RETRACTED ARTICLE: Numerical Solution of Abel′s Integral Equations using Hermite Wavelet
Publié en ligne: 25 oct. 2019
Pages: 395 - 406
Reçu: 21 mars 2019
Accepté: 08 mai 2019
© 2019 R. A. Mundewadi and Kumbinarasaiah S, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 Public License.
Fig. 1
Comparison of numerical solutions with exact solutions of example 1.Fig. 2
Comparison of numerical solutions, exact solutions and existing methods of example 2.Fig. 3
Comparison of error analysis of example 2.Fig. 4
Comparison of Numerical solutions of example 3.Fig. 5
Comparison of Numerical solutions of example 4.Fig. 6
Comparison of Error analysis of example 4.Numerical results of example 4.
| x |
Exact solution |
Present method (k = 1, M = 10) |
Method [22] (k = 0, M = 16) |
Method [23] (m = 16) |
| 0.1 |
0.414059 |
0.411229 |
0.415689 |
0.402472 |
| 0.2 |
0.508352 |
0.507572 |
0.505528 |
0.519751 |
| 0.3 |
0.564309 |
0.563685 |
0.566205 |
0.554755 |
| 0.4 |
0.603347 |
0.602926 |
0.601908 |
0.605031 |
| 0.5 |
0.632868 |
0.632521 |
0.634188 |
0.640487 |
| 0.6 |
0.656323 |
0.656059 |
0.655109 |
0.654785 |
| 0.7 |
0.675601 |
0.675358 |
0.676588 |
0.678700 |
| 0.8 |
0.691842 |
0.691690 |
0.691596 |
0.688860 |
| 0.9 |
0.705787 |
0.705398 |
0.704377 |
0.706495 |
Error analysis of example 3.
| x |
Present method (k = 1, M = 10) |
Method [23] (m = 16) |
| 0.1 |
1.52e-08 |
2.99e-03 |
| 0.2 |
6.92e-10 |
6.86e-03 |
| 0.3 |
1.79e-09 |
6.55e-03 |
| 0.4 |
5.79e-10 |
1.75e-03 |
| 0.5 |
1.04e-09 |
8.27e-03 |
| 0.6 |
3.79e-10 |
1.64e-03 |
| 0.7 |
1.14e-09 |
4.11e-03 |
| 0.8 |
9.93e-10 |
3.98e-03 |
| 0.9 |
8.86e-09 |
1.13e-03 |
Error analysis of example 2.
| x |
Present method (k = 1, M = 10) |
Method [22] (k = 1, M = 8) |
Method [23] (m = 16) |
| 0.1 |
4.73e-04 |
1.18e-03 |
8.57e-04 |
| 0.2 |
3.77e-05 |
1.38e-03 |
1.32e-02 |
| 0.3 |
6.21e-05 |
1.29e-03 |
1.11e-02 |
| 0.4 |
2.37e-05 |
1.52e-03 |
3.13e-03 |
| 0.5 |
2.85e-05 |
5.90e-04 |
1.38e-02 |
| 0.6 |
9.87e-06 |
4.19e-04 |
2.57e-03 |
| 0.7 |
2.72e-05 |
3.19e-04 |
7.08e-03 |
| 0.8 |
3.05e-05 |
2.54e-04 |
6.71e-03 |
| 0.9 |
2.59e-04 |
2.08e-04 |
2.09e-03 |
Error analysis of example 4.
| x |
Present method (k = 1, M = 10) |
Method [22] (k = 0, M = 16) |
Method [23] (m = 16) |
| 0.1 |
2.82e-03 |
1.62e-03 |
1.15e-02 |
| 0.2 |
7.79e-04 |
2.82e-03 |
1.13e-02 |
| 0.3 |
6.23e-04 |
1.89e-03 |
9.55e-03 |
| 0.4 |
4.21e-04 |
1.43e-03 |
1.68e-03 |
| 0.5 |
3.46e-04 |
1.32e-03 |
7.61e-03 |
| 0.6 |
2.63e-04 |
1.21e-03 |
1.53e-03 |
| 0.7 |
2.42e-04 |
9.86e-04 |
3.09e-03 |
| 0.8 |
1.50e-04 |
2.45e-04 |
2.98e-03 |
| 0.9 |
3.87e-04 |
1.40e-03 |
7.08e-04 |
Numerical results of example 1.
| x |
Exact solution |
Present method (k = 1, M = 8) |
Abs. Error |
| 0.1 |
0.9910 |
0.9910 |
1.61e-12 |
| 0.2 |
0.9680 |
0.9680 |
4.71e-13 |
| 0.3 |
0.9370 |
0.9370 |
2.27e-12 |
| 0.4 |
0.9040 |
0.9040 |
3.04e-12 |
| 0.5 |
0.8750 |
0.8750 |
2.52e-12 |
| 0.6 |
0.8560 |
0.8560 |
9.67e-13 |
| 0.7 |
0.8530 |
0.8530 |
9.64e-13 |
| 0.8 |
0.8720 |
0.8720 |
2.36e-12 |
| 0.9 |
0.9190 |
0.9190 |
2.36e-12 |
Numerical results of example 3.
| x |
Exact solution |
Present method (k = 1, M = 10) |
Method [23] (m = 16) |
| 0.1 |
0.953462589245592 |
0.953462604453520 |
0.95646081381695 |
| 0.2 |
0.912870929175277 |
0.912870928482700 |
0.90601007037324 |
| 0.3 |
0.877058019307029 |
0.877058021105882 |
0.88361513925322 |
| 0.4 |
0.845154254728517 |
0.845154255308354 |
0.84340093819493 |
| 0.5 |
0.816496580927726 |
0.816496581976408 |
0.80822420481499 |
| 0.6 |
0.790569415042095 |
0.790569415421744 |
0.79221049469412 |
| 0.7 |
0.766964988847370 |
0.766964989996025 |
0.76284677221990 |
| 0.8 |
0.745355992499930 |
0.745355991505969 |
0.74933888037055 |
| 0.9 |
0.725476250110012 |
0.725476258975992 |
0.72434536240934 |
Numerical results of example 2.
| x |
Exact solution |
Present method (k = 1, M = 10) |
Method [22] (k = 1, M = 8) |
Method [23] (m = 16) |
| 0.1 |
0.201317 |
0.200842 |
0.200128 |
0.200460 |
| 0.2 |
0.284705 |
0.284667 |
0.286092 |
0.297987 |
| 0.3 |
0.348691 |
0.348628 |
0.347394 |
0.337588 |
| 0.4 |
0.402634 |
0.402609 |
0.404161 |
0.405769 |
| 0.5 |
0.450158 |
0.450129 |
0.449568 |
0.464014 |
| 0.6 |
0.493124 |
0.493113 |
0.492704 |
0.490550 |
| 0.7 |
0.532634 |
0.532607 |
0.532315 |
0.539721 |
| 0.8 |
0.569410 |
0.569440 |
0.569156 |
0.562698 |
| 0.9 |
0.603951 |
0.603690 |
0.603742 |
0.606044 |
