Generalized finite difference method for solving two-dimensional inverse Cauchy problems

Figures & data

Table

$a_1=a_1(x,y)$	given coefficient functions
$a_2=a_2(x,y)$	given coefficient functions
$a_3=a_3(x,y)$	given coefficient functions
$a_4=a_4(x,y)$	given coefficient functions
$a_5=a_5(x,y)$	given coefficient functions
$a_6=a_6(x,y)$	given coefficient functions
$a_7=a_7(x,y)$	given coefficient functions
$d{m}_i$	supporting radius of weighting function for the ith node
$f(x,y)$	given boundary condition
$g(x,y)$	given boundary condition
$h(x,y)$	given boundary condition
$l(x,y)$	given boundary condition
$\overrightarrow{n}=(n_x,n_y)$	unit outward normal vector
$N$	number of total nodes
$n_b$	number of boundary nodes
$n_i$	number of interior nodes
$n_s$	number of nodes in a star
rand	random number
$s$	percentage of added noise
$u(x,y)$	(unknown) solution
$u_n(x,y)$	normal derivative of the (unknown) solution
$w(d_{ij})$	weighting function
${\{w{x}_j^i\}}_{j=0}^{n_s}$	weighting coefficients
${\{w{y}_j^i\}}_{j=0}^{n_s}$	weighting coefficients
${\{wx{x}_j^i\}}_{j=0}^{n_s}$	weighting coefficients
${\{wy{y}_j^i\}}_{j=0}^{n_s}$	weighting coefficients
${\{wx{y}_j^i\}}_{j=0}^{n_s}$	weighting coefficients
$(x_i,y_i)$	coordinates of the ith node
$(x_j^i,y_j^i)$	coordinates of the other nodes in the ith star
${\mathrm{\Gamma }}_d$	boundary portions with Dirichlet boundary condition
${\mathrm{\Gamma }}_n$	boundary portions with Neumann boundary condition
${\mathrm{\Gamma }}_o$	boundary portions with over-specified boundary condition
$\mathrm{\Omega }$	computational domain
$\mathit{\gamma }$	boundary portions with no boundary condition

Figure 1. Schematic diagram for the circular shape of the star.

Figure 2. (a) The schematic diagram for example 1 and (b) the distributions of interior and boundary nodes.

Figure 3. The distributions of numerical (solid lines) and analytical solutions (dashed lines). (a) $s=0$ and (b) $s=3$.

Figure 4. The profiles of numerical solutions along ${\mathrm{\Gamma }}_4$ for example 1.

Table 1. The maximum absolute errors by adding different levels of noise for example 1.

Percentages of noise	Maximum absolute errors
s = 0	0.0118
s = 1	0.0242
s = 2	0.0510
s = 3	0.0806

Figure 5. The profiles of numerical solutions along ${\mathrm{\Gamma }}_4$ for example 1 (s = 3).

Figure 6. The profiles of numerical solutions along ${\mathrm{\Gamma }}_4$ by adopting different supporting radii of weighting function for example 1 (s = 3, $n_s=12$).

Figure 7. (a) The schematic diagram for example 2 and (b) the distributions of interior and boundary nodes.

Figure 8. The distributions of numerical (solid lines) and analytical solutions (dashed lines). (a) $s=0$ and (b) $s=3$.

Figure 9. The profiles of numerical solutions along ${\mathrm{\Gamma }}_2$ for example 2.

Table 2. The maximum absolute errors by adding different levels of noise for example 2.

Percentages of noise	Maximum absolute errors
s = 0	7.7658 × 10^−12
s = 1	0.0806
s = 2	0.3427
s = 3	0.5077

Figure 10. The profiles of numerical solutions along ${\mathrm{\Gamma }}_2$ by adopting different weighting functions for example 2.

Figure 11. The schematic diagrams for example 3. (a) Direct problem and (b) inverse problem.

Figure 13. The profiles of numerical solutions along ${\mathrm{\Gamma }}_4$ for example 3.