Theory and Modern Applications
Table 1 Comparing MAEs of the present method at \(\pmb{\theta_{i}=\vartheta _{i}=0}\) , \(\pmb{i=1,2,3}\) , and the method in [ 32 ] for Example 1
Implicit method [32] | \(\Delta_{t} = \Delta_{x}^{2} = \frac{1}{4}\) | \(\Delta_{t} = \Delta_{x}^{2} = \frac{1}{16}\) | \(\Delta_{t} = \Delta_{x}^{2} = \frac {1}{64}\) | \(\Delta_{t} = \Delta_{x}^{2} = \frac{1}{256}\) |
1.39 × 10−2 | 4.49 × 10−3 | 1.18 × 10−3 | 3.06 × 10−4 | |
Explicit method [32] | \(\Delta_{t} =\frac{1}{15}\), \(\Delta_{x}^{2} = \frac {1}{2}\) | \(\Delta_{t} =\frac{1}{54}\), \(\Delta_{x}^{2} = \frac{1}{3}\) | \(\Delta _{t} =\frac{1}{114}\), \(\Delta_{x}^{2} = \frac{1}{4}\) | \(\Delta_{t} =\frac {1}{192}\), \(\Delta_{x}^{2} = \frac{1}{5}\) |
4.57 × 10−2 | 2.78 × 10−3 | 1.19 × 10−3 | 4.98 × 10−4 | |
Our method | N = 4 | N = 6 | N = 8 | N = 10 |
1.30 × 10−4 | 2.39 × 10−7 | 2.08 × 10−10 | 2.13 × 10−13 |