Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions

Figures & data

Figure 1. Local support domains for an arbitrary nodal point ${\mathbf{x}}_i$ for two-dimensional hypothesis domain.

Figure 2. Considered domains in the current paper.

Figure 3. $u_{approx}$ and ${\mathit{\epsilon }}_{\infty }(u)$ profiles for different noise levels with $N=625$, $\mathit{\delta }t=0.01$ and $T=1$ on ${[0,1]}^2$ for Example 1.

Figure 4. The exact solutions $\{r(t),u(x,0.5,T)\}$ in comparison with the regularized numerical solutions for $N=625$, $\mathit{\delta }t=0.01$, $\mathit{\sigma }\in \{1,2,3\}\%$ and $T=1$ on ${[0,1]}^2$ for Example 1.

Figure 5. Condition number and regularization parameter values for $k=\overline{1:T/\mathit{\delta }t}$ with $N=121$, $\mathit{\delta }t=0.01$ and $T=1$ on ${[0,1]}^2$ for Example 1.

Figure 6. Graphs of approximate solution and absolute error of $u(\mathbf{x},t)$ (a) and (b) with approximate solution and absolute error of r(t) (c) and (d) in the case of no regularization and $N=441$, $\mathit{\delta }t=0.01$ and $T=1$ on domain ${\mathrm{\Omega }}_2$ for Example 1.

Figure 7. $u_{approx}$ (left) and ${\mathit{\epsilon }}_{\infty }(u)$ (right) profiles for different noise levels with $N=625$, $\mathit{\delta }t=0.01$ and $T=1$ on ${[0,1]}^2$ for Example 2.

Table 1. Numerical results of absolute errors and relative errors on ${[0,1]}^2$ with $N=529$, $\mathit{\delta }t=0.02$, $T=1$ and condition number $5.5427\times {10}^3$ for Example 1.

$\mathit{\sigma }$ (%)	${\mathit{\epsilon }}_{\infty }(r)$	${\mathit{\epsilon }}_R(r)$	${\mathit{\epsilon }}_{\infty }(u)$	${\mathit{\epsilon }}_R(u)$
1	5.9904e$-$002	5.9904e$-$002	2.5846e$-$002	3.0563e$-$003
3	6.2430e$-$002	6.2430e$-$002	2.6260e$-$002	3.0864e$-$003
5	7.6729e$-$002	7.6729e$-$002	3.8895e$-$002	5.0006e$-$003

Table 2. Numerical results of absolute errors, relative errors and condition numbers on ${[0,1]}^2$ with $\mathit{\sigma }=1\%$ and $T=1$ for Example 1.

	${\mathit{\epsilon }}_{\infty }(r)$	${\mathit{\epsilon }}_R(r)$	${\mathit{\epsilon }}_{\infty }(u)$	${\mathit{\epsilon }}_R(u)$	cond
$h=\mathit{\delta }t=1/10$	9.1063e$-$001	9.1063e$-$001	6.6331e$-$00	9.4847e$-$001	3.0202e$-$002
$h=1/14,\mathit{\delta }t=1/20$	7.6473e$-$001	7.6473e$-$001	4.7625e$-$001	7.3015e$-$002	5.8425e$-$002
$h=1/20,\mathit{\delta }t=1/50$	6.8627e$-$002	6.8627e$-$002	1.0667e$-$002	1.2570e$-$003	1.7197e$-$003
$h=1/24,\mathit{\delta }t=1/100$	5.7520e$-$002	5.7520e$-$002	2.7169e$-$003	3.2620e$-$004	4.7351e$-$003

Figure 8. Graphs of absolute errors r(t) and u(x, y, T) using the SMRPI with $\mathit{\sigma }\in \{0,3,5\}\%$, $N\in \{121,225,441,625,1089,1849,2209\}$, $\mathit{\delta }t=0.02$ and $T=1$ on ${[0,1]}^2$ for Example 2.

Figure 9. Condition number and regularization parameter values for $k=\overline{1:T/\mathit{\delta }t}$ with $N=2209$, $\mathit{\delta }t=0.02$ and $T=1$ on ${[0,1]}^2$ for Example 2.

Figure 10. Graphs of approximate solution and absolute error of $u(\mathbf{x},t)$ (a,b) with approximate solution and absolute error of r(t) (c,d) in the case of no regularization and $N=441$, $\mathit{\delta }t=0.01$ and $T=1$ on domain ${\mathrm{\Omega }}_1$ for Example 2.

Table 3. Numerical results of absolute errors and relative errors on ${[0,1]}^2$ with $N=1849$, $\mathit{\delta }t=0.02$, $T=1$ and condition number $4.7605\times {10}^4$ for Example 2.

$\mathit{\sigma }$ (%)	${\mathit{\epsilon }}_{\infty }(r)$	${\mathit{\epsilon }}_R(r)$	${\mathit{\epsilon }}_{\infty }(u)$	${\mathit{\epsilon }}_R(u)$
1	1.7999e$-$001	1.7999e$-$001	1.1730e$-$002	7.4711e$-$003
3	1.8026e$-$001	1.8026e$-$001	1.1737e$-$002	7.4766e$-$003
5	1.8212e$-$001	1.8212e$-$001	1.2048e$-$002	7.6859e$-$003

Figure 11. $u_{approx}$ (left) and ${\mathit{\epsilon }}_{\infty }(u)$ (right) profiles for different noise levels with $N=625$, $\mathit{\delta }t=0.01$ and $T=1$ on ${[0,1]}^2$ for Example 3.

Figure 12. Graphs of relative errors r(t) and u(x, y, T) using the SMRPI with $\mathit{\sigma }\in \{0,3,5\}\%$, $N\in \{121,225,441,625,1089,1681\}$, $\mathit{\delta }t=0.02$ and $T=1$ on ${[0,1]}^2$ for Example 3.

Figure 13. Condition number and regularization parameter values for $k=\overline{1:T/\mathit{\delta }t}$ with $N=441$, $\mathit{\delta }t=0.02$ and $T=1$ on ${[0,1]}^2$ for Example 3.

Figure 14. Graphs of approximate solution and absolute error of $u(\mathbf{x},t)$ (a,b) with approximate solution and absolute error of r(t) (c,d) in the case of no regularization and $N=2209$, $\mathit{\delta }t=0.01$ and $T=1$ on domain ${\mathrm{\Omega }}_3$ for Example 3.

Table 4. Numerical results of absolute errors with no regularization and $T=1$, on irregular domains ${\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2$ and ${\mathrm{\Omega }}_3$ for Example 2.

	${\mathrm{\Omega }}_1$		${\mathrm{\Omega }}_2$		${\mathrm{\Omega }}_3$
	${\mathit{\epsilon }}_{\infty }(r)$	${\mathit{\epsilon }}_{\infty }(u)$	${\mathit{\epsilon }}_{\infty }(r)$	${\mathit{\epsilon }}_{\infty }(u)$	${\mathit{\epsilon }}_{\infty }(r)$	${\mathit{\epsilon }}_{\infty }(u)$
$N=225$, $\mathit{\delta }t=1/10$	3.4436e$-$004	8.9253e$-$006	3.2252e$-$004	5.6662e$-$004	5.4894e$-$003	1.0634e$-$003
$N=441$, $\mathit{\delta }t=1/50$	8.2893e$-$005	2.3996e$-$006	4.8099e$-$006	1.1507e$-$006	1.7490e$-$003	7.3119e$-$004
$N=729$ , $\mathit{\delta }t=1/80$	5.4471e$-$005	1.1960e$-$006	2.6571e$-$006	1.8344e$-$007	1.3912e$-$003	2.1163e$-$004
$N=1089$ , $\mathit{\delta }t=1/100$	1.0076e$-$005	3.5422e$-$007	1.5345e$-$006	4.7612e$-$008	2.0823e$-$004	4.0857e$-$005
$N=1225$ , $\mathit{\delta }t=1/150$	5.5202e$-$006	1.5043e$-$007	1.4544e$-$006	3.5298e$-$008	1.3900e$-$005	2.9496e$-$005

Table 5. Numerical results of absolute errors and relative errors on ${[0,1]}^2$ with $N=2209$, $\mathit{\delta }t=0.02$, $T=1$ and condition number $2.8050\times {10}^4$ for Example 3.

$\mathit{\sigma }$ (%)	${\mathit{\epsilon }}_{\infty }(r)$	${\mathit{\epsilon }}_R(r)$	${\mathit{\epsilon }}_{\infty }(u)$	${\mathit{\epsilon }}_R(u)$
0	1.3175e$-$009	4.8467e$-$010	1.6392e$-$007	1.6617e$-$007
1	1.2140e$-$000	4.4659e$-$001	1.7974e$-$001	1.9101e$-$001
3	1.2179e$-$000	4.48060e$-$001	1.8048e$-$001	1.9180e$-$001
5	1.2168e$-$000	4.4762e$-$001	1.8026e$-$001	1.9156e$-$001