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

TL;DR

This paper develops an explicit integral solution for a linear convection-diffusion equation with time-dependent coefficients using the Unified Transform Method, and extends the approach to other evolution equations, analyzing well-posedness and regularity.

Contribution

It adapts the Unified Transform Method to handle time-dependent coefficients and nonzero boundary data, providing explicit solutions and regularity results for evolution equations.

Findings

Explicit integral formula for the convection-diffusion equation solution.

Well-posedness and regularity estimates in fractional Sobolev spaces.

Extension of the method to other evolution equations with time-dependent coefficients.

Abstract

In this paper, we study a linear convection-diffusion equation with time-dependent coefficients on a bounded interval. The problem includes inhomogeneous Dirichlet boundary conditions and is motivated by physical models where the diffusivity and transport change with time, such as heat or mass transfer in non-stationary environments. We apply and adapt the Unified Transform Method (UTM), which handles both time-varying coefficients and nonzero boundary data, to obtain an explicit integral formula for the solution. %The method allows us to handle both time-varying coefficients and nonzero boundary data by removing unknown boundary traces emerging upon finite line Fourier transform (FLFT) through a spectral symmetry argument. Next, we study well-posedness of the model in fractional Sobolev spaces and prove spatial and temporal regularity estimates. We show that the smoothing effect of the heat operator is still prevalent even when coefficients depend on time under suitable sign and growth assumptions on the coefficients. Finally, we extend this approach to obtain the solution for several evolution equations with time-dependent coefficients, in one space variable.