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 4 Comparisons for solving Example 5.1 by the PBiCGSTAB/PGPBiCOR(2, 1) 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	\(\mathrm{PBiCGSTAB(S)}\)		\(\mathrm{PBiCGSTAB(C)}\)		\(\mathrm{PGPBiCOR(S)}\)		\(\mathrm{PGPBiCOR(C)}\)
β	M	N	\(\mathrm{CPU(s)}\)	Iter	\(\mathrm{CPU(s)}\)	Iter	\(\mathrm{CPU(s)}\)	Iter	\(\mathrm{CPU(s)}\)	Iter
1.3	2⁶	2²	0.02	14.0	0.02	14.0	0.02	13.0	0.02	12.3
	2⁷	2³	0.04	17.6	0.04	17.4	0.04	15.0	0.04	15.0
	2⁸	2⁴	0.09	21.1	0.09	20.8	0.09	17.3	0.09	17.0
	2⁹	2⁵	0.39	24.8	0.38	24.1	0.36	19.3	0.36	19.9
	2¹⁰	2⁶	1.27	28.0	1.25	27.7	1.07	21.0	1.18	23.3
	2¹¹	2⁷	8.48	31.0	15.92	60.5	6.64	22.6	11.73	41.1
1.5	2⁶	2²	0.01	12.0	0.01	11.8	0.01	11.0	0.02	10.0
	2⁷	2³	0.03	14.0	0.03	14.0	0.03	12.0	0.03	12.0
	2⁸	2⁴	0.07	15.6	0.07	15.9	0.07	13.0	0.08	14.3
	2⁹	2⁵	0.28	17.4	0.29	18.2	0.26	14.0	0.29	16.0
	2¹⁰	2⁶	0.88	19.1	0.90	19.7	0.80	15.0	0.89	17.3
	2¹¹	2⁷	5.59	20.2	5.75	21.1	4.75	15.8	5.37	18.0
1.8	2⁶	2²	0.01	10.0	0.01	10.0	0.01	8.0	0.01	9.0
	2⁷	2³	0.02	9.8	0.02	11.0	0.02	8.0	0.03	11.0
	2⁸	2⁴	0.05	10.2	0.07	13.9	0.05	9.0	0.07	12.8
	2⁹	2⁵	0.19	11.0	0.26	15.6	0.18	9.0	0.28	15.0
	2¹⁰	2⁶	0.56	11.0	0.85	17.8	0.55	9.3	0.94	17.9
	2¹¹	2⁷	3.28	11.0	5.93	21.5	3.24	10.0	6.19	20.8