Theory and Modern Applications
Table 8 Improved graded mesh, \(\lambda =2\), \(\tau _{\epsilon }= \min \{0.5,-\epsilon (p+1)\ln \epsilon \}\), \(\epsilon =10^{-8}\)
From: The coupled method for singularly perturbed Volterra integro-differential equations
N | p = 1 | p = 2 | p = 3 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\|\cdot \|_{\infty }\) | order | \(\|\cdot \|_{L^{2}}\) | order | \(\|\cdot \|_{\infty }\) | order | \(\|\cdot \|_{L^{2}}\) | order | \(\|\cdot \|_{\infty }\) | order | \(\|\cdot \|_{L^{2}}\) | order | |
32 | 1.32e − 2 | 2.17 | 9.57e − 5 | 2.00 | 4.69e − 4 | 3.98 | 5.77e − 7 | 3.00 | 2.67e − 7 | 5.99 | 2.37e − 9 | 3.97 |
64 | 3.22e − 3 | 2.04 | 2.39e − 5 | 2.00 | 2.85e − 5 | 4.04 | 7.23e − 8 | 3.00 | 4.33e − 9 | 5.95 | 1.49e − 10 | 3.99 |
128 | 8.13e − 4 | 1.98 | 5.98e − 6 | 2.00 | 1.82e − 6 | 3.97 | 9.05e − 9 | 3.00 | 6.77e − 11 | 6.00 | 9.31e − 12 | 4.00 |
256 | 2.03e − 4 | 2.00 | 1.50e − 6 | 2.00 | 1.13e − 7 | 4.00 | 1.13e − 9 | 3.00 | 1.06e − 12 | 6.00 | 5.82e − 13 | 4.00 |