A non-iterative method for identifying multiple unknown time-dependent sources compactly supported occurring in a 2D parabolic equation

Figures & data

Figure 1. Functions $g_{incl}$(a) and $g_{cont}$(b) corresponding to ${\mathit{\omega }}_1$ defined by $G_1$ in Table 3.

Figure 2. Functions $g_{incl}$(a) and $g_{cont}$(b) corresponding to ${\mathit{\omega }}_2$ defined by $G_2$ in Table 3.

Table 1. Source detection: simulation using $G_1=(97,70);G_2=(273,7);G_3=(514,96)$.

${\mathrm{\Omega }}_i=((i-1)\times 200,i\times 200)\times (0,\ell )$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_3$	${\mathrm{\Omega }}_4$
Detection coefficient: $P_1^i/T$	42.22	115.23	21.35	-0.05
Identified geometric centre $G_n$	(96.88, 69.99)	(272.90, 6.97)	(513.89, 95.99)	No source

Table 2. Source detection: simulation using $G_1=(43,6);G_2=(327,86);G_3=(716,3)$.

${\mathrm{\Omega }}_i=((i-1)\times 200,i\times 200)\times (0,\ell )$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_3$	${\mathrm{\Omega }}_4$
Detection coefficient: $P_1^i/T$	42.19	115.25	-0.03	21.35
Identified geometric centre $G_n$	(42.99, 5.99)	(326.95, 86.01)	No source	(715.87, 2.99)

Table 3. Source detection: simulation using $G_1=(160,84);G_2=(415,20);G_3=(695,40)$.

${\mathrm{\Omega }}_i=((i-1)\times 200,i\times 200)\times (0,\ell )$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_3$	${\mathrm{\Omega }}_4$
Detection coefficient: $P_1^i/T$	42.25	0.036	115.18	21.26
Identified geometric centre $G_n$	(159.99, 83.99)	No source	(414.95, 20.01)	(694.95, 39.99)

Table 4. Source detection: simulation using $G_1=(304,61);G_2=(496,7);G_3=(623,97)$.

${\mathrm{\Omega }}_i=((i-1)\times 200,i\times 200)\times (0,\ell )$	${\mathrm{\Omega }}_1$	${\mathrm{\Omega }}_2$	${\mathrm{\Omega }}_3$	${\mathrm{\Omega }}_4$
Detection coefficient: $P_1^i/T$	$-7.0E-5$	42.22	115.20	21.30
Identified geometric centre $G_n$	No source	(303.96, 60.98)	(495.91, 6.95)	(622.97, 96.98)

Figure 3. Functions $g_{incl}$(a) and $g_{cont}$(b) corresponding to ${\mathit{\omega }}_3$ defined by $G_3$ in Table 3.

Figure 4. Identification of ${\mathit{\lambda }}_1$: (a) Noise $3\%$; Error 0.18, (b) Noise $10\%$; Error 0.20.

Figure 5. Identification of ${\mathit{\lambda }}_2$: (a) Noise $3\%$; Error 0.45, (b) Noise $10\%$; Error 0.48.

Figure 6. Identification of ${\mathit{\lambda }}_3$: (a) Noise $3\%$; Error 0.29, (b) Noise $10\%$; Error 0.32.