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. 2026 Feb 23;16:7471. doi: 10.1038/s41598-026-38505-8

Table 2.

Presents a numerical comparison of Eqs. (1) and (26).

t NS of the nonlinear Eq. (1) NS of the advanced NPA Eq. 26 Absolute error Relative absolute error Relative absolute error (%)
0 0.2 0.2 0 0 0
5  − 0.0971513  − 0.0929275 0.00422382 0.0434768 4.34768
10 0.0414185 0.0356473 0.00577126 0.13934 13.934
15  − 0.0195691  − 0.0164479 0.00312118 0.159496 15.9496
20  − 0.00169614  − 0.00156325 0.000132892 0.0783498 7.83498
25 0.00936973 0.00718634 0.00218339 0.233026 23.3026
30  − 0.00104883 0.00175433 0.00280316 2.67265 267.265
35 0.00535094 0.00284964 0.00250131 0.467452 46.7452
40  − 0.00767451  − 0.00584803 0.00182648 0.237993 23.7993
45  − 0.00279337  − 0.00390287 0.00110949 0.397187 39.7187
50  − 0.000428967 0.000123265 0.000552232 1.28735 128.735