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Table 2 Absolute errors for Example 1 given by \(\Vert y_e-y_a\Vert\)

From: An efficient algorithm to solve damped forced oscillator problems by Bernoulli operational matrix of integration

t

\(m=2\)

\(m=3\)

\(m=5\)

\(m=7\)

\(m=9\)

0.0

7.91342855 \(\times 10^{-2}\)

8.67203112 \(\times 10^{-2}\)

1.7122077 \(\times 10^{-3}\)

1.36462 \(\times 10^{-5}\)

9.350\(\times 10^{-7}\)

0.1

1.488264874 \(\times 10^{-1}\)

1.99981028 \(\times 10^{-2}\)

5.938054 \(\times 10^{-4}\)

2.8730 \(\times 10^{-6}\)

2.862\(\times 10^{-7}\)

0.2

1.315948258 \(\times 10^{-1}\)

4.17514750 \(\times 10^{-2}\)

4.789498 \(\times 10^{-4}\)

5.7607\(\times 10^{-6}\)

2.362\(\times 10^{-7}\)

0.3

6.77587069\(\times 10^{-2}\)

1.88592116 \(\times 10^{-2}\)

3.553027 \(\times 10^{-4}\)

3.5319 \(\times 10^{-6}\)

9.92\(\times 10^{-8}\)

0.4

9.8506587 \(\times 10^{-3}\)

1.58474766 \(\times 10^{-2}\)

3.899000 \(\times 10^{-4}\)

4.1422 \(\times 10^{-6}\)

1.205\(\times 10^{-7}\)

0.5

7.90561751 \(\times 10^{-2}\)

4.01914937 \(\times 10^{-2}\)

5.036652 \(\times 10^{-4}\)

4.9631\(\times 10^{-6}\)

2.401\(\times 10^{-7}\)

0.6

1.288352991\(\times 10^{-1}\)

4.31502967 \(\times 10^{-2}\)

1.387573 \(\times 10^{-4}\)

2.8589 \(\times 10^{-6}\)

1.303\(\times 10^{-7}\)

0.7

1.579359764\(\times 10^{-1}\)

2.34718312 \(\times 10^{-2}\)

6.119877 \(\times 10^{-4}\)

5.8336 \(\times 10^{-6}\)

8.99\(\times 10^{-8}\)

0.8

1.724335492 \(\times 10^{-1}\)

1.27685605 \(\times 10^{-2}\)

7.76566 \(\times 10^{-5}\)

9.540\(\times 10^{-7}\)

2.330\(\times 10^{-7}\)

0.9

1.829509214\(\times 10^{-1}\)

5.49479747 \(\times 10^{-2}\)

7.417110 \(\times 10^{-4}\)

2.7400 \(\times 10^{-6}\)

2.851\(\times 10^{-7}\)

1.0

2.020789096 \(\times 10^{-1}\)

9.04755948 \(\times 10^{-2}\)

1.7110015 \(\times 10^{-3}\)

1.36480 \(\times 10^{-5}\)

9.350\(\times 10^{-7}\)