Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On time-fractional partial differential equations of time-dependent piecewise constant order

Yavar Kian, Marián Slodička, Éric Soccorsi, Karel Van Bockstal

TL;DR

This paper addresses the well-posedness of time-fractional diffusion with a time-dependent order $\beta(t)$ that is piecewise constant with a finite number of jumps. It develops a Fourier-based approach that leverages the spectral decomposition of the elliptic operator and the Mittag-Leffler representation for constant-order problems to handle each subinterval, then assembles a global solution. Under suitable data regularity and jump-compatibility conditions, the authors prove existence and uniqueness of a solution in $\mathcal{C}^0(\overline{I},D(\mathcal{L}))\cap W^{1,1}(I,\mathscr{H})$ with explicit data-dependent energy estimates. The results extend constant-order theories to a finite-jump variable-order setting, accommodating non-monotone, non-positive kernels and offering a framework relevant to viscoelastic and regime-switching models, while noting limitations to finite jumps and piecewise-constant $\beta(t)$.

Abstract

This contribution considers the time-fractional subdiffusion with a time-dependent variable-order fractional operator of order $β(t)$. It is assumed that $β(t)$ is a piecewise constant function with a finite number of jumps. A proof technique based on the Fourier method and results from constant-order fractional subdiffusion equations has been designed. This novel approach results in the well-posedness of the problem.

On time-fractional partial differential equations of time-dependent piecewise constant order

TL;DR

This paper addresses the well-posedness of time-fractional diffusion with a time-dependent order that is piecewise constant with a finite number of jumps. It develops a Fourier-based approach that leverages the spectral decomposition of the elliptic operator and the Mittag-Leffler representation for constant-order problems to handle each subinterval, then assembles a global solution. Under suitable data regularity and jump-compatibility conditions, the authors prove existence and uniqueness of a solution in with explicit data-dependent energy estimates. The results extend constant-order theories to a finite-jump variable-order setting, accommodating non-monotone, non-positive kernels and offering a framework relevant to viscoelastic and regime-switching models, while noting limitations to finite jumps and piecewise-constant .

Abstract

This contribution considers the time-fractional subdiffusion with a time-dependent variable-order fractional operator of order . It is assumed that is a piecewise constant function with a finite number of jumps. A proof technique based on the Fourier method and results from constant-order fractional subdiffusion equations has been designed. This novel approach results in the well-posedness of the problem.
Paper Structure (20 sections, 2 theorems, 80 equations)

This paper contains 20 sections, 2 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Highlights of the paper
  3. Formulation of the problem
  4. Results
  5. Definitions and notations
  6. Main result
  7. Outline and comments
  8. Proof of Theorem \ref{['thm-main']}
  9. The Fourier method
  10. Auxiliary result
  11. End of the proof of Theorem \ref{['thm-main']}
  12. Proof of Proposition \ref{['pr1']}
  13. Inductive step
  14. Proof of (i)
  15. Step 1
  16. ...and 5 more sections

Key Result

Theorem 2.2

Let $u_0 \in D(\mathcal{L})$, let $f \in W^{1,1} \left( \cup_{j=0}^{M-1} I_j,\mathscr{H} \right)$, and assume that for all $j=0,\ldots,M-1$, Then, the IBVP eq:problem admits a unique solution in the sense of Definition def-sol. Moreover, there exists a positive constant $C$, depending only on $\Omega$, $T$ and $\{ (t_j,\beta_j, \varepsilon_j),\ j=0,\ldots,M-1 \}$, such that

Theorems & Definitions (3)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 3.1