MMP.9:5.3 - Decide whether a fast transient can be omitted

For a dimensionless model with epsilon>0,

x_dot=y, epsilon*y_dot=-y,

the solutions for t>=0 are y(t)=y0*exp(-t/epsilon) and x(t)=x0+epsilon*y0*(1-exp(-t/epsilon)).

Suppose the admitted initial values satisfy abs(y0)<=Y. Replacing the model by constant x0 gives an error at most epsilon*Y for every t>=0. With epsilon=0.01 and Y=1, that is 0.01. It can meet a tolerance 0.02 while failing to establish tolerance 0.001. The coefficient alone is insufficient if the initial class changes: y0=1/epsilon leaves a later change approaching one.

When y0 is known and the use concerns only later times, a different approximation is constant x0+epsilon*y0. Its error is at most epsilon*Y*exp(-t/epsilon). For tolerance 0<eta<epsilon*Y, it meets that tolerance at all times starting from t_start=epsilon*log(epsilon*Y/eta). With eta=0.001 in the preceding case, t_start is about 0.0231. The adjusted initial value has retained the transient’s later effect; it does not reproduce the initial interval.

If y0 is unknown, the adjusted value is also unknown. Its stated range can still yield a useful interval for x. The choice between an early transient calculation, a later approximation and a bound follows the requested result and available initial information.