An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

Kim, Hyunju; Lee, Junseo; Jang, Bongsoo

doi:10.1186/s13662-021-03355-3

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 12 The uniform two-mesh difference error and computed rates of convergence obtained by the proposed methods, Newton’s method, and Halley’s method. We set \(s_{0} = 0.2\), \(\mathit{Tol}= 10^{-10}\)

From: An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

\(\alpha _{2}\)		N = 64	N = 128	N = 256	N = 512	N = 1024	N = 2048
Finite difference method [22]
1.3	error	6.557E−002	4.240E−002	2.509E−002	1.387E−002	7.337E−003	3.782E−003
1.3	rate	0.629	0.757	0.855	0.919	0.956	0.977
1.5	error	2.271E−002	1.265E−002	6.749E−003	3.509E−003	1.795E−003	9.104E−004
1.5	rate	0.844	0.906	0.944	0.967	0.980	0.988
1.7	error	1.110E−002	5.968E−003	3.140E−003	1.628E−003	8.356E−004	4.262E−004
1.7	rate	0.895	0.927	0.948	0.962	0.971	0.978
1.9	error	5.650E−003	2.978E−003	1.557E−003	8.090E−004	4.184E−004	2.156E−004
1.9	rate	0.017	0.050	0.084	0.117	0.168	0.255
Newton’s method with linear explicit
1.3	error	5.35E−03	2.55E−03	1.14E−03	4.95E−04	2.09E−04	8.68E−05
1.3	rate	0.935	1.068	1.157	1.211	1.244	1.265
1.5	error	5.35E−03	2.55E−03	1.14E−03	4.95E−04	2.09E−04	8.68E−05
1.5	rate	0.935	1.068	1.157	1.211	1.244	1.265
1.7	error	4.19E−04	1.39E−04	4.51E−05	1.45E−05	4.62E−06	1.46E−06
1.7	rate	1.562	1.594	1.618	1.636	1.650	1.660
1.9	error	1.00E−04	2.85E−05	8.07E−06	2.27E−06	6.35E−07	1.77E−07
1.9	rate	1.798	1.810	1.821	1.830	1.837	1.844
Newton’s method
1.3	error	4.04E−03	6.03E−04	4.95E−05	8.12E−06	5.88E−06	2.17E−06
1.3	rate	2.238	2.745	3.607	2.606	0.465	1.438
1.5	error	2.41E−04	8.57E−05	2.50E−05	6.74E−06	1.74E−06	4.42E−07
1.5	rate	0.329	1.489	1.775	1.893	1.949	1.980
1.7	error	1.53E−04	4.05E−05	1.03E−05	2.58E−06	6.40E−07	1.58E−07
1.7	rate	1.779	1.919	1.975	1.998	2.010	2.017
1.9	error	9.08E−05	2.29E−05	5.74E−06	1.43E−06	3.58E−07	8.92E−08
1.9	rate	1.941	1.984	1.997	2.001	2.003	2.004
Halley’s method
1.3	error	1.00E−03	9.19E−05	6.67E−06	3.25E−07	1.60E−08	8.44E−09
1.3	rate	3.056	3.456	3.782	4.358	4.348	0.920
1.5	error	8.24E−06	2.48E−06	8.70E−07	3.02E−07	1.05E−07	3.64E−08
1.5	rate	2.111	1.732	1.511	1.524	1.526	1.525
1.7	error	1.26E−05	3.60E−06	1.07E−06	3.26E−07	9.97E−08	3.06E−08
1.7	rate	1.862	1.802	1.746	1.719	1.707	1.703
1.9	error	4.87E−06	1.06E−06	2.53E−07	6.42E−08	1.68E−08	4.44E−09
1.9	rate	2.365	2.200	2.063	1.980	1.937	1.916

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