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RETRACTED ARTICLE: Numerical Solution of Abel′s Integral Equations using Hermite Wavelet

R. A. Mundewadi
R. A. Mundewadi
 and 
Kumbinarasaiah S
Kumbinarasaiah S
   | Oct 25, 2019
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Published Online: Oct 25, 2019
Page range: 395 - 406
Received: Mar 21, 2019
Accepted: May 08, 2019
DOI: https://doi.org/10.2478/AMNS.2019.2.00037
Keywords
© 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.
Comparison of numerical solutions with exact solutions of example 1.

Fig. 2

Comparison of numerical solutions, exact solutions and existing methods of example 2.
Comparison of numerical solutions, exact solutions and existing methods of example 2.

Fig. 3

Comparison of error analysis of example 2.
Comparison of error analysis of example 2.

Fig. 4

Comparison of Numerical solutions of example 3.
Comparison of Numerical solutions of example 3.

Fig. 5

Comparison of Numerical solutions of example 4.
Comparison of Numerical solutions of example 4.

Fig. 6

Comparison of Error analysis of example 4.
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
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