Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Spectral Localization Method for Time-Fractional Integro-Differential Equations with Nonsmooth Data

Lijing Zhao, Rui Zhao, Wenyi Tian, Yufeng Nie

TL;DR

The paper addresses time-fractional integro-differential equations with nonsmooth data by introducing a spectral localization method that combines a contour integral method (CIM) in time with a Galerkin finite element discretization in space. It provides a rigorous analysis of well-posedness and error bounds, showing spectral accuracy in time and second-order spatial convergence even for nonsmooth initial data or vanishing data, and introduces acceleration techniques to reduce computational cost. Numerical experiments in 1D and 2D validate the theoretical results and demonstrate robustness across a range of fractional orders and kernel parameters. The approach enables efficient, parallelizable simulations of nonlocal, memory-driven processes and lays groundwork for extensions to nonlinear or semi-linear problems.

Abstract

In this work, we develop a localized numerical scheme with low regularity requirements for solving time-fractional integro-differential equations. First, a fully discrete numerical scheme is constructed. Specifically, for temporal discretization, we employ the contour integral method (CIM) with parameterized hyperbolic contours to approximate the nonlocal operators. For spatial discretization, the standard piecewise linear Galerkin finite element method (FEM) is used. We then provide a rigorous error analysis, demonstrating that the proposed scheme achieves high accuracy even for problems with nonsmooth/vanishing initial values or low-regularity solutions, featuring spectral accuracy in time and second-order convergence in space. Finally, a series of numerical experiments in both 1-D and 2-D validate the theoretical findings and confirm that the algorithm combines the advantages of spectral accuracy, low computational cost, and efficient memory usage.

A Spectral Localization Method for Time-Fractional Integro-Differential Equations with Nonsmooth Data

TL;DR

The paper addresses time-fractional integro-differential equations with nonsmooth data by introducing a spectral localization method that combines a contour integral method (CIM) in time with a Galerkin finite element discretization in space. It provides a rigorous analysis of well-posedness and error bounds, showing spectral accuracy in time and second-order spatial convergence even for nonsmooth initial data or vanishing data, and introduces acceleration techniques to reduce computational cost. Numerical experiments in 1D and 2D validate the theoretical results and demonstrate robustness across a range of fractional orders and kernel parameters. The approach enables efficient, parallelizable simulations of nonlocal, memory-driven processes and lays groundwork for extensions to nonlinear or semi-linear problems.

Abstract

In this work, we develop a localized numerical scheme with low regularity requirements for solving time-fractional integro-differential equations. First, a fully discrete numerical scheme is constructed. Specifically, for temporal discretization, we employ the contour integral method (CIM) with parameterized hyperbolic contours to approximate the nonlocal operators. For spatial discretization, the standard piecewise linear Galerkin finite element method (FEM) is used. We then provide a rigorous error analysis, demonstrating that the proposed scheme achieves high accuracy even for problems with nonsmooth/vanishing initial values or low-regularity solutions, featuring spectral accuracy in time and second-order convergence in space. Finally, a series of numerical experiments in both 1-D and 2-D validate the theoretical findings and confirm that the algorithm combines the advantages of spectral accuracy, low computational cost, and efficient memory usage.

Paper Structure

This paper contains 16 sections, 9 theorems, 86 equations, 8 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discretization of problem (\ref{['eq1.1']})
  3. Contour Integral Method (CIM)
  4. Temporal Semi-discrete CIM Scheme
  5. Spatial Semi-discrete Galerkin Scheme
  6. Fully Discrete Scheme and Acceleration of Algorithm
  7. Solution Theory and Error Analysis
  8. Solution Theory
  9. Error Analysis in Temporal Semi-discrete CIM Scheme
  10. Error Analysis of the Fully Discrete Scheme
  11. Numerical Results
  12. Scalar Problem
  13. Homogeneous Problems with Nonsmooth Initial Values in 1-D and 2-D
  14. Nonhomogeneous Problems with Vanishing Initial Values in 1-D and 2-D
  15. Nonhomogeneous Problem with Nonsmooth Solution in the 1-D Case
  16. ...and 1 more sections

Key Result

Lemma 1

Let $j\in\{0,1\}$, $m\in\mathbb{N}$, and denote $u^{(m)}(t) := \frac{\partial^m u(t)}{\partial t^m}, f^{(m)}(t) := \frac{\partial^m f(t)}{\partial t^m}$. The solution $u(t)$ of problem (eq1.1) satisfies the following properties.

Figures (8)

  • Figure 1: Illustration of the integration path transformation.
  • Figure 2: A schematic mapping relationship between the horizontal region $K$ in the $\phi$-plane and the region $S$ in the $z$-plane.
  • Figure 3: Numerical performances of the CIM for problem (\ref{['ex1']}).
  • Figure 4: Numerical performance for the 1-D homogeneous problem with nonsmooth initial (\ref{['eq4.2']}) at $t=0.4$ with $h=1/2^7$, $\alpha=0.6$, $\beta=0.75$, and $\Lambda=10$.
  • Figure 5: Reference solutions for the 1-D homogeneous problem with nonsmooth initial data (\ref{['eq4.2']}) at different times, with $h=1/2^7$, $\alpha=0.6$, $\beta=0.75$, and $\Lambda=1000$, $t_0=0.001$.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Lemma 1: Solution Theory
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Lemma 3: cf. Lemma 1 in LopezFernndez2004OnTN
  • Theorem 2
  • proof
  • ...and 9 more