Theory and Modern Applications
Table 2 Shishkin mesh, \(\tau _{N}=\min \{0.5,\epsilon (2p+1)\ln (N+1) \}\), \(\epsilon =10^{-8}\)
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 | 1.58 | 9.57e − 5 | 2.00 | 7.77e − 4 | 2.66 | 7.05e − 7 | 2.66 | 2.18e − 6 | 4.40 | 1.54e − 8 | 2.84 |
64 | 4.71e − 3 | 1.49 | 2.39e − 5 | 2.00 | 9.12e − 5 | 3.09 | 1.27e − 7 | 2.48 | 8.31e − 8 | 4.71 | 1.84e − 9 | 3.07 |
128 | 1.59e − 3 | 1.56 | 5.99e − 6 | 2.00 | 1.06e − 5 | 3.10 | 2.34e − 8 | 2.43 | 2.85e − 9 | 4.87 | 1.98e − 10 | 3.22 |
256 | 5.19e − 4 | 1.62 | 1.50e − 6 | 2.00 | 1.13e − 6 | 3.23 | 4.24e − 9 | 2.46 | 9.02e − 11 | 4.98 | 1.99e − 11 | 3.32 |