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

Konstantinos Kalimeris, Türker Özsarı

TL;DR

The paper addresses a linear convection-diffusion equation with time-dependent coefficients on a finite interval and inhomogeneous boundary data, formulating explicit integral representations via the Unified Transform Method (UTM). It introduces a time-dependent global relation with phase \omega(k,t) = k^2B(t) - i k C(t) and uses an invariant map to eliminate unknown boundary traces, yielding clear integral expressions on a deformed contour. The authors establish Hadamard-type well-posedness in fractional Sobolev spaces, prove smoothing estimates consistent with parabolic behavior, and extend the methodology to other one-dimensional evolution equations including KdV and Schrödinger-type models. They also outline reductions to the half-line and discuss nonlinear extensions and inverse-problem perspectives, highlighting the broad applicability of the UTM to time-varying-coefficient IBVPs and related higher-order PDEs.

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.

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

TL;DR

The paper addresses a linear convection-diffusion equation with time-dependent coefficients on a finite interval and inhomogeneous boundary data, formulating explicit integral representations via the Unified Transform Method (UTM). It introduces a time-dependent global relation with phase \omega(k,t) = k^2B(t) - i k C(t) and uses an invariant map to eliminate unknown boundary traces, yielding clear integral expressions on a deformed contour. The authors establish Hadamard-type well-posedness in fractional Sobolev spaces, prove smoothing estimates consistent with parabolic behavior, and extend the methodology to other one-dimensional evolution equations including KdV and Schrödinger-type models. They also outline reductions to the half-line and discuss nonlinear extensions and inverse-problem perspectives, highlighting the broad applicability of the UTM to time-varying-coefficient IBVPs and related higher-order PDEs.

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.
Paper Structure (17 sections, 9 theorems, 173 equations, 8 figures)

This paper contains 17 sections, 9 theorems, 173 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Integral representation formula for weak solutions
  3. Derivation of representation formula
  4. Reductions
  5. Illustration of the solution
  6. Well-posedness in fractional Sobolev spaces
  7. Homogeneous Cauchy problem with extended data
  8. Reduced initial boundary value problem
  9. Local and global well-posedness results for the full ibvp
  10. Other evolution PDEs
  11. Linear KdV on the half-line
  12. Linear Schrödinger
  13. Biharmonic Schrödinger
  14. A fourth order PDE
  15. Some remarks
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $b=b(t)$ and $c=c(t)$ be two smooth functions on $[0,T]$ (or more globally on $\mathbb{R}$), where $b$ is bounded from below by a strictly positive constant. Let $u=u(x,t)$ satisfy the linear convection-diffusion model maineq on the domain $Q_T= \Omega\times (0,T)$. Assume $u$ is a sufficiently

Figures (8)

  • Figure 1: The contours of integration $\partial D_t^+$.
  • Figure 2: $u(x,t)$ by \ref{['sol-rep-heat-hl']} for $b(t)=\frac{6 \pi }{t+1}+1$ in the domain $(0,8)\times(0,6\pi)$
  • Figure 3: $u(4,t)$ by \ref{['sol-rep-heat-hl']} for $b(t)=\frac{6 \pi }{t+1}+1, \ t\in (0,12\pi)$.
  • Figure 4: $u(x,t)$ by \ref{['sol-rep-hl']} for $b(t)=1+2t$ and $c(t)=1$ in the domain $(0,8)\times(0,6\pi)$
  • Figure 5: $u(4,t)$ by \ref{['sol-rep-hl']} for $b(t)=1+2t$ and $c(t)=1, \ t\in (0,12\pi)$.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • Lemma 3.1: Space estimates and smoothing
  • proof
  • Remark 3.1
  • Lemma 3.2: Time estimate
  • proof
  • Remark 3.2
  • Lemma 3.3: Ibvp space estimate with zero drift
  • ...and 10 more