Three Landweber iterative methods for solving the initial value problem of time-fractional diffusion-wave equation on spherically symmetric domain

Figures & data

Figure 1. The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.1. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 2. The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.1. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 3. The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.1. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 4. The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.2. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 5. The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.2. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 6. The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.2. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Table 1. The iteration steps of Example 5.1 for different regularization method.

ε		0.01	0.008	0.005
Landweber	$\alpha =1.2$	597	766	1362
	$\alpha =1.5$	272	386	687
	$\alpha =1.8$	262	378	679
Fractional Landweber	$\alpha =1.2$	75	93	138
	$\alpha =1.5$	41	52	85
	$\alpha =1.8$	42	52	83
Modified iterative	$\alpha =1.2$	137	170	261
	$\alpha =1.5$	67	91	155
	$\alpha =1.8$	67	89	153

Table 2. The CPU time of Example 5.1 for different regularization method.

α		1.2	1.5	1.8
Landweber	$ϵ=0.01$	11.9304	5.8485	5.7179
	$ϵ=0.008$	15.4281	7.4576	7.5216
	$ϵ=0.005$	27.4445	13.4199	13.4263
Fractional Landweber	$ϵ=0.01$	1.7742	0.6786	0.7475
	$ϵ=0.008$	1.5056	1.4258	1.4617
	$ϵ=0.005$	2.4426	1.4684	1.5987
Modified iterative	$ϵ=0.01$	2.8694	1.6987	1.7893
	$ϵ=0.008$	3.4975	1.5797	1.4896
	$ϵ=0.005$	5.4024	3.4720	3.4510

Table 3. Error behaviour of Example 5.1 for different α with $ϵ=0.01,0.005$.

α			1.1	1.3	1.5	1.7	1.9
Landweber	$ϵ=0.01$	Rela	0.7320	0.6001	0.5101	0.3923	0.2431
	$ϵ=0.01$	Abso	0.6078	0.4319	0.3121	0.1920	0.1311
	$ϵ=0.005$	Rela	0.6501	0.5017	0.3725	0.2519	0.0935
	$ϵ=0.005$	Abso	0.5910	0.4270	0.2110	0.1175	0.0734
Fractional Landweber	$ϵ=0.01$	Rela	0.4560	0.3817	0.3039	0.2377	0.2105
	$ϵ=0.01$	Abso	0.2113	0.1352	0.0987	0.0553	0.0098
	$ϵ=0.005$	Rela	0.2715	0.1219	0.0902	0.0835	0.0631
	$ϵ=0.005$	Abso	0.1091	0.0836	0.0430	0.0259	0.0047
Modified iterative	$ϵ=0.01$	Rela	0.5734	0.5180	0.3945	0.3030	0.2509
	$ϵ=0.01$	Abso	0.3978	0.2534	0.2575	0.1114	0.0273
	$ϵ=0.005$	Rela	0.3807	0.3741	0.2370	0.1415	0.0901
	$ϵ=0.005$	Abso	0.2933	0.1717	0.1682	0.0807	0.0109

Table 4. Error behaviour of Example 5.2 for different α with $ϵ=0.01,0.005$.

α			1.1	1.3	1.5	1.7	1.9
Landweber	$ϵ=0.01$	Rela	0.9845	0.7516	0.5322	0.3303	0.2730
	$ϵ=0.01$	Abso	0.7814	0.5831	0.4911	0.2510	0.1425
	$ϵ=0.005$	Rela	0.9013	0.6289	0.4516	0.2035	0.1013
	$ϵ=0.005$	Abso	0.6317	0.5521	0.3849	0.1228	0.0616
Fractional Landweber	$ϵ=0.01$	Rela	0.4532	0.3907	0.3015	0.2072	0.1723
	$ϵ=0.01$	Abso	0.3017	0.2125	0.1526	0.0821	0.0101
	$ϵ=0.005$	Rela	0.2853	0.2036	0.1489	0.0956	0.0745
	$ϵ=0.005$	Abso	0.1350	0.1049	0.0731	0.0232	0.0059
Modified iterative	$ϵ=0.01$	Rela	0.6711	0.5332	0.4136	0.2871	0.2020
	$ϵ=0.01$	Abso	0.3506	0.3029	0.2207	0.1019	0.0310
	$ϵ=0.005$	Rela	0.3421	0.2521	0.1987	0.1115	0.0109
	$ϵ=0.005$	Abso	0.1823	0.1352	0.0908	0.0673	0.0456

Figure 7. The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.3. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 8. The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.3. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 9. The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.3. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 10. The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.4. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 11. The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.4. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 12. The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.4. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 13. The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.5. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 14. The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.5. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 15. The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.5. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 16. The exact solution and regularization solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.6. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 17. The exact solution and regularization solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.6. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 18. The exact solution and regularization solution of modified iterative method by using the a posteriori parameter choice rule for Example 5.6. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.

Figure 19. The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.6. (a) $\alpha =1.2$, (b) $\alpha =1.5$, (c) $\alpha =1.8$.