Solving Lane–Emden equations with boundary conditions of various types using high-order compact finite differences

Figures & data

Figure 1. Comparison of the exact and approximate solution plots for Example 6.1, showing a good agreement between the two.

Figure 2. Error plots for the approximation of Example 6.1 for varying values of N.

Figure 3. Convergence plots for the approximation of Example 6.1 for varying values of N, showing convergence after 6 iterations.

Table 1. Maximum absolute error ($L_{\mathrm{\infty }}$) for Example 6.1.

Present method (CFDM)			Singh et al. [29]		Singh et al. [33]
N	Error norm ($L_{\mathrm{\infty }}$)	ROC	N	Error norm ($L_{\mathrm{\infty }}$)	N	Error norm ($L_{\mathrm{\infty }}$)
8	5.627e−05		4	4.69144e−02	8	7.80e−04
16	1.45823e−07	8.16582	6	6.92128e−03	16	1.94e−04
32	1.37249e−09	6.42723	8	8.25260e−04	32	4.84e−05
64	5.62594e−12	7.75156	10	1.08089e−04	64	1.21e−06
					128	3.02e−06
					256	7.55e−07
					512	1.89e−07
					1024	4.72e−08

Table 2. Maximum absolute error ($L_{\mathrm{\infty }}$) for Example 6.2.

Present method (CFDM)				Singh et al. [33]
N	h	Error norm ($L_{\mathrm{\infty }}$)	Time(s)	N	Error norm ($L_{\mathrm{\infty }}$)
8	0.125	5.17898e−05	0.0081017	8	3.96e−04
16	0.0625	3.97123e−08	0.0103628	16	5.71e−05
32	0.03125	5.65056e−10	0.0245085	32	1.44e−06
64	0.015625	2.79043e−12	0.0813154	64	3.61e−06
			0.279279	128	9.04e−07
				256	2.26e−07
				512	5.65e−08
				1024	1.41e−08

Figure 4. Comparison of the exact and approximate solution plots for Example 6.2, showing a good agreement between the two.

Figure 5. Error plots for the approximation of Example 6.2 for varying values of N.

Figure 6. Convergence plots for the approximation of Example 6.2 for varying values of N.

Figure 7. Comparison of the exact and approximate solution plots for Example 6.3 showing a good agreement between the two.

Table 3. Maximum absolute error ($L_{\mathrm{\infty }}$) for Example 6.3.

Present method (CFDM)				Singh et al. [33]
N	h	Error norm ($L_{\mathrm{\infty }}$)	ROC	N	Error norm ($L_{\mathrm{\infty }}$)
10	0.125	2.8657e−06		8	1.95e−03
20	0.05	2.16368e−08	6.53918	16	4.89e−04
40	0.025	1.30177e−10	7.11041	32	1.22e−04
80	0.0125	4.01679e−13	8.1897	64	3.06e−05
				128	7.65e−06
				256	1.91e−06
				512	4.78e−07
				1024	1.20e−07

Figure 8. Error plots for the approximation of Example 6.3 for varying values of N.

Figure 9. Convergence plots for the approximation of Example 6.3 for varying values of N showing convergence after 5 iterations.

Figure 10. Comparison of the exact and approximate solution plots for Example 6.4, showing a good agreement between the two.

Figure 11. Error plots for the approximation of Example 6.4 for varying values of N.

Figure 12. Convergence plots for the approximation of Example 6.4 for varying values of N, showing convergence after 5 iterations.

Table 4. Maximum absolute error ($L_{\mathrm{\infty }}$) and $\text{ROC}$ for Example 6.4.

Present method (CFDM)			Gümgüm [60]	Kanth et al. [17]	Chawla et al. [21]
N	Error norm ($L_{\mathrm{\infty }}$)	ROC	Error norm ($L_{\mathrm{\infty }}$)	Error norm ($L_{\mathrm{\infty }}$)	N	Error norm ($L_{\mathrm{\infty }}$)
10	2.79012e−07		3.24e−06	3.92e−04	16	3.64e−04
20	6.79228e−09	5.91062			32	2.49e−05
30	3.39322e−10	7.01049			64	1.6e−06
40	3.87994e−11	7.26498			128	1.01e−07
50	7.28173e−12	7.28756

Table 5. Maximum absolute error ($L_{\mathrm{\infty }}$) and $\text{ROC}$ for Example 6.5.

Present method (CFDM)			Chawla et al. [21]		Alam et al. [27]
N	Error norm ($L_{\mathrm{\infty }}$)	ROC	N	Error norm ($L_{\mathrm{\infty }}$)	N	Error norm ($L_{\mathrm{\infty }}$)
10	1.62155e−07		16	2.52e−03	16	1.34663e−08
20	8.15177e−10	7.08351	32	1.83e−04	32	7.86881e−10
30	3.50054e−11	7.44437	64	1.28e−05	64	4.7697e−11
40	3.81284e−12	7.48358	128	8.33e−07	128	2.94776e−12
50	6.66689e−13	7.6396			256	1.67977e−13

Figure 13. Comparison of the exact and approximate solution plots for Example 6.5, showing good agreement between the two.

Figure 14. Error plots for the approximation of Example 6.5 for varying values of N.

Figure 15. Convergence plots for the approximation of Example 6.5 for varying values of N, showing convergence after 6 iterations.

Figure 16. Approximate solution plots for Example 6.6 for N = 20.

Table 6. Maximum absolute error ($L_{\mathrm{\infty }}$) for Example 6.6.

Present method (CFDM)
x	N = 10	N = 20	Alam et al. [27]	Singh et al. [61]	Singh et al. [33]
0.1	0.268756901799	0.268756900638	0.2697159	0.26349926	0.268755437
0.2	0.264932818575	0.264932817546	0.2687104	0.25965525	0.264931354
0.3	.2585397904233	0.258539789390	0.2649118	0.25322880	0.258538326
0.4	0.249548181308	0.249548180263	0.2585270	0.24418997	0.249546718
0.5	0.237915888665	0.237915887598	0.2495394	0.23249622	0.237914425
0.6	0.223587708284	0.223587707191	0.2440578	0.21809176	0.223586244
0.7	0.206494484164	0.206494483034	0.2310872	0.20090669	0.206493019
0.8	0.186552015349	0.186552014177	0.2153874	0.18085589	0.186550548
0.9	0.163659682834	0.163659681592	0.1968819	0.15783764	0.163658211
1.0	0.137698747815	0.137698746625	0.1754788	0.13173190	0.137697271

Figure 17. Approximate solution plots for Example 6.7 for N = 20.

Table 7. Maximum absolute error ($L_{\mathrm{\infty }}$) for Example 6.7.

Present method (CFDM)
x	N = 10	N = 20	Gümgüm [60]	Singh [33]
0.1	0.829706092436	0.829706092434	0.8297063594	0.829706118
0.2	0.833374733596	0.833374733591	0.8333750006	0.833374759
0.3	0.839489913959	0.839489913954	0.8394901811	0.839489939
0.4	0.848052785001	0.848052784996	0.8480530523	0.848052810
0.5	0.859064927173	0.859064927169	0.8590651943	0.859064951
0.6	0.872528319962	0.872528319958	0.8725285872	0.872528343
0.7	0.888445305626	0.888445305623	0.8884455738	0.888445328
0.8	0.906818548069	0.906818548067	0.9068188164	0.906818569
0.9	0.927650988367	0.927650988366	0.9276512446	0.927651008
1.0	0.950945798498	0.950945798497	0.9509459977	0.950945816

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