Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives
Abstract
1. Introduction
2. Preliminaries
- 1.
- 2.
3. Generalized FFWP Model
4. Existence of the Solutions
4.1. Existence
4.2. Uniqueness
4.3. Stability Analysis
5. Numerical Scheme Based on Proposed Model
6. Graphical Representations and Discussion
Comparative Study
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
ABC | Atangana–Baleanu–Caputo |
CW | Contaminated water |
DEs | Differential equations |
EUS | Existence and uniqueness of solution |
FPT | Fixed-point theory |
FC | Fractional Calculus |
FDEs | Fractional differential equations |
FFDs | Fractal–fractional derivatives |
FFI | Fractal–fractional integral |
LC | Lipschitz continuous |
LP | Lagrange polynomial |
MM | Mathematical modeling |
NP | Newton polynomial |
ODEs | Ordinary differential equations |
RK4 | Runge–Kutta |
WP | Water pollution |
WSs | Water sources |
WPs | Water pollutants |
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Time Range t | RK-4 CPU Time (s) | EM CPU Time (s) | NP (AB) CPU Time (s) | |
---|---|---|---|---|
100 | 50 | 0.129 | 0.114 | 0.092 |
200 | 100 | 0.251 | 0.228 | 0.188 |
3000 | 150 | 0.372 | 0.340 | 0.284 |
400 | 200 | 0.503 | 0.451 | 0.369 |
500 | 250 | 0.634 | 0.567 | 0.457 |
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Loudahi, L.; Ali, A.; Yuan, J.; Ahmad, J.; Amin, L.G.; Wei, Y. Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives. Fractal Fract. 2025, 9, 321. https://doi.org/10.3390/fractalfract9050321
Loudahi L, Ali A, Yuan J, Ahmad J, Amin LG, Wei Y. Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives. Fractal and Fractional. 2025; 9(5):321. https://doi.org/10.3390/fractalfract9050321
Chicago/Turabian StyleLoudahi, Lamia, Amjad Ali, Jing Yuan, Jalil Ahmad, Lamiaa Galal Amin, and Yunlan Wei. 2025. "Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives" Fractal and Fractional 9, no. 5: 321. https://doi.org/10.3390/fractalfract9050321
APA StyleLoudahi, L., Ali, A., Yuan, J., Ahmad, J., Amin, L. G., & Wei, Y. (2025). Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives. Fractal and Fractional, 9(5), 321. https://doi.org/10.3390/fractalfract9050321