Figure 1.
(a) Sparsity of matrix W, when and , and (b) centre distribution in domain corresponding to Example 1.
Figure 1.
(a) Sparsity of matrix W, when and , and (b) centre distribution in domain corresponding to Example 1.
Figure 2.
(a) Nodes M against MAE and RMSE norms for , and (b) stability plots of our numerical scheme corresponding to Example 1.
Figure 2.
(a) Nodes M against MAE and RMSE norms for , and (b) stability plots of our numerical scheme corresponding to Example 1.
Figure 3.
(a) Numerical solution at different times t, and (b) point wise error when , and when corresponding to Example 1.
Figure 3.
(a) Numerical solution at different times t, and (b) point wise error when , and when corresponding to Example 1.
Figure 4.
(a) Exact against (b) numerical solutions, when , , , , corresponding to Example 2.
Figure 4.
(a) Exact against (b) numerical solutions, when , , , , corresponding to Example 2.
Figure 5.
Numerical solution at different times t for , corresponding to Example 2.
Figure 5.
Numerical solution at different times t for , corresponding to Example 2.
Figure 6.
(a) Exact against (b) numerical solutions, when , , , corresponding to Example 3.
Figure 6.
(a) Exact against (b) numerical solutions, when , , , corresponding to Example 3.
Figure 7.
(a) Point wise error, when , and , and (b) numerical solution, corresponding to Example 3.
Figure 7.
(a) Point wise error, when , and , and (b) numerical solution, corresponding to Example 3.
Figure 8.
(a) Ameoba and (b) Cassini-shaped domain with interior (red) and boundary (green) stencil.
Figure 8.
(a) Ameoba and (b) Cassini-shaped domain with interior (red) and boundary (green) stencil.
Figure 9.
(a) Exact against (b) numerical solution of Ameoba-shaped domain when , , , to Example 4.
Figure 9.
(a) Exact against (b) numerical solution of Ameoba-shaped domain when , , , to Example 4.
Figure 10.
(a) Exact against (b) numerical solution of Cassini-shaped domain when , , , to Example 4.
Figure 10.
(a) Exact against (b) numerical solution of Cassini-shaped domain when , , , to Example 4.
Table 1.
Numerical results when for domain (Example 1).
Table 1.
Numerical results when for domain (Example 1).
| ns | MCN | MAE | RMSE | C-Time | Shape (c) |
---|
RK4 (MQ RBF) | 5 | 3.099 | 1.931 | 8.336 | 0.950 | 0.010 |
11 | 1.241 | 1.915 | 7.935 | 0.858 | 0.520 |
21 | 7.732 | 1.215 | 7.643 | 2.319 | 1.160 |
49 | 2.929 | 4.109 | 1.269 | 1.869 | 1.990 |
60 | 2.941 | 2.725 | 1.121 | 2.504 | 2.150 |
80 | 5.096 | 2.961 | 1.287 | 4.199 | 2.320 |
Table 2.
Numerical solution of
at
compared with exact solution and Zhang et al. [
5] (Example 1).
Table 2.
Numerical solution of
at
compared with exact solution and Zhang et al. [
5] (Example 1).
Number of Nodes = 13 × 13 | | | | | |
---|
Present Solution | 1.2628 | 1.1426 | 1.0339 | 0.9355 | 0.8465 |
Zhang et al. Solution [5] | 1.2574 | 1.1382 | 1.0302 | 0.9325 | 0.844 |
Exact Solution | 1.2628 | 1.1426 | 1.0339 | 0.9355 | 0.8465 |
Table 3.
Numerical results corresponding to Example 3 in .
Table 3.
Numerical results corresponding to Example 3 in .
| ns | MCN | MAE | RMSE | C-Time | Shape (c) |
---|
RK4 (MQ RBF) | 5 | 3.099 | 1.35 | 8.85 | 0.14538 | 0.01 |
10 | 5.136 | 6.93 | 4.70 | 0.76946 | 0.43 |
15 | 4.549 | 6.90 | 4.75 | 0.77773 | 0.85 |
25 | 1.367 | 7.50 | 5.47 | 0.88658 | 1.41 |
50 | 3.225 | 7.34 | 4.67 | 1.26577 | 1.99 |
70 | 3.558 | 6.92 | 4.72 | 1.57271 | 2.26 |
80 | 5.096 | 7.53 | 4.53 | 1.74231 | 2.32 |
Table 4.
Numerical solution of
at
compared with exact solution and Zhang et al. [
5] (Example 3).
Table 4.
Numerical solution of
at
compared with exact solution and Zhang et al. [
5] (Example 3).
Number of Nodes = 13 × 13 | | | | | |
---|
Present Solution | 1.2628 | 1.1426 | 1.0339 | 0.9355 | 0.8465 |
Zhang et al. Solution [5] | 1.2574 | 1.1382 | 1.0302 | 0.9325 | 0.844 |
Exact Solution | 1.2628 | 1.1426 | 1.0339 | 0.9355 | 0.8465 |
Table 5.
MAE and RMSE for Ameoba and Cassini-shaped domains at with different stencil sizes.
Table 5.
MAE and RMSE for Ameoba and Cassini-shaped domains at with different stencil sizes.
| Ameoba | Cassini |
---|
ns | MAE | RMSE | MAE | RMSE |
---|
5 | 9.40 | 3.52 | 1.63 | 1.02 |
15 | 2.40 | 2.21 | 7.33 | 4.32 |
30 | 3.20 | 9.12 | 6.36 | 3.82 |
45 | 1.78 | 1.51 | 2.51 | 1.56 |
60 | 1.20 | 2.12 | 4.45 | 2.22 |