A computational method for solving a class of singular boundary value problems arising in science and engineering

Figures & data

Table 1 Computed values and comparison to the exact solutions for Example 4.1.

x	Exact	BOM $n=2$	Absolute error
0.1	0.9950	0.9954	0.0004
0.2	0.9801	0.9816	0.0015
0.3	0.9553	0.9586	0.0033
0.4	0.9211	0.9264	0.0053
0.5	0.8776	0.8851	0.0075
0.6	0.8253	0.8345	0.0092
0.7	0.7648	0.7747	0.0099
0.8	0.6967	0.7058	0.0091
0.9	0.6216	0.6276	0.0060
1	0.5403	0.5403	0

Fig. 1 Comparison between exact and BOM solutions with $n=2$ for Example 4.1.

Table 2 Errors of the present method compared with B-Spline results for the Example 4.2.

x	Exact	BOM $n=2$ (E1)	B-Spline (E2)	Error by E1	Error by E2
0	1.0000	1.0000	0.9990	0	0.001
0.1	0.9900	0.9900	0.9899	0	1E-04
0.2	0.9600	0.9600	0.9599	0	1E-04
0.3	0.9100	0.9100	0.9099	0	1E-04
0.4	0.8400	0.8400	0.8399	0	1E-04
0.5	0.7500	0.7500	0.7499	0	1E-04
0.6	0.6400	0.6400	0.6399	0	1E-04
0.7	0.5100	0.5100	0.5099	0	1E-04
0.8	0.3600	0.3600	0.3599	0	1E-04
0.9	0.1900	0.1900	0.1900	0	0
1	0	0	0	0	0

Table 3 Computed values and comparison to the exact solutions for the Example 4.3.

x	Exact	BOM $n=3$	Absolute error
0.1	1.3036	1.3128	0.0092
0.2	1.2939	1.3015	0.0076
0.3	1.2777	1.2833	0.0056
0.4	1.2552	1.2585	0.0033
0.5	1.2265	1.2277	0.0012
0.6	1.1919	1.1914	0.0005
0.7	1.1516	1.1500	0.0016
0.8	1.1060	1.1040	0.0020
0.9	1.0554	1.0538	0.0016
1	1	1	0

Table 4 Computed values and comparison to the exact solutions for the Example 4.4.

x	Exact	BOM $n=3$	Absolute Error
0.1	3.2756	3.3143	0.0387
0.2	3.3313	3.605	0.0292
0.3	3.4256	3.4438	0.0182
0.4	3.5608	3.5694	0.0086
0.5	3.7402	3.7425	0.0023
0.6	3.9682	3.9682	0
0.7	4.2503	4.2516	0.0013
0.8	4.5937	4.5979	0.0042
0.9	5.0067	5.0123	0.0056
1	5.5000	5.5000	0

Fig. 2 Comparison between exact and BOM solutions with $n=3$ for Example 4.4.

Fig. 3 Comparison between the BOM, HPM, and exact solutions of Example 4.5.

Table 5 Computed values and comparison to the exact solutions for the Example 4.6.

x	Exact	BOM $n=2$	Absolute error
0.1	–0.2646	–0.2636	0.0010
0.2	–0.257	–0.2556	0.0014
0.3	–0.2444	–0.2423	0.0021
0.4	–0.2267	–0.2237	0.0030
0.5	–0.2036	–0.1997	0.0039
0.6	–0.175	–0.1704	0.0046
0.7	–0.1407	–0.1358	0.0049
0.8	–0.1003	–0.0959	0.0044
0.9	–0.0536	–0.0506	0.0030
1	0	0	0

Fig. 4 Comparison between the results obtained from BOM and exact solution of Example 6.