Bio-inspired computational heuristics to study Lane–Emden systems arising in astrophysics model

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Table 9 Comparative studies of the results of proposed methodologies for Problem III

X	Exact	Reported solution		Proposed approximate solution \(\hat{y}({\text{x}})\)
X	y(x)	Analytical results	ChNN (Mall and Chakraverty 2014)	SQP	PS	GA	PS-SQP	GA-SQP
0	1.000	1.000	1.000	1.000	1.0013	1.0003	1.000	1.000
0.1	0.998	0.998	0.998	0.99835	1.001	0.9987	0.9983	0.99834
0.2	0.993	0.993	0.994	0.99342	0.9964	0.994	0.9935	0.99341
0.3	0.985	0.985	0.990	0.98535	0.9878	0.9861	0.9854	0.98534
0.4	0.973	0.974	0.971	0.97437	0.977	0.9751	0.9744	0.97436
0.5	0.958	0.961	0.968	0.96078	0.9635	0.9615	0.9608	0.96077
0.6	0.940	0.945	0.941	0.94492	0.9471	0.9455	0.9449	0.94492
0.7	0.918	0.927	0.930	0.92716	0.9289	0.9276	0.9271	0.92715
0.8	0.893	0.908	0.908	0.90785	0.9097	0.9082	0.9078	0.90785
0.9	0.865	0.887	0.883	0.88737	0.8888	0.8876	0.8873	0.88736
1	0.833	0.866	0.865	0.86603	0.8673	0.8662	0.866	0.86603