In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_t+{\int }_0^t\beta (t-s)Au\left(s\right)ds=0,u\left(0\right)=v,t>0,$ where A is an elliptic partial-differential operator and $\beta \left(t\right)$ is positive, nonincreasing and log-convex on $(0,\infty )$ with $0\le \beta \left(\infty \right)<\beta \left(0^{+}\right)\le \infty$. Error estimates are derived in the norm of $L_t^1(0,\infty ;L_x^2)$, and some estimates for the first order time derivatives of the errors are also given.

This article focuses on the heat radiation intensity optimization on the surface of an aluminium shell mould. The outer mould surface is heated by infrared heaters located above the mould and the inner mould surface is sprinkled with a special PVC powder. This is an economic way of producing artificial leathers in the automotive industry (e.g. the artificial leather on car dashboards). The article includes a description of a mathematical model that allows us to calculate the heat radiation intensity...