Design Feed Forward Neural Network to Solve Singular Boundary Value Problems

Table 1

Analytic and neural solutions of Example 1.

Input	Analytic solution	Out of suggested FFNN 𝑦𝑡(𝑥) for different training algorithms
𝑥	𝑦𝑎(𝑥)	Trainlm	Trainbfg	Trainbr
0.0	1	1	1.00026931058869	1.00000028044679
0.1	0.995004165278026	0.994992000703617	0.995004165151447	0.995003016942414
0.2	0.980066577841242	0.980066577841242	0.980038331595444	0.980068202697384
0.3	0.955336489125606	0.955336489125606	0.955326539544390	0.955327719921018
0.4	0.921060994002885	0.921060994002885	0.921060993873107	0.921057767471849
0.5	0.877582561890373	0.877582561890373	0.877583333411869	0.877587507015204
0.6	0.825335614909678	0.825335614909678	0.825335614867241	0.825332390929038
0.7	0.764842187284489	0.764838404395133	0.764842187227892	0.764831348535762
0.8	0.696706709347165	0.696656420761032	0.696706709263878	0.696708274506624
0.9	0.621609968270664	0.621454965504409	0.621609968278774	0.621608898908218
1.0	0.540302305868140	0.540302305868140	0.540302305877365	0.540302558954826