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

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Table 3 Comparative studies of the results of proposed methodologies for Problem I

X	Exact	Reported solution		Proposed approximated solution \(\hat{y}({\text{x}})\)
X	y(x)	Analytical result	ChNN (Mall and Chakraverty 2014)	SQP	PS	GA	PS-SQP	GA-SQP
0	1.000000	1.0000	1.0000	1.000000	1.000001	1.000037	1.000000	1.000000
0.1	0.998333	0.9983	0.9993	0.998349	0.998322	0.998451	0.998224	0.998339
0.2	0.993333	0.9933	0.9901	0.993376	0.993513	0.993512	0.993197	0.993341
0.3	0.985000	0.9850	0.9822	0.985053	0.985395	0.985151	0.984859	0.985006
0.4	0.973333	0.9733	0.9766	0.973380	0.973868	0.973433	0.973186	0.973341
0.5	0.958333	0.9583	0.9602	0.958369	0.958901	0.958420	0.958184	0.958341
0.6	0.940000	0.9400	0.9454	0.940032	0.940514	0.940122	0.939851	0.940007
0.7	0.918333	0.9183	0.9134	0.918369	0.918761	0.918511	0.918183	0.918340
0.8	0.893333	0.8933	0.8892	0.893375	0.893700	0.893543	0.893182	0.893340
0.9	0.865000	0.8650	0.8633	0.865044	0.865370	0.865192	0.864848	0.865007
1	0.833333	0.8333	0.8322	0.833373	0.833750	0.833480	0.833181	0.833340