Table 1 Summary of how to obtain numerical integration, semi-analytical, and full analytical solutions

$\tau$

Numerical integration of Eq (1)

Eq (3)

(Eqs (7)-(8) are numerically solved)

Eq (9)

${\epsilon }_i$

Numerical integration of Eq (1)

Eq (4)

(Eqs (7)-(8) are numerically solved)

Eq (10)

${\text{Φ}}_{\tau }$

Numerical computation (see Methods)

Numerical computation (see Methods)

Eq (12)

${\text{Φ}}_{{\epsilon }_i}$

Numerical computation (see Methods)

Numerical computation (see Methods)

Eq (13)