The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation

Figures & data

Figure 1. The analytical f(t) and its approximation $f^{\ast }(t)$ when there is no regularization method for Example 1.

Figure 2. The GCV function obtained for various levels of noise added into the measured data, namely $\mathit{\delta }=1\%(....)$, $\mathit{\delta }=3\%(--)$ and $\mathit{\delta }=5\%(-)$, with $n=m=s=20$, $T=2.5$, $X=-1$ for Example 1.

Figure 3. The analytical f(t) and its approximation $f^{\ast }(t)$ with $n=m=s=20$, $T=2.5$, $X=-1$, and various levels of noise added into the measured data, namely $\mathit{\delta }=1\%$, $\mathit{\delta }=3\%$ and $\mathit{\delta }=5\%$ for Example 1.

Figure 4. The analytical f(t) and its approximation $f^{\ast }(t)$ with $n=m=s=20$, $T=2.5$, $X=-1$ for fractional MFS ($\mathit{\alpha }=2$) and $n=m=s=20$, $T=2.5$ for classical MFS when level of noise added into the measured data $\mathit{\delta }=1\%$ for Example 1.

Figure 5. (a) The RMSE(f) and RRMSE(f) of the numerical solutions for Example 1 with $n=m=s=20$, $\mathit{\delta }=1\%$, $T=2.5$, $X=-1$ and $x_0=0.5$ with respect to the fractional order $\mathit{\alpha }$; (b) The RMSE(u) and RRMSE(u) of the numerical solutions for Example 1 with $n=m=s=20$, $\mathit{\delta }=1\%$, $T=2.5$, $X=-1$ and $x_0=0.5$ with respect to the fractional order $\mathit{\alpha }$; (c) The condition number cond(A) of A with $n=m=s=20$, $T=2.5$, $X=-1$ and $x_0=0.5$ with respect to the fractional order $\mathit{\alpha }$.

Figure 6. (a) The RMSE(u) and RRMSE(u) of the numerical solutions for Example 1 with $\mathit{\delta }=1\%$, $x_0=0.2$, $X=-1$ with respect to the parameter T; (b) The RMSE(u) and RRMSE(u) of the numerical solutions for Example 1 with $\mathit{\delta }=1\%$, $x_0=0.5$, $T=2.5$ with respect to the parameter X.

Figure 7. (a) The RMSE(f) and RRMSE(f) of the numerical solutions for Example 1 with $\mathit{\delta }=1\%$, $x_0=0.5$, $X=-1$ with respect to the parameter T; (b) The RMSE(f) and RRMSE(f) of the numerical solutions for Example 1 with $\mathit{\delta }=1\%$, $x_0=0.5$, $T=2.5$ with respect to the parameter X.

Figure 8. (a) The analytical f(t) and its approximation $f^{\ast }(t)$ with $n=m=s=25$, $T=1.5$, $X=-1$, and various levels of noise added into the measured data, namely $\mathit{\delta }=1\%$, $\mathit{\delta }=3\%$ and $\mathit{\delta }=5\%$  for  Example 2; (b) The analytical f(t) and its approximation $f^{\ast }(t)$ with $n=m=s=30$, $T=2$, $X=-1$, and various levels of noise added into the measured data,namely $\mathit{\delta }=1\%$, $\mathit{\delta }=3\%$ and $\mathit{\delta }=5\%$ for Example 2.

Figure 9. The GCV function obtained for various levels of noise added into the measured data, namely $\mathit{\delta }=1\%(....)$, $\mathit{\delta }=3\%(--)$ and $\mathit{\delta }=5\%(-)$, with $n=m=s=25$, $T=2.5$, $X=-1$ for Example 2.

Figure 10. (a) The accuracy of the numerical solutions for Example 2 with $\mathit{\delta }=1\%,x_0=0.9,X=-1$ and $T=2.5$ with respect to the parameter n; (b) The accuracy of the numerical solutions for Example 2 with $\mathit{\delta }=1\%$, $x_0=0.9$, $X=-1$ with respect to the parameter T; (c) The accuracy of the numerical solutions for Example 2 with $\mathit{\delta }=1\%$, $x_0=0.9$, $T=1.5$ with respect to the parameter X.

Table 1. The values of cond(A), RMSE(u), RRMSE(u), RMSE(f), RRMSE(f) for various values of X in Example 1, $\mathit{\delta }=1\%$, $T=2.5$.

X	RMSE(u)	RRMSE(u)	RMSE(f)	RRMSE(f)	cond(A)
$X=-1$	0.5008	0.1127	0.0170	0.0083	5.9324e+18
$X=-1.3$	0.2198	0.0495	0.0091	0.0044	2.9995e+18
$X=-1.6$	0.5007	0.1127	0.0159	0.0077	2.3000e+18
$X=-2$	0.4007	0.0902	0.0102	0.0050	4.3280e+18
$X=-2.3$	0.5693	0.1281	0.0136	0.0066	2.5548e+19
$X=-2.5$	0.5898	0.1327	0.0091	0.0044	7.6651e+19

Table 2. The values of cond(A), RMSE(u), RRMSE(u), RMSE(f), RRMSE(f) for various values of T in Example 1, $\mathit{\delta }=1\%$, $X=-1$.

T	RMSE(u)	RRMSE(u)	RMSE(f)	RRMSE(f)	cond(A)
$T=1.5$	0.9622	0.2166	0.0373	0.0182	3.2758e+22
$T=1.8$	0.6423	0.1446	0.0118	0.0057	6.2945e+19
$T=2.0$	0.4672	0.1052	0.0124	0.0060	2.2338e+18
$T=2.3$	0.6369	0.1434	0.0157	0.0076	1.7224e+18
$T=2.5$	0.1999	0.0450	0.0137	0.0066	5.9324e+18
$T=2.8$	0.1358	0.0306	0.0079	0.0038	2.9637e+18

Table 3. The values of RRMSE(f) for various values of $\mathit{\delta }$ and $x_0$ in Example 1, $T=2.5$, $X=-1$.

$x_0$	$\mathit{\delta }=1\%$	$\mathit{\delta }=3\%$	$\mathit{\delta }=5\%$
$x_0=0.1$	0.2020	0.2127	0.2749
$x_0=0.2$	0.1228	0.1437	0.1687
$x_0=0.3$	0.0566	0.0859	0.1405
$x_0=0.4$	0.0480	0.0676	0.1178
$x_0=0.5$	0.0379	0.0546	0.0697
$x_0=0.6$	0.0807	0.1244	0.1481
$x_0=0.7$	0.1251	0.1435	0.1558
$x_0=0.8$	0.1405	0.1847	0.2610
$x_0=0.9$	0.1429	0.2581	0.3010