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Local modification of subdiffusion by initial Fickian diffusion: Multiscale modeling, analysis and computation

Xiangcheng Zheng, Yiqun Li, Wenlin Qiu

TL;DR

The paper develops a local modification of the subdiffusion model by enforcing $\alpha(0)=0$, yielding a multiscale diffusion equation that transitions from Fickian behavior near the initial time to subdiffusion at later times while preserving the heavy-tail characteristics. A reformulation to a parabolic system with a time-dependent Volterra operator $B(t)$ enables rigorous well-posedness and high-order regularity analysis via resolvent estimates. It then constructs and analyzes a fully discrete finite element scheme, proving optimal-order error bounds and providing detailed stability and discretization error estimates. Numerical experiments in one spatial dimension confirm the theoretical convergence rates and demonstrate the model’s multiscale dynamics, validating both the analytical framework and the computational method. Overall, the work offers a robust mathematical and numerical treatment of variable-exponent fractional diffusion with local initial data, enabling accurate simulations of heterogeneous media with dual diffusion scales.

Abstract

We propose a local modification of the standard subdiffusion model by introducing the initial Fickian diffusion, which results in a multiscale diffusion model. The developed model resolves the incompatibility between the nonlocal operators in subdiffusion and the local initial conditions and thus eliminates the initial singularity of the solutions of the subdiffusion, while retaining its heavy tail behavior away from the initial time. The well-posedness of the model and high-order regularity estimates of its solutions are analyzed by resolvent estimates, based on which the numerical discretization and analysis are performed. Numerical experiments are carried out to substantiate the theoretical findings.

Local modification of subdiffusion by initial Fickian diffusion: Multiscale modeling, analysis and computation

TL;DR

The paper develops a local modification of the subdiffusion model by enforcing , yielding a multiscale diffusion equation that transitions from Fickian behavior near the initial time to subdiffusion at later times while preserving the heavy-tail characteristics. A reformulation to a parabolic system with a time-dependent Volterra operator enables rigorous well-posedness and high-order regularity analysis via resolvent estimates. It then constructs and analyzes a fully discrete finite element scheme, proving optimal-order error bounds and providing detailed stability and discretization error estimates. Numerical experiments in one spatial dimension confirm the theoretical convergence rates and demonstrate the model’s multiscale dynamics, validating both the analytical framework and the computational method. Overall, the work offers a robust mathematical and numerical treatment of variable-exponent fractional diffusion with local initial data, enabling accurate simulations of heterogeneous media with dual diffusion scales.

Abstract

We propose a local modification of the standard subdiffusion model by introducing the initial Fickian diffusion, which results in a multiscale diffusion model. The developed model resolves the incompatibility between the nonlocal operators in subdiffusion and the local initial conditions and thus eliminates the initial singularity of the solutions of the subdiffusion, while retaining its heavy tail behavior away from the initial time. The well-posedness of the model and high-order regularity estimates of its solutions are analyzed by resolvent estimates, based on which the numerical discretization and analysis are performed. Numerical experiments are carried out to substantiate the theoretical findings.
Paper Structure (19 sections, 9 theorems, 126 equations, 1 figure, 2 tables)

This paper contains 19 sections, 9 theorems, 126 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Modeling issues
  3. Analysis issues
  4. Preliminaries
  5. Notations
  6. Solution representation and resolvent estimates
  7. Well-posedness
  8. Model reformulation
  9. Auxiliary estimates
  10. Well-posedness of model (\ref{['VtFDEs']})
  11. Step 1: Well-posedness and contraction property of $\mathcal{M}$
  12. Step 2. Stability estimate of $u$
  13. High-order regularity estimate
  14. Numerical computation
  15. Numerical discretization
  16. ...and 4 more sections

Key Result

Lemma 2.1

\newlabellem:LpAkr If $f\in L^p(L^2)$ for $1<p<\infty$, problem (HeatPDE) has a unique solution $u\in W^{1,p}(L^2)\cap L^p(\check{H}^2)$ given by (HeatSoln) such that where $Q$ is independent of $f$, $t$ or $T$.

Figures (1)

  • Figure 1.1: Plots of the solutions $u(0.5,t)$ for three models under $\Omega=(0,1)$, $T=8$, $f\equiv 0$, $u_0=\sin(\pi x)$. Here the variable exponent is chosen as $\alpha(t)=\alpha(T) + (\alpha(0) - \alpha(T))(1 - \frac{t}{T} -\frac{\sin(2\pi (1-t/T))}{2\pi} )$, a smooth and monotonic function on $[0,T]$ with end values $\alpha(0) = 0$ and $\alpha(T) = 0.4$. We accordingly fix $\bar{\alpha}=0.4$.

Theorems & Definitions (17)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 7 more