100 | 1.005738026 | 0.008464752889 + r |
1000 | 1.000178662 | 0.0002556471280 + r |
2000 | 1.000062538 | 0.00008970896566 + r |
Caputo Derivative Method | ||
---|---|---|
Estimate for | Estimate for | |
5 | 0.9999999998 | |
10 | 0.9999999999 | |
15 | 1.000000000 | |
20 | 1.000000000 | r |
25 | 1.000000000 | r |
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Abolhassanifar, M.S.; Saadati, R.; Ghaemi, M.B.; O’Regan, D. Existence and Mittag–Leffler Stability for the Solution of a Fuzzy Fractional System with Application of Laplace Transforms to Solve Fractional Differential Systems. Algorithms 2025, 18, 264. https://doi.org/10.3390/a18050264
Abolhassanifar MS, Saadati R, Ghaemi MB, O’Regan D. Existence and Mittag–Leffler Stability for the Solution of a Fuzzy Fractional System with Application of Laplace Transforms to Solve Fractional Differential Systems. Algorithms. 2025; 18(5):264. https://doi.org/10.3390/a18050264
Chicago/Turabian StyleAbolhassanifar, Mohammad Saeid, Reza Saadati, Mohammad Bagher Ghaemi, and Donal O’Regan. 2025. "Existence and Mittag–Leffler Stability for the Solution of a Fuzzy Fractional System with Application of Laplace Transforms to Solve Fractional Differential Systems" Algorithms 18, no. 5: 264. https://doi.org/10.3390/a18050264
APA StyleAbolhassanifar, M. S., Saadati, R., Ghaemi, M. B., & O’Regan, D. (2025). Existence and Mittag–Leffler Stability for the Solution of a Fuzzy Fractional System with Application of Laplace Transforms to Solve Fractional Differential Systems. Algorithms, 18(5), 264. https://doi.org/10.3390/a18050264