Parameter Estimation of Fractional Uncertain Differential Equations
Abstract
1. Introduction
- (1)
- A novel approach for estimating parameters of FUDEs is introduced. In particular, the rectangular and trapezoidal methods are utilized for the numerical approximation of optimization problems.
- (2)
- The prediction–correction technique is employed to solve fractional-order uncertain differential equations, with expected values being derived through the -path method. Numerical simulations are carried out along various -paths, yielding corresponding numerical solutions.
- (3)
- Finally, the proposed method is applied to practical models for prediction, demonstrating its applicability and robustness.
2. Preliminaries
- (A1) Normality axiom. for the universal set Γ.
- (A2) Duality axiom. for any event Λ.
- (A3) Subadditivity axiom. For every countable sequence of events , we have
- (A4) Product axiom. Let be uncertain spaces for . Then, the product uncertain measure satisfieswhere is an arbitrarily chosen event from for , respectively.
- (i)
- , and almost all simple paths are Lipschitz continuous;
- (ii)
- has stationary and independent increments;
- (iii)
- The increment has a normal uncertain distribution
- (H1)
- Lipschitz condition ,
- (H2)
- linear growth condition ,
3. Parameter Estimation
3.1. Discretization Model of the Fractional-Order Rectangular Method
Algorithm 1 Parameter Estimation for FUDE (1) Using the Fractional-Order Rectangular Method |
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3.2. The Discretization Model of Fractional Stepwise Normal Method
Algorithm 2 Parameter Estimation for FUDE (1) Using the Fractional-Order Trapezoidal Method |
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3.3. Some Example
4. -Path Solutions of Fractional Uncertain Differential Equation
4.1. Theoretical Explanation
4.2. Example 3
5. An Example: Tencent Holdings Ltd. Stock
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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0.0333 | 0.2673 | 0.3000 | 0.6757 | 0.5667 | 0.8316 | 0.8333 | 1.0767 |
0.0667 | 0.3383 | 0.3333 | 0.6709 | 0.6000 | 0.8287 | 0.8667 | 0.9615 |
0.1000 | 0.4051 | 0.3667 | 0.6738 | 0.6333 | 0.8575 | 0.9000 | 0.9844 |
0.1333 | 0.5149 | 0.4000 | 0.6922 | 0.6667 | 0.9440 | 0.9333 | 1.0370 |
0.1667 | 0.3382 | 0.4333 | 0.8186 | 0.7000 | 0.9088 | 0.9667 | 1.0895 |
0.2000 | 0.5763 | 0.4667 | 0.7060 | 0.7333 | 0.9489 | 1.0000 | 1.0381 |
0.2333 | 0.5676 | 0.5000 | 0.7738 | 0.7667 | 0.9622 | ||
0.2667 | 0.4241 | 0.5333 | 0.7627 | 0.8000 | 0.8669 |
0.0333 | 1.3678 | 0.3000 | 2.3468 | 0.5667 | 4.1754 | 0.8333 | 6.3859 |
0.0667 | 1.5076 | 0.3333 | 2.7597 | 0.6000 | 3.3747 | 0.8667 | 5.5457 |
0.1000 | 1.8188 | 0.3667 | 2.8815 | 0.6333 | 3.8666 | 0.9000 | 6.3509 |
0.1333 | 1.7297 | 0.4000 | 2.5305 | 0.6667 | 3.8233 | 0.9333 | 4.4128 |
0.1667 | 1.9184 | 0.4333 | 3.1536 | 0.7000 | 3.7180 | 0.9667 | 5.1034 |
0.2000 | 1.9780 | 0.4667 | 2.9979 | 0.7333 | 3.9068 | 1.0000 | 7.1114 |
0.2333 | 2.2190 | 0.5000 | 3.4912 | 0.7667 | 4.8796 | ||
0.2667 | 2.2190 | 0.5333 | 3.3200 | 0.8000 | 4.6056 |
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Ning, J.; Li, Z.; Xu, L. Parameter Estimation of Fractional Uncertain Differential Equations. Fractal Fract. 2025, 9, 138. https://doi.org/10.3390/fractalfract9030138
Ning J, Li Z, Xu L. Parameter Estimation of Fractional Uncertain Differential Equations. Fractal and Fractional. 2025; 9(3):138. https://doi.org/10.3390/fractalfract9030138
Chicago/Turabian StyleNing, Jing, Zhi Li, and Liping Xu. 2025. "Parameter Estimation of Fractional Uncertain Differential Equations" Fractal and Fractional 9, no. 3: 138. https://doi.org/10.3390/fractalfract9030138
APA StyleNing, J., Li, Z., & Xu, L. (2025). Parameter Estimation of Fractional Uncertain Differential Equations. Fractal and Fractional, 9(3), 138. https://doi.org/10.3390/fractalfract9030138