Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation
Abstract
:1. Introduction
2. Computational Solutions vs. Accuracy
2.1. Analytical Solutions
- Through the ESE method’s steps gets the next values:Set ISet IISet IIISet IV
- 2.
- Through the MKud method’s steps gets the next valuesSet ISet IIConsequently, the exact solutions of the fractional nonlinear equation are constructed in the following
2.2. Solutions’ Accuracy
3. Results’ Explanation
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Value of | Absolute Error | Value of | Absolute Error | ||||
---|---|---|---|---|---|---|---|
0 | 0.666666667 | 0.666666667 | 0 | 0.26 | 0.537334231 | 0.540427702 | 0.003093471 |
0.01 | 0.66646668 | 0.666466687 | 6.66684 × 10 | 0.27 | 0.527670559 | 0.531271644 | 0.003601085 |
0.02 | 0.66586688 | 0.665866987 | 1.06678 × 10 | 0.28 | 0.517712301 | 0.521881361 | 0.00416906 |
0.03 | 0.664867746 | 0.664868286 | 5.40129 × 10 | 0.29 | 0.507466562 | 0.512268611 | 0.004802049 |
0.04 | 0.663470077 | 0.663471784 | 1.70739 × 10 | 0.3 | 0.49694059 | 0.502445467 | 0.005504876 |
0.05 | 0.661674988 | 0.661679158 | 4.16943 × 10 | 0.31 | 0.486141768 | 0.492424307 | 0.006282539 |
0.06 | 0.659483911 | 0.65949256 | 8.64823 × 10 | 0.32 | 0.475077598 | 0.482217803 | 0.007140204 |
0.07 | 0.656898591 | 0.656914619 | 1.60274 × 10 | 0.33 | 0.463755696 | 0.471838905 | 0.008083209 |
0.08 | 0.653921083 | 0.653948435 | 2.73526 × 10 | 0.34 | 0.452183778 | 0.461300833 | 0.009117055 |
0.09 | 0.650553747 | 0.65059758 | 4.38328 × 10 | 0.35 | 0.440369653 | 0.450617058 | 0.010247405 |
0.1 | 0.646799248 | 0.646866089 | 6.68405 × 10 | 0.36 | 0.428321208 | 0.43980129 | 0.011480082 |
0.11 | 0.64266055 | 0.642758463 | 9.79132 × 10 | 0.37 | 0.416046402 | 0.428867467 | 0.012821065 |
0.12 | 0.638140907 | 0.638279661 | 0.000138754 | 0.38 | 0.403553253 | 0.417829735 | 0.014276482 |
0.13 | 0.633243865 | 0.633435098 | 0.000191233 | 0.39 | 0.390849829 | 0.406702435 | 0.015852606 |
0.14 | 0.627973249 | 0.628230636 | 0.000257388 | 0.4 | 0.377944239 | 0.395500089 | 0.01755585 |
0.15 | 0.62233316 | 0.622672585 | 0.000339425 | 0.41 | 0.36484462 | 0.384237382 | 0.019392762 |
0.16 | 0.616327971 | 0.616767693 | 0.000439722 | 0.42 | 0.351559131 | 0.372929146 | 0.021370015 |
0.17 | 0.609962313 | 0.610523141 | 0.000560827 | 0.43 | 0.33809594 | 0.361590344 | 0.023494403 |
0.18 | 0.603241075 | 0.603946535 | 0.00070546 | 0.44 | 0.324463218 | 0.35023605 | 0.025772832 |
0.19 | 0.596169391 | 0.597045905 | 0.000876514 | 0.45 | 0.310669125 | 0.338881435 | 0.02821231 |
0.2 | 0.588752633 | 0.589829689 | 0.001077056 | 0.46 | 0.296721809 | 0.327541748 | 0.03081994 |
0.21 | 0.580996403 | 0.582306733 | 0.00131033 | 0.47 | 0.282629388 | 0.316232296 | 0.033602908 |
0.22 | 0.572906525 | 0.574486278 | 0.001579754 | 0.48 | 0.268399947 | 0.304968423 | 0.036568476 |
0.23 | 0.564489035 | 0.566377955 | 0.00188892 | 0.49 | 0.254041531 | 0.293765499 | 0.039723967 |
0.24 | 0.555750171 | 0.557991772 | 0.002241601 | 0.5 | 0.239562133 | 0.282638889 | 0.043076756 |
0.25 | 0.546696364 | 0.549338108 | 0.002641743 | 0.51 | 0.224969687 | 0.271603943 | 0.046634256 |
Value of | Absolute Error | Value of | Absolute Error | ||||
---|---|---|---|---|---|---|---|
0 | 0.603474 | 0.603474 | 0 | 0.26 | 0.591331 | 0.591583 | 0.000252 |
0.01 | 0.603456 | 0.603456 | 5.49 × 10 | 0.27 | 0.590392 | 0.590686 | 0.000294 |
0.02 | 0.603402 | 0.603402 | 8.79 × 10 | 0.28 | 0.589421 | 0.589761 | 0.00034 |
0.03 | 0.603311 | 0.603311 | 4.45 × 10 | 0.29 | 0.588416 | 0.588807 | 0.000391 |
0.04 | 0.603183 | 0.603183 | 1.41 × 10 | 0.3 | 0.587379 | 0.587827 | 0.000448 |
0.05 | 0.603019 | 0.60302 | 3.43 × 10 | 0.31 | 0.586308 | 0.58682 | 0.000511 |
0.06 | 0.602819 | 0.60282 | 7.12 × 10 | 0.32 | 0.585206 | 0.585787 | 0.000581 |
0.07 | 0.602583 | 0.602584 | 1.32 × 10 | 0.33 | 0.584071 | 0.584728 | 0.000657 |
0.08 | 0.602311 | 0.602313 | 2.25 × 10 | 0.34 | 0.582905 | 0.583646 | 0.000741 |
0.09 | 0.602002 | 0.602006 | 3.61 × 10 | 0.35 | 0.581707 | 0.582539 | 0.000832 |
0.1 | 0.601657 | 0.601663 | 5.5 × 10 | 0.36 | 0.580477 | 0.58141 | 0.000932 |
0.11 | 0.601277 | 0.601285 | 8.05 × 10 | 0.37 | 0.579217 | 0.580258 | 0.001041 |
0.12 | 0.60086 | 0.600871 | 1.14 × 10 | 0.38 | 0.577926 | 0.579084 | 0.001159 |
0.13 | 0.600408 | 0.600423 | 1.57 × 10 | 0.39 | 0.576604 | 0.57789 | 0.001286 |
0.14 | 0.59992 | 0.599941 | 2.11 × 10 | 0.4 | 0.575253 | 0.576677 | 0.001424 |
0.15 | 0.599396 | 0.599424 | 2.79 × 10 | 0.41 | 0.573871 | 0.575444 | 0.001573 |
0.16 | 0.598837 | 0.598873 | 3.61 × 10 | 0.42 | 0.57246 | 0.574193 | 0.001733 |
0.17 | 0.598243 | 0.598288 | 4.6 × 10 | 0.43 | 0.57102 | 0.572925 | 0.001905 |
0.25 | 0.592235 | 0.592451 | 0.000216 | 0.51 | 0.558485 | 0.562275 | 0.00379 |
0.18 | 0.597613 | 0.597671 | 5.78 × 10 | 0.44 | 0.569551 | 0.571641 | 0.00209 |
0.19 | 0.596948 | 0.59702 | 7.18 × 10 | 0.45 | 0.568053 | 0.570341 | 0.002288 |
0.2 | 0.596249 | 0.596337 | 8.82 × 10 | 0.46 | 0.566527 | 0.569027 | 0.0025 |
0.21 | 0.595515 | 0.595622 | 0.000107 | 0.47 | 0.564973 | 0.567699 | 0.002726 |
0.22 | 0.594746 | 0.594876 | 0.000129 | 0.48 | 0.563391 | 0.566359 | 0.002968 |
0.23 | 0.593943 | 0.594098 | 0.000154 | 0.49 | 0.561783 | 0.565008 | 0.003225 |
0.24 | 0.593106 | 0.59329 | 0.000183 | 0.5 | 0.560147 | 0.563646 | 0.003499 |
0.25 | 0.592235 | 0.592451 | 0.000216 | 0.51 | 0.558485 | 0.562275 | 0.00379 |
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Yue, C.; Lu, D.; Khater, M.M.A. Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation. Fluids 2021, 6, 235. https://doi.org/10.3390/fluids6070235
Yue C, Lu D, Khater MMA. Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation. Fluids. 2021; 6(7):235. https://doi.org/10.3390/fluids6070235
Chicago/Turabian StyleYue, Chen, Dianchen Lu, and Mostafa M. A. Khater. 2021. "Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation" Fluids 6, no. 7: 235. https://doi.org/10.3390/fluids6070235