Fading-memory representation of the Oldroyd-A model

Put $S={\sigma }^d-2{\mu }_{r}D$ in the Oldroyd-A model. It obeys ${\mathcal{D}}_{l}S+S/\tau =2(\mu -{\mu }_r)D/\tau$. Pulling back as $F^{T}SF$ converts this to an ordinary linear relaxation equation. With decayed remote-past initial memory, its solution isAt a finite initial time, include the homogeneous transported term $e^{-(t-t_0)/\tau }F(t{)}^{-T}F(t_0{)}^{T}S(t_0)F(t_0)F(t{)}^{-1}$. The integral-only law is not equivalent to all arbitrary initial-value solutions without a memory condition.