From: Bio-inspired computational heuristics to study Lane–Emden systems arising in astrophysics model
X | Exact | Reported solution | Proposed approximate solution \(\hat{y}({\text{x}})\) | |||||
---|---|---|---|---|---|---|---|---|
y(x) | Analytical results | ChNN (Mall and Chakraverty 2014) | SQP | PS | GA | PS-SQP | GA-SQP | |
0 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 0.99998 | 0.99999 | 1.00000 | 0.99999 |
0.1 | 0.99800 | 0.99830 | 1.00180 | 0.99800 | 0.99805 | 0.99828 | 0.99833 | 0.99833 |
0.2 | 0.99300 | 0.99330 | 0.99050 | 0.99300 | 0.99278 | 0.99346 | 0.99335 | 0.99336 |
0.3 | 0.98500 | 0.98510 | 0.98390 | 0.98500 | 0.98470 | 0.98542 | 0.98513 | 0.98513 |
0.4 | 0.97300 | 0.97350 | 0.97340 | 0.97400 | 0.97343 | 0.97418 | 0.97375 | 0.97375 |
0.5 | 0.95800 | 0.95890 | 0.95980 | 0.95900 | 0.95891 | 0.95983 | 0.95935 | 0.95935 |
0.6 | 0.94100 | 0.94110 | 0.94170 | 0.94200 | 0.94169 | 0.94255 | 0.94209 | 0.94209 |
0.7 | 0.91800 | 0.92030 | 0.92100 | 0.92200 | 0.92187 | 0.92256 | 0.92217 | 0.92217 |
0.8 | 0.89300 | 0.89670 | 0.89250 | 0.91200 | 0.89943 | 0.90019 | 0.89979 | 0.89979 |
0.9 | 0.86500 | 0.87040 | 0.87000 | 0.87500 | 0.87489 | 0.87544 | 0.87521 | 0.87521 |
1 | 0.83300 | 0.8415 | 0.8431 | 0.84900 | 0.84832 | 0.84868 | 0.84865 | 0.84865 |