Recovering space-dependent source for a time-space fractional diffusion wave equation by fractional Landweber method

Figures & data

Figure 1. Exact solution and numerical solution $f(x)$ for Example 6.1 with different α, β and ε. (a) Numerical solution for $\alpha =1.3$, $\beta =1.2$. (b) Numerical solution for $\alpha =1.7$, $\beta =1.2$. (c) Numerical solution for $\alpha =1.3$, $\beta =1.6$. (d) Numerical solution for $\alpha =1.7$, $\beta =1.6$.

Figure 2. Exact solution and numerical solution $f(x)$ of Example 6.1 for various value of γ with fixed $\mu =6$, $\epsilon =0.01$. (a) Numerical solution for $\alpha =1.3$, $\beta =1.2$. (b) Numerical solution for $\alpha =1.7$, $\beta =1.2$. (c) Numerical solution for $\alpha =1.3$, $\beta =1.6$. (d) Numerical solution for $\alpha =1.7$, $\beta =1.6$.

Figure 3. Exact solution and numerical solution $f(x)$ for Example 6.2 with different $\alpha ,\beta$ and ε. (a) Numerical solution for $\alpha =1.3$, $\beta =1.2$. (b) Numerical solution for $\alpha =1.7$, $\beta =1.2$. (c) Numerical solution for $\alpha =1.3$, $\beta =1.6$. (d) Numerical solution for $\alpha =1.7$, $\beta =1.6$.

Figure 4. Exact solution and numerical solution $f(x)$ for Example 6.3 with different α and β. (a) Numerical solution for $\alpha =1.3$, $\beta =1.2$. (b) Numerical solution for $\alpha =1.7$, $\beta =1.2$. (c) Numerical solution for $\alpha =1.3$, $\beta =1.6$. (d) Numerical solution for $\alpha =1.7$, $\beta =1.6$. (e) Exact solution.

Figure 5. The absolute error for Example 6.3 with different α and β. (a) $\alpha =1.3$, $\beta =1.2$. (b) $\alpha =1.7$, $\beta =1.2$. (c) $\alpha =1.3$, $\beta =1.6$. (d) $\alpha =1.7$, $\beta =1.6$.

Figure 6. Exact solution and numerical solution $f(x)$ for Example 6.4 with different α and β. (a) Numerical solution for $\alpha =1.3$, $\beta =1.2$. (b) Numerical solution for $\alpha =1.7$, $\beta =1.2$. (c) Numerical solution for $\alpha =1.3$, $\beta =1.6$. (d) Numerical solution for $\alpha =1.7$, $\beta =1.6$. (e) Exact solution.

Figure 7. The absolute error for Example 6.4 with different α and β. (a) $\alpha =1.3$, $\beta =1.2$. (b) $\alpha =1.7$, $\beta =1.2$. (c) $\alpha =1.3$, $\beta =1.6$. (d) $\alpha =1.7$, $\beta =1.6$.

Table 1. The relative errors E and iteration steps of Example 6.1 for various value of μ with fixed $\alpha =1.1$, $\beta =1.6$, $\gamma =0.6$, $\epsilon =0.01$.

$\epsilon \setminus \mu$	0.01	0.1	1	10	20	70	75
0.01	0.039(249)	0.039(104)	0.037(44)	0.033(18)	0.031(14)	0.032(9)	0.231(1)

Table 2. The relative errors E and iteration steps of Example 6.1 for various value of β and γ with fixed $\alpha =1.1$, $\mu =6$, $\epsilon =0.01$.

β	1.1	1.3	1.5	1.7	1.9
$\gamma =0.6$	0.051(7)	0.060(11)	0.083(17)	0.061(27)	0.064(42)
$\gamma =1$	0.0105(15)	0.093(27)	0.097(44)	0.098(79)	0.10(157)

Table 3. The relative errors E and iteration steps of Example 6.2 for various value of β and γ with fixed $\alpha =1.3$, $\mu =6$, $\epsilon =0.01$.

β	1.1	1.3	1.5	1.7	1.9
$\gamma =0.6$	0.094(6)	0.052(10)	0.037(15)	0.045(22)	0.032(34)
$\gamma =1$	0.147(11)	0.0768(21)	0.0798(36)	0.0935(62)	0.1184(107)

Table 4. The relative errors E and iteration steps of Example 6.3 for various value of β and γ with fixed $\alpha =1.7$, $\mu =6$, $\epsilon =0.01$.

β	1.1	1.3	1.5	1.7	1.9
$\gamma =0.6$	0.086(6)	0.019(9)	0.045(12)	0.066(18)	0.083(27)
$\gamma =1$	0.083(11)	0.041(19)	0.056(26)	0.196(47)	0.287(76)

Table 5. The relative errors E and iteration steps of Example 6.4 for various value of β and γ with fixed $\alpha =1.5$, $\mu =6$, $\epsilon =0.01$.

β	1.1	1.3	1.5	1.7	1.9
$\gamma =0.6$	0.086(7)	0.051(12)	0.054(18)	0.048(29)	0.048(45)
$\gamma =1$	0.088(14)	0.065(27)	0.086(49)	0.113(87)	0.10(157)

Table 6. The relative errors E and iteration steps of Example 6.1 for various value of α and γ with fixed $\beta =1.9$, $\mu =6$, $\epsilon =0.01$.

α	1.1	1.3	1.5	1.7	1.9
$\gamma =0.6$	0.064(43)	0.063(43)	0.045(45)	0.0448(45)	0.064(43)
$\gamma =1$	0.96(142)	0.082(143)	0.0804(143)	0.0684(144)	0.1013(172)

Table 7. The relative errors E and iteration steps of Example 6.2 for various value of α and γ with fixed $\beta =1.7$, $\mu =6$, $\epsilon =0.01$.

α	1.1	1.3	1.5	1.7
$\gamma =0.6$	0.0534(22)	0.0437(22)	0.0328(22)	0.0504(23)
$\gamma =1$	0.1341(62)	0.0935(62)	0.0755(64)	0.0109(64)

Table 8. The relative errors E and iteration steps of Example 6.3 for various value of α and γ with fixed $\beta =1.5$, $\mu =6$, $\epsilon =0.01$.

α	1.1	1.3	1.5	1.7	1.9
$\gamma =0.6$	0.024(12)	0.018(12)	0.0177(12)	0.045(12)	0.031(13)
$\gamma =1$	0.042(28)	0.093(29)	0.096(29)	0.084(28)	0.115(29)

Table 9. The relative errors E and iteration steps of Example 6.4 for various value of α and γ with fixed $\beta =1.3$, $\mu =6$, $\epsilon =0.01$.

α	1.1	1.3	1.5	1.7	1.9
$\gamma =0.6$	0.050(12)	0.053(12)	0.055(12)	0.053(11)	0.060(14)
$\gamma =1$	0.063(26)	0.1156(25)	0.1140(25)	0.0946(26)	0.1135(31)