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Numerical Solution of Abel′s Integral Equations using Hermite Wavelet

R. A. Mundewadi

R. A. Mundewadi

Department of Mathematics, M.E.S Degree College, MalleswaramBangalore, India
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Mundewadi, R. A.
 y 
S. Kumbinarasaiah

S. Kumbinarasaiah

Department of Mathematics, Karnatak UniversityDharwad, India
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Kumbinarasaiah, S.
  
28 jun 2019
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Publicado en línea: 28 jun 2019
Páginas: 169 - 180
Recibido: 21 mar 2019
Aceptado: 08 may 2019
DOI: https://doi.org/10.2478/AMNS.2019.1.00017
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© 2019 R. A. Mundewadi et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International 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 Method [22] Method [23]
(k = 1, M = 10) (k = 0, M = 16) (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 Method [22] Method [23]
(k = 1, M = 10) (k = 1, M = 8) (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 Method [22] Method [23]
(k = 1, M = 10) (k = 0, M = 16) (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 Method [23]
(k = 1, M = 10) (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 0.72547625011012 0.725476258975992 0.72434536240934

Numerical results of example 2_

x Exact solution Present method Method [22] Method [23]
(k = 1, M = 10) (k = 1, M = 8) (m=16)
0.1 0.201317 0.200842 0.20128 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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