The Fokas method for linear convection–diffusion equation with time-dependent coefficients and its extension to other evolution equations

Abstract

In this paper, we study a linear convection–diffusion equation with time-dependent coefficients on a bounded interval, motivated by associated physical problems. We apply and adapt the unified transform method (UTM), also known as the Fokas method, which handles both time-varying coefficients and non-zero boundary data, to obtain an explicit integral formula for the solution in a particular setting. Next, we study well-posedness of the model in fractional Sobolev spaces and prove spatial and temporal regularity estimates, showing that the smoothing effect of the heat operator is still prevalent even when coefficients depend on time. Finally, we extend this approach to obtain the solution for several evolution equations with time-dependent coefficients in one space variable.