A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations
, , y | 05 sept 2018
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Publicado en línea: 05 sept 2018
Páginas: 447 - 458
Recibido: 28 abr 2018
Aceptado: 05 sept 2018
DOI: https://doi.org/10.21042/AMNS.2018.2.00035
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© 2018 S.C. Shiralashetti et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
Fig. 1
Fig. 2
Numerical result of Example 4.
| Exact solution | Method [ | Abs. Error (Method [ | HPM ( | Abs. Error (HPM) | |
|---|---|---|---|---|---|
| 0 | 1.00000 | 1.00004 | 4.00e-05 | 0.99999 | 7.39e-06 |
| 0.1 | 1.22140 | 1.22147 | 6.72e-05 | 1.22140 | 6.61e-07 |
| 0.2 | 1.49182 | 1.49180 | 2.46e-05 | 1.49182 | 1.35e-09 |
| 0.3 | 1.82211 | 1.82218 | 6.12e-05 | 1.82211 | 4.61e-08 |
| 0.4 | 2.22554 | 2.22551 | 3.09e-05 | 2.22554 | 1.29e-07 |
| 0.5 | 2.71828 | 2.71822 | 6.18e-05 | 2.71828 | 6.17e-08 |
| 0.6 | 3.32011 | 3.32017 | 5.30e-05 | 3.32011 | 1.73e-07 |
| 0.7 | 4.05519 | 4.05529 | 9.00e-05 | 4.05519 | 1.31e-07 |
| 0.8 | 4.95303 | 4.95306 | 2.75e-05 | 4.95303 | 1.25e-07 |
| 0.9 | 6.04964 | 6.04961 | 3.74e-05 | 6.04964 | 9.70e-07 |
| 1 | 7.38905 | 7.38901 | 4.60e-05 | 7.38905 | 8.99e-06 |
Numerical result of Example 3.
| Exact solution | HWM ( | Abs. Error (HWM) | HPM ( | Abs. Error (HPM) | |
|---|---|---|---|---|---|
| 0.0625 | 1.0645 | 1.0892 | 2.47e-02 | 1.0645 | 5.50e-09 |
| 0.1875 | 1.2062 | 1.2362 | 2.99e-02 | 1.2062 | 6.56e-09 |
| 0.3125 | 1.3668 | 1.4021 | 3.52e-02 | 1.3668 | 7.63e-09 |
| 0.4375 | 1.5488 | 1.5893 | 4.04e-02 | 1.5488 | 8.69e-09 |
| 0.5625 | 1.7551 | 1.8008 | 4.57e-02 | 1.7551 | 9.77e-09 |
| 0.6875 | 1.9887 | 2.0397 | 5.09e-02 | 1.9887 | 1.08e-08 |
| 0.8125 | 2.2535 | 2.3098 | 5.62e-02 | 2.2535 | 1.19e-08 |
| 0.9375 | 2.5536 | 2.6151 | 6.14e-02 | 2.5536 | 1.30e-08 |
Numerical result of Example 5.
| Exact solution | Method [ | Abs. Error (Method [ | HPM ( | Abs. Error (HPM) | |
|---|---|---|---|---|---|
| 0 | 1.00000 | 1.01642 | 1.64e-02 | 1.00000 | 3.79e-06 |
| 0.1 | 1.10517 | 1.11627 | 1.10e-02 | 1.10517 | 1.82e-07 |
| 0.2 | 1.22140 | 1.22593 | 4.52e-03 | 1.22140 | 4.30e-07 |
| 0.3 | 1.34985 | 1.34637 | 3.48e-03 | 1.34985 | 4.56e-07 |
| 0.4 | 1.49182 | 1.47864 | 1.31e-03 | 1.49182 | 4.88e-07 |
| 0.5 | 1.64872 | 1.62391 | 2.48e-02 | 1.64872 | 5.39e-07 |
| 0.6 | 1.82211 | 1.84004 | 1.79e-02 | 1.82211 | 5.80e-07 |
| 0.7 | 2.01375 | 2.02082 | 7.06e-03 | 2.01375 | 6.34e-07 |
| 0.8 | 2.22554 | 2.21936 | 6.18e-03 | 2.22554 | 6.90e-07 |
| 0.9 | 2.45960 | 2.43742 | 2.21e-02 | 2.45960 | 7.18e-07 |
| 1 | 2.71828 | 2.67690 | 4.13e-02 | 2.71828 | 1.10e-06 |
Numerical result of Example 1.
| Exact solution | HWM ( | Abs. Error (HWM) | HPM ( | Abs. Error (HPM) | |
|---|---|---|---|---|---|
| 0.0625 | 0.1945 | 0.2046 | 1.00e-02 | 0.1947 | 1.94e-04 |
| 0.1875 | 0.2332 | 0.2453 | 1.20e-02 | 0.2334 | 2.36e-04 |
| 0.3125 | 0.2709 | 0.2850 | 1.41e-02 | 0.2711 | 2.78e-04 |
| 0.4375 | 0.3049 | 0.3211 | 1.62e-02 | 0.3052 | 3.19e-04 |
| 0.5625 | 0.3311 | 0.3495 | 1.83e-02 | 0.3315 | 3.61e-04 |
| 0.6875 | 0.3485 | 0.3689 | 2.04e-02 | 0.3489 | 4.03e-04 |
| 0.8125 | 0.3670 | 0.3895 | 2.25e-02 | 0.3675 | 4.45e-04 |
| 0.9375 | 0.4216 | 0.4462 | 2.45e-02 | 0.4221 | 4.86e-04 |
Error Analysis.
| Haar Wavelet Method (HWM) | Hosoya Polynomial Method (HPM) | |||||
|---|---|---|---|---|---|---|
| Example 1 | Example 2 | Example 3 | Example 1 | Example 3 | ||
| 8 | 2.46e-02 | 7.2e-02 | 2.17e-02 | 3 | 1.25e-01 | 1.83e-02 |
| 16 | 6.70e-03 | 1.7e-02 | 1.55e-02 | 4 | 4 6.89e-02 | 2.15e-03 |
| 32 | 2.10e-03 | 4.3e-03 | 3.90e-03 | 6 | 6 5.64e-03 | 6.86e-06 |
| 64 | 9.25e-04 | 1.3e-02 | 9.82e-04 | 8 | 8 4.86e-04 | 1.30e-08 |