New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method
Abstract
:1. Introduction
2. Basic Definitions of Fractional Calculus
3. New Novel Analytical Method (NAM) for Fractional Partial Differential Equations (FPDE’S)
4. Convergence Analysis of Novel Analytical Method (NAM)
5. Application
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Exact Solution | Approximate Solution | Absolute Error | Exact Solution | Approximate Solution | Absolute Error | ||||
---|---|---|---|---|---|---|---|---|---|
0.1 | 0.00 | −0.099833 | −0.099833 | 1.388 × 10−17 | 1.0 | 0.00 | −0.841471 | −0.841471 | 1.110 × 10−16 |
0.25 | 0.149438 | 0.149438 | 0 | 0.25 | −0.681639 | −0.681639 | 1.110 × 10−16 | ||
0.50 | 0.389418 | 0.389418 | 0 | 0.50 | −0.479426 | −0.479426 | 2.220 × 10−16 | ||
0.75 | 0.605186 | 0.605186 | 0 | 0.75 | −0.247404 | −0.247404 | 1.943 × 10−16 | ||
1.00 | 0.783327 | 0.783327 | 0 | 1.00 | 0.000000 | 0.000000 | 1.665 × 10−16 | ||
1.25 | 0.912764 | 0.912764 | 1.110 × 10−16 | 1.25 | 0.247404 | 0.247404 | 8.327 × 10−17 | ||
1.50 | 0.985450 | 0.985450 | 1.110 × 10−16 | 1.50 | 0.479426 | 0.479426 | 5.551 × 10−17 | ||
1.75 | 0.996865 | 0.996865 | 0 | 1.75 | 0.681639 | 0.681639 | 0 | ||
2.00 | 0.946300 | 0.946300 | 1.110 × 10−16 | 2.00 | 0.841471 | 0.841471 | 0 | ||
2.25 | 0.836899 | 0.836899 | 0 | 2.25 | 0.948985 | 0.948985 | 0 | ||
2.50 | 0.675463 | 0.675463 | 1.110 × 10−16 | 2.50 | 0.997495 | 0.997495 | 1.110 × 10−16 | ||
2.75 | 0.472031 | 0.472031 | 5.551 × 10−17 | 2.75 | 0.983986 | 0.983986 | 1.110 × 10−16 | ||
3.00 | 0.239249 | 0.239249 | 8.327 × 10−17 | 3.00 | 0.909297 | 0.909297 | 1.110 × 10−16 | ||
3.25 | −0.008407 | −0.008407 | 6.072 × 10−17 | 3.25 | 0.778073 | 0.778073 | 2.220 × 10−16 | ||
3.50 | −0.255541 | −0.255541 | 1.110 × 10−16 | 3.50 | 0.598472 | 0.598472 | 0 | ||
3.75 | −0.486787 | −0.486787 | 5.551 × 10−17 | 3.75 | 0.381661 | 0.381661 | 2.220 × 10−16 | ||
4.00 | −0.687766 | −0.687766 | 1.110 × 10−16 | 4.00 | 0.141120 | 0.141120 | 1.943 × 10−16 | ||
4.25 | −0.845984 | −0.845984 | 2.220 × 10−16 | 4.25 | −0.108195 | −0.108195 | 1.665 × 10−16 | ||
4.50 | −0.951602 | −0.951602 | 0 | 4.50 | −0.350783 | −0.350783 | 1.665 × 10−16 | ||
4.75 | −0.998054 | −0.998054 | 1.110 × 10−16 | 4.75 | −0.571561 | −0.571561 | 0 | ||
5.00 | −0.982453 | −0.982453 | 1.110 × 10−16 | 5.00 | −0.756802 | −0.756802 | 0 | ||
5.25 | −0.905767 | −0.905767 | 2.220 × 10−16 | 5.25 | −0.894989 | −0.894989 | 0 | ||
5.50 | −0.772764 | −0.772764 | 3.331 × 10−16 | 5.50 | −0.977530 | −0.977530 | 0 | ||
5.75 | −0.591716 | −0.591716 | 3.331 × 10−16 | 5.75 | −0.999293 | −0.999293 | 0 | ||
6.00 | −0.373877 | −0.373877 | 3.331 × 10−16 | 6.00 | −0.958924 | −0.958924 | 0 | ||
2.0 | 0.00 | −0.909297 | −0.909297 | 3.331 × 10−16 | 4.0 | 0.00 | 0.756802 | 0.756802 | 2.651 × 10−09 |
0.25 | −0.983986 | −0.983986 | 6.661 × 10−16 | 0.25 | 0.571561 | 0.571561 | 1.194 × 10−09 | ||
0.50 | −0.997495 | −0.997495 | 1.443 × 10−15 | 0.50 | 0.350783 | 0.350783 | 4.965 × 10−09 | ||
0.75 | −0.948985 | −0.948985 | 2.220 × 10−15 | 0.75 | 0.108195 | 0.108195 | 8.427 × 10−09 | ||
1.00 | −0.841471 | −0.841471 | 2.887 × 10−15 | 1.00 | −0.141120 | −0.141120 | 1.137 × 10−08 | ||
1.25 | −0.681639 | −0.681639 | 3.331 × 10−15 | 1.25 | −0.381661 | −0.381661 | 1.360 × 10−08 | ||
1.50 | −0.479426 | −0.479426 | 3.664 × 10−15 | 1.50 | −0.598472 | −0.598472 | 1.498 × 10−08 | ||
1.75 | −0.247404 | −0.247404 | 3.691 × 10−15 | 1.75 | −0.778073 | −0.778073 | 1.544 × 10−08 | ||
2.00 | 0.000000 | 0.000000 | 3.497 × 10−15 | 2.00 | −0.909297 | −0.909297 | 1.493 × 10−08 | ||
2.25 | 0.247404 | 0.247404 | 3.136 × 10−15 | 2.25 | −0.983986 | −0.983986 | 1.350 × 10−08 | ||
2.50 | 0.479426 | 0.479426 | 2.498 × 10−15 | 2.50 | −0.997495 | −0.997495 | 1.123 × 10−08 | ||
2.75 | 0.681639 | 0.681639 | 1.887 × 10−15 | 2.75 | −0.948985 | −0.948985 | 8.255 × 10−09 | ||
3.00 | 0.841471 | 0.841471 | 7.772 × 10−16 | 3.00 | −0.841471 | −0.841471 | 4.771 × 10−09 | ||
3.25 | 0.948985 | 0.948985 | 0 | 3.25 | −0.681639 | −0.681639 | 9.900 × 10−10 | ||
3.50 | 0.997495 | 0.997495 | 9.992 × 10−16 | 3.50 | −0.479426 | −0.479426 | 2.852 × 10−09 | ||
3.75 | 0.983986 | 0.983986 | 1.776 × 10−15 | 3.75 | −0.247404 | −0.247404 | 6.517 × 10−09 | ||
4.00 | 0.909297 | 0.909297 | 2.665 × 10−15 | 4.00 | 0.000000 | 0.000000 | 9.777 × 10−09 | ||
4.25 | 0.778073 | 0.778073 | 3.109 × 10−15 | 4.25 | 0.247404 | 0.247404 | 1.243 × 10−08 | ||
4.50 | 0.598472 | 0.598472 | 3.664 × 10−15 | 4.50 | 0.479426 | 0.479426 | 1.431 × 10−08 | ||
4.75 | 0.381661 | 0.381661 | 3.719 × 10−15 | 4.75 | 0.681639 | 0.681639 | 1.530 × 10−08 | ||
5.00 | 0.141120 | 0.141120 | 3.636 × 10−15 | 5.00 | 0.841471 | 0.841471 | 1.534 × 10−08 | ||
5.25 | −0.108195 | −0.108195 | 3.386 × 10−15 | 5.25 | 0.948985 | 0.948985 | 1.442 × 10−08 | ||
5.50 | −0.350783 | −0.350783 | 2.887 × 10−15 | 5.50 | 0.997495 | 0.997495 | 1.261 × 10−08 | ||
5.75 | −0.571561 | −0.571561 | 2.109 × 10−15 | 5.75 | 0.983986 | 0.983986 | 1.001 × 10−08 | ||
6.00 | −0.756802 | −0.756802 | 1.332 × 10−15 | 6.00 | 0.909297 | 0.909297 | 6.795 × 10−09 |
Exact Solution | Approximate Solution | Absolute Error | Exact Solution | Approximate Solution | Absolute Error | ||||
---|---|---|---|---|---|---|---|---|---|
0.01 | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.05 | 0.00 | 0.000000 | 0.000000 | 0.000000 |
0.10 | 0.309002 | 0.309002 | 0.000000 | 0.10 | 0.308631 | 0.308631 | 0.000000 | ||
0.20 | 0.587756 | 0.587756 | 0.000000 | 0.20 | 0.587051 | 0.587051 | 0.000000 | ||
0.30 | 0.808977 | 0.808977 | 0.000000 | 0.30 | 0.808006 | 0.808006 | 0.000000 | ||
0.40 | 0.951009 | 0.951009 | 0.000000 | 0.40 | 0.949868 | 0.949868 | 0.000000 | ||
0.50 | 0.999950 | 0.999950 | 0.000000 | 0.50 | 0.998750 | 0.998750 | 0.000000 | ||
0.60 | 0.951009 | 0.951009 | 0.000000 | 0.60 | 0.949868 | 0.949868 | 0.000000 | ||
0.70 | 0.808977 | 0.808977 | 0.000000 | 0.70 | 0.808006 | 0.808006 | 0.000000 | ||
0.80 | 0.587756 | 0.587756 | 0.000000 | 0.80 | 0.587051 | 0.587051 | 0.000000 | ||
0.90 | 0.309002 | 0.309002 | 0.000000 | 0.90 | 0.308631 | 0.308631 | 0.000000 | ||
0.07 | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.10 | 0.00 | 0.000000 | 0.000000 | 0.000000 |
0.10 | 0.308260 | 0.308260 | 0.000000 | 0.10 | 0.307473 | 0.307473 | 0.000000 | ||
0.20 | 0.586346 | 0.586346 | 0.000000 | 0.20 | 0.584849 | 0.584849 | 0.000000 | ||
0.30 | 0.807036 | 0.807036 | 0.000000 | 0.30 | 0.804975 | 0.804975 | 0.000000 | ||
0.40 | 0.948727 | 0.948727 | 0.000000 | 0.40 | 0.946305 | 0.946305 | 0.000000 | ||
0.50 | 0.997551 | 0.997551 | 0.000000 | 0.50 | 0.995004 | 0.995004 | 0.000000 | ||
0.60 | 0.948727 | 0.948727 | 0.000000 | 0.60 | 0.946305 | 0.946305 | 0.000000 | ||
0.70 | 0.807036 | 0.807036 | 0.000000 | 0.70 | 0.804975 | 0.804975 | 0.000000 | ||
0.80 | 0.586346 | 0.586346 | 0.000000 | 0.80 | 0.584849 | 0.584849 | 0.000000 | ||
0.90 | 0.308260 | 0.308260 | 0.000000 | 0.90 | 0.307473 | 0.307473 | 0.000000 | ||
1.00 | 1.222 × 10−16 | 1.222 × 10−16 | 0.000000 | 1.00 | 1.219 × 10−16 | 1.219 × 10−16 | 0.000000 |
Exact Solution | Approximate Solution | Absolute Error | Exact Solution | Approximate Solution | Absolute Error | ||||
---|---|---|---|---|---|---|---|---|---|
0.01 | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.05 | 0.00 | 0.000000 | 0.000000 | 0.000000 |
0.10 | 0.200000 | 0.200000 | 0.000000 | 0.10 | 0.200013 | 0.200013 | 0.000000 | ||
0.20 | 0.400000 | 0.400000 | 0.000000 | 0.20 | 0.400025 | 0.400025 | 0.000000 | ||
0.30 | 0.600000 | 0.600000 | 0.000000 | 0.30 | 0.600038 | 0.600038 | 0.000000 | ||
0.40 | 0.800000 | 0.800000 | 0.000000 | 0.40 | 0.800050 | 0.800050 | 0.000000 | ||
0.50 | 1.000000 | 1.000000 | 0.000000 | 0.50 | 1.000060 | 1.000060 | 0.000000 | ||
0.60 | 1.200000 | 1.200000 | 0.000000 | 0.60 | 1.200080 | 1.200080 | 0.000000 | ||
0.70 | 1.400000 | 1.400000 | 0.000000 | 0.70 | 1.400090 | 1.400090 | 0.000000 | ||
0.80 | 1.600000 | 1.600000 | 0.000000 | 0.80 | 1.600100 | 1.600100 | 0.000000 | ||
0.90 | 1.800000 | 1.800000 | 0.000000 | 0.90 | 1.800110 | 1.800110 | 0.000000 | ||
0.07 | 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.10 | 0.00 | 0.000000 | 0.000000 | 0.000000 |
0.10 | 0.200034 | 0.200034 | 0.000000 | 0.10 | 0.300000 | 0.300000 | 0.000000 | ||
0.20 | 0.400069 | 0.400069 | 0.000000 | 0.20 | 0.600000 | 0.600000 | 0.000000 | ||
0.30 | 0.600103 | 0.600103 | 0.000000 | 0.30 | 0.900000 | 0.900000 | 0.000000 | ||
0.40 | 0.800137 | 0.800137 | 0.000000 | 0.40 | 1.200000 | 1.200000 | 0.000000 | ||
0.50 | 1.000170 | 1.000170 | 0.000000 | 0.50 | 1.500000 | 1.500000 | 0.000000 | ||
0.60 | 1.200210 | 1.200210 | 0.000000 | 0.60 | 1.800000 | 1.800000 | 0.000000 | ||
0.70 | 1.400240 | 1.400240 | 0.000000 | 0.70 | 2.100000 | 2.100000 | 0.000000 | ||
0.80 | 1.600270 | 1.600270 | 0.000000 | 0.80 | 2.400000 | 2.400000 | 0.000000 | ||
0.90 | 1.800310 | 1.800310 | 0.000000 | 0.90 | 2.700000 | 2.700000 | 0.000000 | ||
1.00 | 2.000340 | 2.000340 | 0.000000 | 1.00 | 3.000000 | 3.000000 | 0.000000 |
Exact Solution | Approximate Solution | Absolute Error | Exact Solution | Approximate Solution | Absolute Error | |||||
---|---|---|---|---|---|---|---|---|---|---|
0.0 | 0.01 | 0.0 | 0.999800 | 0.999800 | 0 | 0.1 | 0.0 | 0.980067 | 0.980067 | 0 |
1.0 | 0.523366 | 0.523366 | 1.110 × 10−16 | 1.0 | 0.362358 | 0.362358 | 1.110 × 10−16 | |||
2.0 | −0.434248 | −0.434248 | 0 | 2.0 | −0.588501 | −0.588501 | 1.110 × 10−16 | |||
3.0 | −0.992617 | −0.992617 | 0 | 3.0 | −0.998295 | −0.998295 | 1.110 × 10−16 | |||
4.0 | −0.638378 | −0.638378 | 2.220 × 10−16 | 4.0 | −0.490261 | −0.490261 | 1.665 × 10−16 | |||
5.0 | 0.302783 | 0.302783 | 3.886 × 10−16 | 5.0 | 0.468517 | 0.468517 | 1.110 × 10−16 | |||
6.0 | 0.965566 | 0.965566 | 0 | 6.0 | 0.996542 | 0.996542 | 1.110 × 10−16 | |||
7.0 | 0.740613 | 0.740613 | 3.331 × 10−16 | 7.0 | 0.608351 | 0.608351 | 0 | |||
8.0 | −0.165257 | −0.165257 | 4.441 × 10−16 | 8.0 | −0.339155 | −0.339155 | 7.772 × 10−16 | |||
9.0 | −0.919190 | −0.919190 | 2.220 × 10−16 | 9.0 | −0.974844 | −0.974844 | 1.110 × 10−16 | |||
10.0 | −0.828024 | −0.828024 | 2.220 × 10−16 | 10.0 | −0.714266 | −0.714266 | 5.551 × 10−16 | |||
0.5 | 0.01 | 0.0 | 0.867819 | 0.867819 | 0 | 0.1 | 0.0 | 0.764842 | 0.764842 | 0 |
1.0 | 0.050775 | 0.050775 | 1.388 × 10−17 | 1.0 | −0.128844 | −0.128844 | 8.327 × 10−17 | |||
2.0 | −0.812952 | −0.812952 | 0 | 2.0 | −0.904072 | −0.904072 | 1.110 × 10−16 | |||
3.0 | −0.929254 | −0.929254 | 0 | 3.0 | −0.848100 | −0.848100 | 1.110 × 10−16 | |||
4.0 | −0.191204 | −0.191204 | 4.441 × 10−16 | 4.0 | −0.012389 | −0.012389 | 1.596 × 10−16 | |||
5.0 | 0.722638 | 0.722638 | 3.331 × 10−16 | 5.0 | 0.834713 | 0.834713 | 0 | |||
6.0 | 0.972090 | 0.972090 | 2.220 × 10−16 | 6.0 | 0.914383 | 0.914383 | 0 | |||
7.0 | 0.327807 | 0.327807 | 4.441 × 10−16 | 7.0 | 0.153374 | 0.153374 | 1.388 × 10−16 | |||
8.0 | −0.617860 | −0.617860 | 3.331 × 10−16 | 8.0 | −0.748647 | −0.748647 | 5.551 × 10−16 | |||
9.0 | −0.995470 | −0.995470 | 1.110 × 10−16 | 9.0 | −0.962365 | −0.962365 | 2.220 × 10−16 | |||
10.0 | −0.457849 | −0.457849 | 4.441 × 10−16 | 10.0 | −0.291289 | −0.291289 | 6.661 × 10−16 | |||
1.0 | 0.01 | 0.0 | 0.523366 | 0.523366 | 1.110 × 10−16 | 0.1 | 0.0 | 0.362358 | 0.362358 | 1.110 × 10−16 |
1.0 | −0.434248 | −0.434248 | 0 | 1.0 | −0.588501 | −0.588501 | 1.110 × 10−16 | |||
2.0 | −0.992617 | −0.992617 | 0 | 2.0 | −0.998295 | −0.998295 | 1.110 × 10−16 | |||
3.0 | −0.638378 | −0.638378 | 2.220 × 10−16 | 3.0 | −0.490261 | −0.490261 | 1.665 × 10−16 | |||
4.0 | 0.302783 | 0.302783 | 3.886 × 10−16 | 4.0 | 0.468517 | 0.468517 | 1.110 × 10−16 | |||
5.0 | 0.965566 | 0.965566 | 0 | 5.0 | 0.996542 | 0.996542 | 1.110 × 10−16 | |||
6.0 | 0.740613 | 0.740613 | 3.331 × 10−16 | 6.0 | 0.608351 | 0.608351 | 0 | |||
7.0 | −0.165257 | −0.165257 | 4.441 × 10−16 | 7.0 | −0.339155 | −0.339155 | 7.772 × 10−16 | |||
8.0 | −0.919190 | −0.919190 | 2.220 × 10−16 | 8.0 | −0.974844 | −0.974844 | 1.110 × 10−16 | |||
9.0 | −0.828024 | −0.828024 | 2.220 × 10−16 | 9.0 | −0.714266 | −0.714266 | 5.551 × 10−16 | |||
10.0 | 0.024423 | 0.024423 | 4.267 × 10−16 | 10.0 | 0.203005 | 0.203005 | 7.216 × 10−16 |
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Sultana, M.; Arshad, U.; Alam, M.N.; Bazighifan, O.; Askar, S.; Awrejcewicz, J. New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method. Symmetry 2021, 13, 2296. https://doi.org/10.3390/sym13122296
Sultana M, Arshad U, Alam MN, Bazighifan O, Askar S, Awrejcewicz J. New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method. Symmetry. 2021; 13(12):2296. https://doi.org/10.3390/sym13122296
Chicago/Turabian StyleSultana, Mariam, Uroosa Arshad, Md. Nur Alam, Omar Bazighifan, Sameh Askar, and Jan Awrejcewicz. 2021. "New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method" Symmetry 13, no. 12: 2296. https://doi.org/10.3390/sym13122296
APA StyleSultana, M., Arshad, U., Alam, M. N., Bazighifan, O., Askar, S., & Awrejcewicz, J. (2021). New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method. Symmetry, 13(12), 2296. https://doi.org/10.3390/sym13122296