Fig. 5.

Comparison between numerical solution and asymptotic (method of multiple scales, or MMS) solution for (a) $\beta =1$, $\tau =1.025$, $a=1\times 10^{-5}$ (numerical) and $c_1=310$ (MMS); (b) $\beta =1$, $\tau =1.05$, $a=1\times 10^{-5}$ (numerical) and $c_1=200.5$ (MMS). The initial condition used for integrating Eq. (18) is $P\left(t\right)=a\left(1+\frac{t}{1+\tau }\right),t\le 0$. The arbitrary time-shift $c_1$ of the multiple scales solution (Eq. (47)) is chosen here to obtain a good visual match.