Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context
Abstract
1. Introduction
2. Fractional Calculus Fundamental Concepts
- ,
- ,
- ,
- .
3. Methodologies
3.1. ARPSM Technique
3.2. ATIM Technique
4. Application of the Proposed Methods
4.1. Solution of the Problem Using ARPSM
4.2. Solution of the Problem Using ATIM
5. Results and Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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0.01 | 02 | 0.0221098 | 0.0211957 | 0.0220352 | 0.0223319 | 2.967220 × |
04 | 0.0478296 | 0.0480287 | 0.0480221 | 0.0479905 | 3.152320 × | |
06 | 0.0579054 | 0.0577927 | 0.0577084 | 0.0576779 | 3.045560 × | |
08 | 0.0604411 | 0.0603879 | 0.0603586 | 0.0603492 | 9.415964 × | |
10 | 0.0610519 | 0.0610369 | 0.0610291 | 0.0610266 | 2.447684 × | |
0.05 | 02 | 0.0321943 | 0.0220065 | 0.0210241 | 0.0221707 | 1.146590 × |
04 | 0.0467622 | 0.0478049 | 0.0480399 | 0.0479098 | 1.301040 × | |
06 | 0.0579458 | 0.0579430 | 0.0577994 | 0.0576540 | 1.453130 × | |
08 | 0.0605275 | 0.0604561 | 0.0603895 | 0.0603430 | 4.649000 × | |
10 | 0.0610793 | 0.0610560 | 0.0610373 | 0.0610251 | 1.218940 × | |
0.10 | 02 | 0.0467014 | 0.0263525 | 0.0205183 | 0.0219690 | 1.450630 × |
04 | 0.0453932 | 0.0473046 | 0.0479997 | 0.0478083 | 1.914300 × | |
06 | 0.0578265 | 0.0580034 | 0.0578972 | 0.0576240 | 2.732150 × | |
08 | 0.0605779 | 0.0605112 | 0.0604268 | 0.0603352 | 9.150590 × | |
10 | 0.0610981 | 0.0610727 | 0.0610474 | 0.0610231 | 2.425630 × |
0.01 | 02 | 0.532107 | 0.545468 | 0.554964 | 0.557983 | 3.019230 × |
04 | 0.314878 | 0.320959 | 0.324483 | 0.325693 | 1.209400 × | |
06 | 0.165805 | 0.167696 | 0.168655 | 0.169047 | 3.916230 × | |
08 | 0.083746 | 0.084402 | 0.084733 | 0.084892 | 1.587480 × | |
10 | 0.041770 | 0.042055 | 0.042199 | 0.042274 | 7.408930 × | |
0.05 | 02 | 0.524115 | 0.527784 | 0.544750 | 0.559138 | 1.438730 × |
04 | 0.306140 | 0.313125 | 0.320750 | 0.326682 | 5.932430 × | |
06 | 0.162208 | 0.165296 | 0.167652 | 0.169610 | 1.958690 × | |
08 | 0.082488 | 0.083570 | 0.084387 | 0.085182 | 7.945760 × | |
10 | 0.041231 | 0.041693 | 0.042048 | 0.042419 | 3.705760 × | |
0.10 | 02 | 0.533186 | 0.519364 | 0.533576 | 0.560578 | 2.700240 × |
04 | 0.302176 | 0.307277 | 0.316344 | 0.327923 | 1.157840 × | |
06 | 0.159580 | 0.163130 | 0.166398 | 0.170317 | 3.918810 × | |
08 | 0.081560 | 0.082815 | 0.083954 | 0.085545 | 1.591250 × | |
10 | 0.040839 | 0.041368 | 0.041859 | 0.042600 | 7.414770 × |
0.01 | 02 | 0.0220278 | 0.0211997 | 0.0220355 | 0.0223319 | 2.963860 × |
04 | 0.0478545 | 0.0480306 | 0.0480221 | 0.0479905 | 3.156600 × | |
06 | 0.0579072 | 0.0577928 | 0.0577084 | 0.0576779 | 3.045750 × | |
08 | 0.0604412 | 0.0603879 | 0.0603586 | 0.0603492 | 9.416050 × | |
10 | 0.0610519 | 0.0610369 | 0.0610291 | 0.0610266 | 2.447689 × | |
0.05 | 02 | 0.0308651 | 0.0219002 | 0.0210279 | 0.0221707 | 1.142720 × |
04 | 0.0469628 | 0.0478345 | 0.0480414 | 0.0479098 | 1.315680 × | |
06 | 0.0579619 | 0.0579451 | 0.0577994 | 0.0576540 | 1.453970 × | |
08 | 0.0605283 | 0.0604562 | 0.0603895 | 0.0603430 | 4.649390 × | |
10 | 0.0610793 | 0.0610560 | 0.0610373 | 0.0610251 | 1.218970 × | |
0.10 | 02 | 0.0427823 | 0.0257535 | 0.0205113 | 0.0219690 | 1.457670 × |
04 | 0.0459034 | 0.0474097 | 0.0480076 | 0.0478083 | 1.992630 × | |
06 | 0.0578686 | 0.0580115 | 0.0578977 | 0.0576240 | 2.737310 × | |
08 | 0.0605800 | 0.0605116 | 0.0604268 | 0.0603352 | 9.153030 × | |
10 | 0.0610983 | 0.0610728 | 0.0610474 | 0.0610231 | 2.425800 × |
0.01 | 02 | 0.528615 | 0.545031 | 0.554950 | 0.557983 | 3.032920 × |
04 | 0.314284 | 0.320883 | 0.324481 | 0.325693 | 1.211790 × | |
06 | 0.165761 | 0.167690 | 0.168655 | 0.169047 | 3.918050 × | |
08 | 0.083742 | 0.084402 | 0.084733 | 0.084892 | 1.587640 × | |
10 | 0.041770 | 0.042055 | 0.042199 | 0.042274 | 7.409110 × | |
0.05 | 02 | 0.506951 | 0.523760 | 0.544404 | 0.559138 | 1.473360 × |
04 | 0.303296 | 0.312441 | 0.320690 | 0.326682 | 5.992400 × | |
06 | 0.162000 | 0.165245 | 0.167647 | 0.169610 | 1.963240 × | |
08 | 0.082470 | 0.083566 | 0.084387 | 0.085182 | 7.949810 × | |
10 | 0.041229 | 0.041693 | 0.042048 | 0.042419 | 3.706200 × | |
0.10 | 02 | 0.498744 | 0.508760 | 0.532171 | 0.560578 | 2.840760 × |
04 | 0.296573 | 0.305501 | 0.316103 | 0.327923 | 1.181980 × | |
06 | 0.159179 | 0.162999 | 0.166380 | 0.170317 | 3.937010 × | |
08 | 0.081525 | 0.082803 | 0.083952 | 0.085545 | 1.592860 × | |
10 | 0.040836 | 0.041367 | 0.041859 | 0.042600 | 7.416530 × |
0.01 | 02 | 0.0223319 | 0.0220352 | 0.0220355 | 2.967220 × | 2.963860 × |
04 | 0.0479905 | 0.0480221 | 0.0480221 | 3.152320 × | 3.156600 × | |
06 | 0.0576779 | 0.0577084 | 0.0577084 | 3.045560 × | 3.045750 × | |
08 | 0.0603492 | 0.0603586 | 0.0603586 | 9.415964 × | 9.416050 × | |
10 | 0.0610266 | 0.0610291 | 0.0610291 | 2.447684 × | 2.447689 × | |
0.05 | 02 | 0.0221707 | 0.0210241 | 0.0210279 | 1.146590 × | 1.142720 × |
04 | 0.0479098 | 0.0480399 | 0.0480414 | 1.301040 × | 1.315680 × | |
06 | 0.0576540 | 0.0577994 | 0.0577994 | 1.453130 × | 1.453970 × | |
08 | 0.0603430 | 0.0603895 | 0.0603895 | 4.649000 × | 4.649390 × | |
10 | 0.0610251 | 0.0610373 | 0.0610373 | 1.218940 × | 1.218970 × | |
0.10 | 02 | 0.0219690 | 0.0205183 | 0.0205113 | 1.450630 × | 1.457670 × |
04 | 0.0478083 | 0.0479997 | 0.0480076 | 1.914300 × | 1.992630 × | |
06 | 0.0576240 | 0.0578972 | 0.0578977 | 2.732150 × | 2.737310 × | |
08 | 0.0603352 | 0.0604268 | 0.0604268 | 9.150590 × | 9.153030 × | |
10 | 0.0610231 | 0.0610474 | 0.0610474 | 2.425630 × | 2.425800 × |
0.01 | 02 | 0.557983 | 0.554964 | 0.554950 | 3.019230 × | 3.032920 × |
04 | 0.325693 | 0.324483 | 0.324481 | 1.209400 × | 1.211790 × | |
06 | 0.169047 | 0.168655 | 0.168655 | 3.916230 × | 3.918050 × | |
08 | 0.084892 | 0.084733 | 0.084733 | 1.587480 × | 1.587640 × | |
10 | 0.042274 | 0.042199 | 0.042199 | 7.408930 × | 7.409110 × | |
0.05 | 02 | 0.559138 | 0.544750 | 0.544404 | 1.438730 × | 1.473360 × |
04 | 0.326682 | 0.320750 | 0.320690 | 5.932430 × | 5.992400 × | |
06 | 0.169610 | 0.167652 | 0.167647 | 1.958690 × | 1.963240 × | |
08 | 0.085182 | 0.084387 | 0.084387 | 7.945760 × | 7.949810 × | |
10 | 0.042419 | 0.042048 | 0.042048 | 3.705760 × | 3.706200 × | |
0.10 | 02 | 0.560578 | 0.533576 | 0.532171 | 2.700240 × | 2.840760 × |
04 | 0.327923 | 0.316344 | 0.316103 | 1.157840 × | 1.181980 × | |
06 | 0.170317 | 0.166398 | 0.166380 | 3.918810 × | 3.937010 × | |
08 | 0.085545 | 0.083954 | 0.083952 | 1.591250 × | 1.592860 × | |
10 | 0.042600 | 0.041859 | 0.041859 | 7.414770 × | 7.416530 × |
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Alkhezi, Y.; Shafee, A. Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context. Fractal Fract. 2025, 9, 50. https://doi.org/10.3390/fractalfract9010050
Alkhezi Y, Shafee A. Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context. Fractal and Fractional. 2025; 9(1):50. https://doi.org/10.3390/fractalfract9010050
Chicago/Turabian StyleAlkhezi, Yousuf, and Ahmad Shafee. 2025. "Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context" Fractal and Fractional 9, no. 1: 50. https://doi.org/10.3390/fractalfract9010050
APA StyleAlkhezi, Y., & Shafee, A. (2025). Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context. Fractal and Fractional, 9(1), 50. https://doi.org/10.3390/fractalfract9010050