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Table 2 x n and y n are numerical solutions by implicit Euler method, \(\tilde{x}_{n}\) and \(\tilde{y}_{n}\) are numerical solutions by BDF method, x(t n and y(t n ) are exact solutions

From: Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays

n \(t_{n} = nh\) x n x n \(\tilde{x}_n\) \(\tilde{y}_n\) x(t n ) y(t n )
0 0 1.0000 0.0500 1.0000 0.5000 1.0000 0.5000
1 0.1 0.8725 0.4363 0.8607 0.4304 0.8607 0.4304
2 0.2 0.7613 0.3806 0.7403 0.3701 0.7408 0.3704
3 0.3 0.6642 0.3321 0.6371 0.3186 0.6376 0.3188
4 0.4 0.5795 0.2898 0.5484 0.2742 0.5488 0.2744
5 0.5 0.5056 0.2528 0.4720 0.2360 0.4724 0.2362
6 0.6 0.4412 0.2206 0.4063 0.2031 0.4066 0.2033
7 0.7 0.3849 0.1925 0.3497 0.1748 0.3499 0.1750
8 0.8 0.3358 0.1679 0.3010 0.1505 0.3012 0.1506
9 0.9 0.2930 0.1465 0.2590 0.1295 0.2592 0.1296
10 1.0 0.2557 0.1278 0.2230 0.1115 0.2231 0.1116
11 1.1 0.2231 0.1115 0.119 0.0960 0.1920 0.0960
12 1.2 0.1946 0.0973 0.1652 0.0826 0.1653 0.0826
13 1.3 0.1698 0.0849 0.1422 0.0711 0.1423 0.0711
14 1.4 0.1482 0.0741 0.1224 0.0612 0.1225 0.0612
15 1.5 0.1293 0.0646 0.1053 0.0527 0.1054 0.0527
16 1.6 0.1128 0.0564 0.0906 0.0453 0.0907 0.0454
17 1.7 0.0984 0.0492 0.0780 0.0390 0.0781 0.0390
18 1.8 0.0859 0.0429 0.0672 0.0336 0.0672 0.0336
19 1.9 0.0749 0.0375 0.0578 0.0289 0.0578 0.0289
20 2.0 0.0654 0.0327 0.0497 0.0249 0.0498 0.0249
  1. All solutions are on [0, 2] with h = 0.1 and nh = 2