A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 3 Comparisons for solving Example 5.1 by the LU method and the PCGNR/PBiCRSTAB method with two circulant preconditioners, where \(\beta = 1.3, 1.5, 1.8\), \(\alpha = 0.5\), \(q = 10\), and \(T =0.75\)

β	M	N	LU	\(\mathrm{PCGNR(S)}\)		\(\mathrm{PCGNR(C)}\)		\(\mathrm{PBiCRSTAB(S)}\)		\(\mathrm{PBiCRSTAB(C)}\)
β	M	N	\(\mathrm{CPU(s)}\)	\(\mathrm{CPU(s)}\)	Iter	\(\mathrm{CPU(s)}\)	Iter	\(\mathrm{CPU(s)}\)	Iter	\(\mathrm{CPU(s)}\)	Iter
1.3	2⁶	2²	0.01	0.02	18.0	0.01	16.8	0.02	14.0	0.02	14.0
	2⁷	2³	0.01	0.04	20.0	0.04	20.4	0.04	18.6	0.04	18.9
	2⁸	2⁴	0.13	0.09	22.9	0.09	24.8	0.11	22.3	0.10	21.7
	2⁹	2⁵	1.36	0.38	25.4	0.44	30.2	0.45	26.3	0.42	25.3
	2¹⁰	2⁶	16.72	1.20	27.5	1.72	40.9	1.47	30.5	1.39	29.0
	2¹¹	2⁷	213.81	7.99	28.8	16.74	63.1	9.60	33.5	16.23	58.7
1.5	2⁶	2²	0.01	0.03	18.0	0.03	14.0	0.01	12.0	0.01	11.8
	2⁷	2³	0.01	0.03	19.3	0.03	17.0	0.03	14.0	0.03	14.0
	2⁸	2⁴	0.14	0.08	21.8	0.08	19.2	0.08	16.4	0.08	16.0
	2⁹	2⁵	1.40	0.34	23.3	0.32	21.6	0.32	18.4	0.32	18.7
	2¹⁰	2⁶	16.86	1.05	24.1	1.06	24.2	1.01	20.6	1.00	20.3
	2¹¹	2⁷	217.19	6.69	24.2	7.20	25.9	6.16	21.3	6.34	22.0
1.8	2⁶	2²	0.01	0.04	18.5	0.03	16.0	0.01	8.0	0.01	9.0
	2⁷	2³	0.01	0.04	22.0	0.03	18.8	0.02	9.0	0.03	10.6
	2⁸	2⁴	0.14	0.09	24.0	0.09	22.9	0.06	10.0	0.07	12.8
	2⁹	2⁵	1.43	0.36	26.1	0.41	27.8	0.21	11.0	0.26	14.9
	2¹⁰	2⁶	17.16	1.21	27.3	1.47	34.4	0.61	11.0	0.86	17.0
	2¹¹	2⁷	221.05	8.22	29.7	12.02	44.7	3.66	11.6	5.86	20.0