Solutions of differential equations using linearly independent Hosoya polynomials of trees
, , and | Oct 31, 2023
About this article
Article Category: Original Study
Published Online: Oct 31, 2023
Page range: 13 - 22
Received: Jun 06, 2023
Accepted: Jul 22, 2023
DOI: https://doi.org/10.2478/ijmce-2024-0002
Keywords
© 2024 Kumbinarasaiah Srinivasa et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Application 2 AE derived by HPM versus Exact solution.
| x | Exact Sol. | AE at n=7 | AE at n=8 | AE at n=9 | AE at n=10 |
|---|---|---|---|---|---|
| 0.1 | 0.990049833749168 | 2.82E-06 | 1.66E-08 | 4.74E-08 | 4.59E-09 |
| 0.2 | 0.960789439152323 | 1.16E-05 | 2.51E-07 | 1.12E-07 | 1.51E-09 |
| 0.3 | 0.913931185271228 | 1.93E-05 | 6.86E-08 | 1.48E-07 | 1.25E-08 |
| 0.4 | 0.852143788966211 | 9.73E-07 | 7.81E-07 | 1.45E-07 | 1.29E-08 |
| 0.5 | 0.778800783071405 | 2.56E-05 | 4.60E-07 | 1.35E-07 | 1.68E-08 |
| 0.6 | 0.697676326071031 | 2.74E-05 | 1.06E-06 | 1.91E-07 | 2.10E-08 |
| 0.7 | 0.612626394184416 | 4.21E-06 | 1.48E-06 | 8.39E-08 | 1.35E-08 |
| 0.8 | 0.527292424043049 | 3.43E-05 | 6.83E-07 | 2.40E-07 | 2.71E-08 |
| 0.9 | 0.444858066222941 | 1.58E-05 | 1.57E-06 | 2.89E-06 | 1.21E-08 |
| 1.0 | 0.367879441171442 | 8.55E-06 | 4.41E-07 | 7.74E-06 | 4.34E-09 |
AE comparison of the solution y1(x) with different values of n.
| x | Exact Sol. | AE at n=7 | AE at n=8 | AE at n=9 | AE at n=10 |
|---|---|---|---|---|---|
| 0.1 | 1.005004168055804 | 1.42E-08 | 1.39E-09 | 8.22E-11 | 3.56E-13 |
| 0.2 | 1.020066755619076 | 1.06E-08 | 1.89E-10 | 1.91E10 | 7.76E-13 |
| 0.3 | 1.045338514128861 | 1.79E-08 | 1.25E-09 | 2.71E-10 | 1.02E-12 |
| 0.4 | 1.081072371838455 | 2.42E-08 | 7.64E-10 | 3.43E-10 | 1.81E-13 |
| 0.5 | 1.127625965206381 | 1.05E-09 | 2.15E-09 | 4.37E-10 | 1.31E-12 |
| 0.6 | 1.185465218242268 | 2.31E-08 | 4.34E-10 | 5.27E-10 | 3.32E-13 |
| 0.7 | 1.255169005630943 | 7.97E-09 | 1.45E-09 | 6.02E-10 | 9.38E-13 |
| 0.8 | 1.337434946304845 | 2.82E-08 | 1.15E-10 | 6.52E-10 | 1.03E-12 |
| 0.9 | 1.433086385448775 | 2.38E-08 | 1.74E-09 | 4.86E-10 | 4.34E-13 |
Judgement of the AE of HPM and other techniques in [29].
| x | Exact Sol. | AE by method [ |
AE at n=10 |
|---|---|---|---|
| 0.1 | 0.990049833749168 | 4.89E-06 | 4.59E-09 |
| 0.2 | 0.960789439152323 | 6.84E-06 | 1.51E-09 |
| 0.3 | 0.913931185271228 | 8.03E-07 | 1.25E-08 |
| 0.4 | 0.852143788966211 | 8.38E-06 | 1.29E-08 |
| 0.5 | 0.778800783071405 | 1.28E-05 | 1.68E-08 |
| 0.6 | 0.697676326071031 | 5.32E-05 | 2.10E-08 |
| 0.7 | 0.612626394184416 | 2.06E-04 | 1.35E-08 |
| 0.8 | 0.527292424043049 | 5.93E-04 | 2.71E-08 |
| 0.9 | 0.444858066222941 | 1.41E-03 | 1.21E-08 |
| 1.0 | 0.367879441171442 | 3.07E-03 | 4.34E-09 |
Application 3, AE derived by HPM versus Exact solution.
| x | Exact Sol. | AE at n=7 | AE at n=8 | AE at n=9 | AE at n=10 |
|---|---|---|---|---|---|
| 0.1 | -0.019900661706336 | 1.87E-05 | 1.87E-06 | 6.40E-06 | 1.75E-07 |
| 0.2 | -0.078441426306563 | 1.63E-05 | 1.63E-05 | 3.14E-07 | 2.71E-07 |
| 0.3 | -0.172355392482105 | 5.57E-05 | 5.57E-05 | 4.67E-07 | 2.95E-07 |
| 0.4 | -0.296840010236547 | 5.03E-05 | 5.03E-06 | 8.74E-07 | 4.93E-07 |
| 0.5 | -0.446287102628420 | 1.90E-05 | 1.90E-05 | 2.09E-07 | 3.04E-07 |
| 0.6 | -0.614969399495921 | 3.88E-06 | 3.88E-07 | 9.45E-07 | 1.10E-08 |
| 0.7 | -0.797552239914736 | 1.05E-05 | 1.05E-05 | 7.97E-07 | 1.06E-07 |
| 0.8 | -0.989392483672214 | 1.15E-05 | 1.15E-06 | 7.48E-07 | 8.61E-08 |
| 0.9 | -1.186653690555469 | 8.85E-06 | 8.85E-06 | 1.62E-06 | 7.51E-09 |
| 1.0 | -1.386294361119891 | 5.00E-06 | 5.00E-06 | 5.18E-05 | 1.54E-08 |