Theory and Modern Applications
Table 2 Comparison of exact and approximate solution of Example 1 using the boundary condition as defined in \(\pmb{\mathbf{S_{1}}}\)
| Â | \(\boldsymbol {S_{1}}\) | \(\boldsymbol {S_{2}}\) | ||
|---|---|---|---|---|
| Â | \(\boldsymbol {\|E_{N}\|_{2}}\) | \(\boldsymbol {\|E_{N}\|_{\infty}}\) | \(\boldsymbol {\|E_{N}\| _{2}}\) | \(\boldsymbol {\|E_{N}\|_{\infty}}\) |
N = 5 | 7.23 × 10−2 | 6.53 × 10−1 | 4.72 × 10−1 | 3.68 × 10−1 |
N = 8 | 1.003 × 10−4 | 2.913 × 10−2 | 6.53 × 10−2 | 0.58 × 10−3 |
N = 10 | 4.82 × 10−6 | 3.21 × 10−5 | 0.587 × 10−3 | 0.778 × 10−4 |
N = 15 | 3.92 × 10−11 | 7.84 × 10−9 | 9.51 × 10−5 | 9.510 × 10−8 |
N = 20 | 6.22 × 10−16 | 5.61 × 10−14 | 1.51 × 10−10 | 1.456 × 10−12 |
N = 30 | 9.36 × 10−19 | 8.22 × 10−18 | 9.510 × 10−11 | 9.108 × 10−13 |
N = 40 | 9.28 × 10−21 | 0.781 × 10−18 | 5.87 × 10−16 | 5.33 × 10−18 |