A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator
Abstract
:1. Introduction
2. Preliminaries
3. Configuration for HPTM
4. Configuration for YTDM
5. Numerical Applications
6. Numerical Simulation Studies
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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(Approx) | (Exact) | |||||
---|---|---|---|---|---|---|
0.2 | 0.002999 | 0.003012 | 0.003025 | 0.003037 | 0.003037 | |
0.4 | 0.004747 | 0.004776 | 0.004805 | 0.004835 | 0.004835 | |
0.01 | 0.6 | 0.007327 | 0.007392 | 0.007457 | 0.007524 | 0.007524 |
0.8 | 0.011097 | 0.011245 | 0.011392 | 0.011541 | 0.011541 | |
1 | 0.016543 | 0.016873 | 0.017203 | 0.017538 | 0.017538 | |
0.2 | 0.002999 | 0.003012 | 0.003024 | 0.003037 | 0.003037 | |
0.4 | 0.004746 | 0.004775 | 0.004805 | 0.004835 | 0.004835 | |
0.02 | 0.6 | 0.007324 | 0.007390 | 0.007456 | 0.007524 | 0.007524 |
0.8 | 0.011091 | 0.011240 | 0.011388 | 0.011541 | 0.011541 | |
1 | 0.016529 | 0.016863 | 0.017195 | 0.017538 | 0.017538 | |
0.2 | 0.002998 | 0.003011 | 0.003024 | 0.003037 | 0.003037 | |
0.4 | 0.004745 | 0.004775 | 0.004804 | 0.004835 | 0.004835 | |
0.03 | 0.6 | 0.007322 | 0.007388 | 0.007454 | 0.007524 | 0.007524 |
0.8 | 0.011086 | 0.011236 | 0.011384 | 0.011541 | 0.011541 | |
1 | 0.016517 | 0.016853 | 0.017187 | 0.017538 | 0.017538 | |
0.2 | 0.002998 | 0.003011 | 0.003024 | 0.003037 | 0.003037 | |
0.4 | 0.004744 | 0.004774 | 0.004803 | 0.004835 | 0.004835 | |
0.04 | 0.6 | 0.007319 | 0.007386 | 0.007453 | 0.007524 | 0.007524 |
0.8 | 0.011081 | 0.011231 | 0.011381 | 0.011541 | 0.011541 | |
1 | 0.016505 | 0.016843 | 0.017179 | 0.017538 | 0.017538 | |
0.2 | 0.002997 | 0.003010 | 0.003023 | 0.003037 | 0.003037 | |
0.4 | 0.004743 | 0.004773 | 0.004802 | 0.004835 | 0.004835 | |
0.05 | 0.6 | 0.004743 | 0.007384 | 0.007451 | 0.007524 | 0.007524 |
0.8 | 0.011076 | 0.011227 | 0.011377 | 0.011541 | 0.011541 | |
1 | 0.016494 | 0.016834 | 0.017171 | 0.017538 | 0.017538 |
PIA Error | RPSM Error | Our Method Error | |||
---|---|---|---|---|---|
0.2 | 0.1 | 0.1 | 3.85217 | 3.85217 | 3.8519486000 |
0.3 | 0.1 | 0.1 | 5.75911 | 5.75912 | 5.7583616000 |
0.4 | 0.1 | 0.1 | 7.65359 | 7.65352 | 7.6517330000 |
0.2 | 0.6 | 0.6 | 4.66337 | 4.66389 | 4.6473505000 |
0.3 | 0.6 | 0.6 | 6.86056 | 6.86314 | 6.8073230000 |
0.4 | 0.6 | 0.6 | 8.98263 | 8.99046 | 8.8581596000 |
0.2 | 0.9 | 0.9 | 5.12131 | 5.14241 | 4.9248718000 |
0.3 | 0.9 | 0.9 | 7.38186 | 7.48450 | 6.7502883000 |
0.4 | 0.9 | 0.9 | 9.57942 | 9.89139 | 8.1510249000 |
(Approx) | (Exact) | |||||
---|---|---|---|---|---|---|
0.2 | 0.030120 | 0.030147 | 0.030173 | 0.030197 | 0.030197 | |
0.4 | 0.035230 | 0.035259 | 0.035288 | 0.035315 | 0.035315 | |
0.01 | 0.6 | 0.040377 | 0.040410 | 0.040442 | 0.040473 | 0.040473 |
0.8 | 0.045568 | 0.045604 | 0.045641 | 0.045675 | 0.045675 | |
1 | 0.050807 | 0.050848 | 0.050889 | 0.050928 | 0.050928 | |
0.2 | 0.030119 | 0.030146 | 0.030172 | 0.030195 | 0.030195 | |
0.4 | 0.035229 | 0.035258 | 0.035288 | 0.035313 | 0.035313 | |
0.02 | 0.6 | 0.040376 | 0.040409 | 0.040441 | 0.040470 | 0.040470 |
0.8 | 0.045566 | 0.045603 | 0.045640 | 0.045672 | 0.045672 | |
1 | 0.050806 | 0.050847 | 0.050888 | 0.050925 | 0.050925 | |
0.2 | 0.030118 | 0.030145 | 0.030172 | 0.030192 | 0.030192 | |
0.4 | 0.035228 | 0.035257 | 0.035287 | 0.035310 | 0.035310 | |
0.03 | 0.6 | 0.040375 | 0.040408 | 0.040441 | 0.040467 | 0.040467 |
0.8 | 0.045565 | 0.045602 | 0.045639 | 0.045670 | 0.045670 | |
1 | 0.050804 | 0.050846 | 0.050887 | 0.050923 | 0.050923 | |
0.2 | 0.030117 | 0.030144 | 0.030171 | 0.030190 | 0.030190 | |
0.4 | 0.035227 | 0.035256 | 0.035286 | 0.035308 | 0.035308 | |
0.04 | 0.6 | 0.040374 | 0.040407 | 0.040440 | 0.040465 | 0.040465 |
0.8 | 0.045564 | 0.045601 | 0.045638 | 0.045667 | 0.045667 | |
1 | 0.050803 | 0.050845 | 0.050886 | 0.050920 | 0.050920 | |
0.2 | 0.030116 | 0.030144 | 0.030170 | 0.030187 | 0.030187 | |
0.4 | 0.035226 | 0.035256 | 0.035285 | 0.035305 | 0.035305 | |
0.05 | 0.6 | 0.040372 | 0.040406 | 0.040439 | 0.040462 | 0.040462 |
0.8 | 0.045563 | 0.045600 | 0.045637 | 0.045664 | 0.045664 | |
1 | 0.050802 | 0.050843 | 0.050885 | 0.050917 | 0.050917 |
VIM Error | Our Method Error | |||
---|---|---|---|---|
0.2 | 0.1 | 0.1 | 5.00091 | 4.9951950000 |
0.3 | 0.1 | 0.1 | 5.00091 | 7.4927920000 |
0.4 | 0.1 | 0.1 | 5.00091 | 9.9903860000 |
0.2 | 0.6 | 0.6 | 3.02003 | 5.0898600000 |
0.3 | 0.6 | 0.6 | 3.02003 | 7.6348000000 |
0.4 | 0.6 | 0.6 | 3.02003 | 1.0179720000 |
0.2 | 0.9 | 0.9 | 4.56780 | 5.2122800000 |
0.3 | 0.9 | 0.9 | 4.56780 | 7.8184000000 |
0.4 | 0.9 | 0.9 | 4.56780 | 1.0424500000 |
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Ganie, A.H.; Mofarreh, F.; Khan, A. A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator. Axioms 2023, 12, 609. https://doi.org/10.3390/axioms12060609
Ganie AH, Mofarreh F, Khan A. A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator. Axioms. 2023; 12(6):609. https://doi.org/10.3390/axioms12060609
Chicago/Turabian StyleGanie, Abdul Hamid, Fatemah Mofarreh, and Adnan Khan. 2023. "A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator" Axioms 12, no. 6: 609. https://doi.org/10.3390/axioms12060609