A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy
Abstract
:1. Introduction
2. Higher-Order Numerical Scheme of Two-Dimensional Nonlinear Fractional VIEs
3. Estimation of the Truncation Errors
4. Convergence Analysis
5. Numerical Examples
6. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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N | Order | CPU Time | |
---|---|---|---|
8 | 1.9973659945 | - | 0.1438033000 s |
16 | 2.0439893391 | 3.2886391376 | 1.7739478000 s |
32 | 2.0217387802 | 3.3377191614 | 27.3948994000 s |
64 | 1.9623458838 | 3.3649453461 | 7.1728836283 min |
128 | 1.8839724669 | 3.3807295678 | 1.8958612181 h |
N | Order | CPU Time | |
---|---|---|---|
8 | 1.0362949128 | - | 0.2546744000 s |
16 | 1.1195268795 | 3.2104735569 | 1.8277023000 s |
32 | 1.1762915084 | 3.2505716291 | 27.406621300 s |
64 | 1.2176430484 | 3.2720824624 | 7.1583191750 min |
128 | 1.2502313451 | 3.2838242818 | 1.8919108256 h |
M | Order | Order | ||
---|---|---|---|---|
2.7093547741 | − | 8.5837427821 | − | |
2.4017036964 | 3.4958192397 | 6.7887046124 | 3.6603986275 | |
2.0350262447 | 3.5609388681 | 5.0238828559 | 3.7562615829 | |
1.6641178999 | 3.6122178456 | 3.6303832318 | 3.7906090690 | |
128 | 1.3366509233 | 3.6380630008 | 2.5643266843 | 3.8234698786 |
N | Order | Order | ||
---|---|---|---|---|
1.4497410528 | − | 6.7268209314 | − | |
1.4126072922 | 3.3593628803 | 6.9945531354 | 3.2656105874 | |
1.3572217464 | 3.3796320927 | 7.1896243538 | 3.2822435908 | |
1.2887160530 | 3.3966501211 | 7.3370971254 | 3.2926351059 | |
128 | 1.2153830342 | 3.4064514616 | 7.4615217116 | 3.2976675925 |
M | Order | Order | ||
---|---|---|---|---|
4.1299227631 | − | 1.2444932111 | − | |
3.5190750852 | 3.5528465991 | 9.1544255179 | 3.7649451968 | |
2.9677317197 | 3.5677637121 | 6.6471665752 | 3.7836579403 | |
2.4390547260 | 3.6049666456 | 4.7740052334 | 3.7994674546 | |
128 | 1.9706716024 | 3.6295628422 | 3.3836403013 | 3.8185520322 |
Exact Solution | Haar Solution in [24] | Numerical Solution | |
---|---|---|---|
0 | 0 | 0 | |
−0.000900 | −0.000931 | −0.000899999999999 | |
−0.006400 | −0.006458 | −0.006400000000000 | |
−0.018900 | −0.018932 | −0.018900000000000 | |
−0.038400 | −0.038412 | −0.038400000000000 | |
−0.062500 | −0.062504 | −0.062499999999999 | |
−0.086400 | −0.086475 | −0.086399999999997 | |
−0.102900 | −0.102941 | −0.102899999999998 | |
−0.102400 | −0.102482 | −0.102399999999991 | |
−0.072900 | −0.072970 | −0.072899999999974 | |
0 | 0 | 0.000000000000137 |
Exact Solution | Method Solution in [25] | Numerical Solution | |||
---|---|---|---|---|---|
0 | 0 | 0.018452 | 0.002162 | 0 | 0 |
0.1 | 0.057735 | 0.031135 | 0.049679 | 0.062761 | 0.057660 |
0.2 | 0.11547 | 0.13261 | 0.117302 | 0.114850 | 0.115427 |
0.3 | 0.173205 | 0.147605 | 0.168105 | 0.171241 | 0.173187 |
0.4 | 0.23094 | 0.246768 | 0.232442 | 0.230698 | 0.230858 |
0.5 | 0.288675 | 0.262075 | 0.285535 | 0.288662 | 0.288662 |
0.6 | 0.34641 | 0.360925 | 0.347581 | 0.346202 | 0.346324 |
0.7 | 0.404145 | 0.378535 | 0.398155 | 0.404047 | 0.404057 |
0.8 | 0.46188 | 0.475083 | 0.462721 | 0.461606 | 0.461757 |
0.9 | 0.519615 | 0.501015 | 0.513525 | 0.519441 | 0.519463 |
0.99 | 0.571577 | 0.583533 | 0.572104 | 0.570976 | 0.571549 |
Max error | 0 | 2.96 | 8.09 | 5.02 | 1.51 |
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Wang, Z.-Q.; Liu, Q.; Cao, J.-Y. A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy. Fractal Fract. 2022, 6, 314. https://doi.org/10.3390/fractalfract6060314
Wang Z-Q, Liu Q, Cao J-Y. A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy. Fractal and Fractional. 2022; 6(6):314. https://doi.org/10.3390/fractalfract6060314
Chicago/Turabian StyleWang, Zi-Qiang, Qin Liu, and Jun-Ying Cao. 2022. "A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy" Fractal and Fractional 6, no. 6: 314. https://doi.org/10.3390/fractalfract6060314