Theory and Modern Applications
Table 6 Improved graded mesh, \(\lambda =2\), \(\tau _{N}=\min \{0.5, \epsilon (2p+1)\ln (N+1)\}\), \(\epsilon =10^{-8}\)
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 | 3.73e − 3 | 1.73 | 9.57e − 5 | 2.00 | 4.62e − 5 | 3.41 | 5.08e − 7 | 3.00 | 1.72e − 8 | 5.17 | 3.94e − 10 | 3.55 |
64 | 1.11e − 3 | 1.75 | 2.39e − 5 | 2.00 | 4.14e − 6 | 3.48 | 6.37e − 8 | 2.99 | 4.26e − 10 | 5.34 | 3.26e − 11 | 3.59 |
128 | 3.21e − 4 | 1.78 | 5.98e − 6 | 2.00 | 3.50e − 7 | 3.57 | 8.00e − 9 | 2.99 | 9.92e − 12 | 5.43 | 2.63e − 12 | 3.63 |
256 | 9.17e − 5 | 1.81 | 1.50e − 6 | 2.00 | 2.86e − 8 | 3.61 | 1.01e − 9 | 2.99 | 2.20e − 13 | 5.50 | 2.07e − 13 | 3.67 |