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A new approach to the Benjamin-Bona-Mahony equation via ultraspherical wavelets collocation method

Mallanagoud Mulimani
Mallanagoud Mulimani
 e 
Kumbinarasaiah Srinivasa
Kumbinarasaiah Srinivasa
   | 10 gen 2024
International Journal of Mathematics and Computer in Engineering's Cover Image
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Article Category: Original Study
Pubblicato online: 10 gen 2024
Pagine: 179 - 192
Ricevuto: 27 lug 2023
Accettato: 06 set 2023
DOI: https://doi.org/10.2478/ijmce-2024-0014
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© 2024 Mallanagoud Mulimani et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Comparison of the projected scheme solution with other solutions for Problem 1 at t=19 t = {1 \over 9} .
Comparison of the projected scheme solution with other solutions for Problem 1 at t=19 t = {1 \over 9} .

Fig. 2

Graphical demonstration of the solution of present and other methods for Problem 1.
Graphical demonstration of the solution of present and other methods for Problem 1.

Fig. 3

Comparison of the projected scheme solution with other solutions for Problem 2 at t=16 t = {1 \over 6} .
Comparison of the projected scheme solution with other solutions for Problem 2 at t=16 t = {1 \over 6} .

Fig. 4

Graphical demonstration of the solution of present and other methods for Problem 2.
Graphical demonstration of the solution of present and other methods for Problem 2.

Fig. 5

Comparison of the projected scheme solution with other solutions for Problem 3 at t=16 t = {1 \over 6} .
Comparison of the projected scheme solution with other solutions for Problem 3 at t=16 t = {1 \over 6} .

Fig. 6

Graphical demonstration of the solution of present and other methods for Problem 3.
Graphical demonstration of the solution of present and other methods for Problem 3.

Comparison of the projected scheme with other techniques for N = M = 8 of Problem 2.

x = (/16) FDM [14] HWM [14] UWM Exact
1 1.15741935 1.20750077 1.20586582 1.20623024
3 1.35750043 1.36926498 1.3658529 1.36683794
5 1.57881432 1.55109039 1.5473850 1.54883029
7 1.82434828 1.75639170 1.7533308 1.75505465
9 2.09744632 1.98892088 1.9869409 1.98873746
11 2.40185758 2.25280954 2.2518987 2.25353478
13 2.74179096 2.55257492 2.5523783 2.55358945
15 3.12197659 2.89308458 2.8931091 2.89359594

Comparison of the projected scheme with other techniques for N = M = 8 of Problem 3.

x = (/16) FDM [14] UWM Exact
1 0.05887280 0.05658619 0.05555555
3 0.17667011 0.16966762 0.16666666
5 0.29462366 0.28247422 0.27777777
7 0.41284152 0.39481645 0.38888888
9 0.53143772 0.50649507 0.50000000
11 0.65053500 0.61729700 0.61111111
13 0.77026763 0.72699079 0.72222222
15 0.89078469 0.83532175 0.83333333

Comparison of the projected scheme with other techniques for N = M = 8 of Problem 1.

x = (/16) FDM [14] HWM [14] UWM Exact
1 0.82942254 0.82957675 0.82887868 0.82902911
3 0.73277957 0.73244494 0.73124041 0.73161562
5 0.64756878 0.64621771 0.64514042 0.64564852
7 0.57245382 0.56995069 0.56922344 0.56978282
9 0.50625604 0.50267060 0.50229344 0.50283157
11 0.44793598 0.44342497 0.44329494 0.44374731
13 0.39657711 0.39130681 0.39129648 0.39160562
15 0.35137142 0.34546659 0.34547605 0.34559075

AE of the present scheme for Problem 3.

x AE at M = 6 AE at M = 7 AE at M = 8 AE at M = 9
0.1 2.1089 × 10−3 1.8448 × 10−3 1.6396 × 10−3 1.5825 × 10−3
0.2 4.0983 × 10−3 3.5849 × 10−3 3.1859 × 10−3 2.9895 × 10−3
0.3 5.8473 ×10−3 5.1141 ×10−3 4.5446 ×10−3 3.1664 ×10−3
0.4 7.2314 × 10−3 6.3236 × 10−3 5.6186 × 10−3 4.8969 × 10−3
0.5 8.1208 × 10−3 7.0998 ×10−3 6.3072 × 10−3 5.9892 ×10−3
0.6 8.3788 ×10−3 7.3234 ×10−3 6.5045 ×10−3 6.3058 ×10−3
0.7 7.8595 ×10−3 6.8674 × 10−3 6.0980 × 10−3 5.2771 ×10−3
0.8 6.4062 ×10−3 5.5955 × 10−3 4.9672 × 10−3 3.7137 ×10−3
0.9 3.8484 ×10−3 3.3601 × 10−3 2.9818 × 10−3 1.0753 ×10−3

AE of the present scheme for Problem 1.

x AE at M = 6 AE at M = 7 AE at M = 8 AE at M = 9
0.1 3.9376 × 10−4 2.9425 × 10−4 2.2818 × 10−4 1.8209 × 10−4
0.2 6.7769 × 10−4 5.0624 × 10−4 3.9246 × 10−4 3.1312 × 10−4
0.3 8.6162 ×10−4 6.4339 ×10−4 4.9864 ×10−4 3.9775 ×10−4
0.4 9.5446 × 10−4 7.1244 × 10−4 5.5200 × 10−4 4.4021 × 10−4
0.5 9.6427 × 10−4 7.1949 ×10−4 5.5729 × 10−4 4.4433 ×10−4
0.6 8.9835 ×10−4 6.7003 ×10−4 5.1883 ×10−4 4.1357 ×10−4
0.7 7.6329 ×10−4 5.6907 × 10−4 4.4052 × 10−4 3.5106 ×10−4
0.8 5.6505 ×10−4 4.2109 × 10−4 3.2587 × 10−4 2.5963 ×10−4
0.9 3.0900 ×10−4 2.3018 × 10−4 1.7807 × 10−4 1.4184 ×10−4

AE of the present scheme for Problem 2.

x AE at M = 6 AE at M = 7 AE at M = 8 AE at M = 9
0.1 1.0354 × 10−3 7.4856 × 10−4 5.6626 × 10−4 4.4325 × 10−4
0.2 1.8990 × 10−3 1.3729 × 10−3 1.0386 × 10−3 8.1295 × 10−4
0.3 2.5727 ×10−3 1.8600 ×10−3 1.4070 ×10−3 1.1014 ×10−3
0.4 3.0365 × 10−3 2.1953 × 10−3 1.6607 × 10−3 1.2999 × 10−3
0.5 3.2684 × 10−3 2.3630 ×10−3 1.7875 × 10−3 1.3992 ×10−3
0.6 3.2441 ×10−3 2.3454 ×10−3 1.7742 ×10−3 1.3888 ×10−3
0.7 2.9364 ×10−3 2.1230 × 10−3 1.6060 × 10−3 1.2571 ×10−3
0.8 2.3157 ×10−3 1.6742 × 10−3 1.2665 × 10−3 9.9137 ×10−4
0.9 1.3490 ×10−3 9.7533 × 10−4 7.3780 × 10−4 5.7753 ×10−4
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