Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications
Abstract
:1. Introduction
2. Preliminaries
- (i)
- ;
- (ii)
- and ;
- (iii)
- ;
- (iv)
- .
- (v)
- For , and , we haveand
- (vi)
- It can be noted, for , thatfor .
3. Solution of Difference Systems
4. Numerical Tests
5. Concluding Remarks and Future Works
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Numerical Result | ||
---|---|---|
1.000000000000000 | 1.003804458469754 | 0.003804458469754 |
1.486763397673679 | 1.492441614080931 | 0.005678216407252 |
1.799017244188177 | 1.799775787083807 | 0.000758542895631 |
2.107699203837270 | 2.108238705023936 | 0.000539501186666 |
2.430043141497662 | 2.435351117027752 | 0.005307975530090 |
2.774285957670009 | 2.782077629971030 | 0.007791672301020 |
3.146213032210335 | 3.155553139052627 | 0.009340106842292 |
3.550802683646889 | 3.552101745731627 | 0.001299062084737 |
3.992835834192708 | 3.998524070801430 | 0.005688236608722 |
4.477184810795735 | 4.481878717206317 | 0.004693906410582 |
5.008980080762283 | 5.009099101457296 | 0.000119020695013 |
Numerical Result | ||
---|---|---|
1.000000000000000 | 1.000711215780434 | 0.000711215780434 |
1.486763397673679 | 1.486985144407696 | 0.000221746734017 |
1.799017244188177 | 1.799134661839032 | 0.000117417650856 |
2.107699203837270 | 2.107995879710488 | 0.000296675873218 |
2.430043141497662 | 2.430361919799588 | 0.000318778301926 |
2.774285957670009 | 2.774710124429723 | 0.000424166759714 |
3.146213032210335 | 3.146720890494996 | 0.000507858284661 |
3.550802683646889 | 3.550888199443980 | 0.000085515797090 |
3.992835834192708 | 3.993098316427406 | 0.000262482234698 |
4.477184810795735 | 4.477985825418505 | 0.000801014622770 |
5.008980080762283 | 5.009009301039845 | 0.000029220277562 |
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Mohammed, P.O.; Lupas, A.A.; Agarwal, R.P.; Yousif, M.A.; Al-Sarairah, E.; Abdelwahed, M. Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications. Axioms 2024, 13, 570. https://doi.org/10.3390/axioms13080570
Mohammed PO, Lupas AA, Agarwal RP, Yousif MA, Al-Sarairah E, Abdelwahed M. Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications. Axioms. 2024; 13(8):570. https://doi.org/10.3390/axioms13080570
Chicago/Turabian StyleMohammed, Pshtiwan Othman, Alina Alb Lupas, Ravi P. Agarwal, Majeed A. Yousif, Eman Al-Sarairah, and Mohamed Abdelwahed. 2024. "Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications" Axioms 13, no. 8: 570. https://doi.org/10.3390/axioms13080570