= Fading-memory representation of the Oldroyd-A model
Put $S=\sigma^d-2\mu_rD$ in the <Oldroyd-A model>. It obeys $\mathcal D_lS+S/\tau=2(\mu-\mu_r)D/\tau$. Pulling back as $F^TSF$ converts this to an ordinary linear relaxation equation. With decayed remote-past initial memory, its solution is
$$
\sigma^d(t)=2\mu_rD(t)+\frac{2(\mu-\mu_r)}\tau\int_{-\infty}^t e^{-(t-s)/\tau}F(t)^{-T}F(s)^TD(s)F(s)F(t)^{-1}\,ds.
$$
At a finite initial time, include the homogeneous transported term $e^{-(t-t_0)/\tau}F(t)^{-T}F(t_0)^TS(t_0)F(t_0)F(t)^{-1}$. The integral-only law is not equivalent to all arbitrary initial-value solutions without a memory condition.
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