Do you know any mathematical model of a physical process such that it satisfies the following equations ?
$$\mathrm{u_t=a(t)u_{xx}},\,\,\,0<x<1,\,\,\,t>0$$ $$\mathrm{-ku_x(0,t)=hu(0,t)},\,\,\,t>0$$ $$\mathrm{u(1,t)=u_0},\,\,\,t>0$$ where $a(t)>0$ is time-dependent thermal diffusivity, $k>0$ constant thermal conductivity and $h>0$ constant heat transfer coefficient.