A Frame Approach for Equations Involving the Fractional Laplacian: Constructing a Solver Based on Frame Properties

Core Concepts

The authors develop a spectral method using frame properties to solve equations involving the fractional Laplacian, achieving expected convergence rates.

Abstract

The content introduces a novel approach to solving equations with the fractional Laplacian using frame properties. It discusses the construction of a solver based on weighted classical orthogonal polynomials and their extensions. The method aims to achieve spectral convergence by expanding solutions in a family of functions that act as frames. Key results include proving convergence rates and applying the method to various examples.

Stats

We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, s ∈ (0, 1), on an unbounded domain in one or two dimensions.
The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to Rd, d ∈ {1, 2}.
We examine the frame properties of this family of functions for the solution expansion and derive an a priori estimate for the stationary equation.
Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation.

Quotes

"In this work, we develop a spectral method to solve equations of the form L[u] := K ∑ k=1 λk(−∆)sk[u] = f."
"We apply our solver to numerous examples including the fractional heat equation."
"Our goal is to construct a flexible spectral method for (1.1) posed on Rd, d ∈ {1, 2}, via a frame approach."

Key Insights Distilled From

A frame approach for equations involving the fractional Laplacian

by Ioan... at arxiv.org 03-01-2024

https://arxiv.org/pdf/2311.12451.pdf

Deeper Inquiries

How does utilizing frame properties enhance the efficiency and accuracy of solving equations involving the fractional Laplacian

Utilizing frame properties enhances the efficiency and accuracy of solving equations involving the fractional Laplacian in several ways. Firstly, by constructing a frame for the solution expansion, we can choose a flexible family of functions that best approximate the solution. This flexibility allows us to capture the algebraic decay or other characteristics of the solutions accurately. Additionally, by formulating the problem in terms of frames, we can efficiently compute expansions using truncated SVD projections. This approach ensures that we obtain optimal coefficients for approximating the solution while controlling errors effectively.

What are some potential limitations or challenges faced when implementing this spectral method in practical applications

Implementing this spectral method in practical applications may face some limitations and challenges. One major challenge is related to computational complexity, especially when dealing with high-dimensional problems or large datasets. The assembly of matrices and evaluation of hypergeometric functions at numerous collocation points can be computationally intensive. Moreover, ensuring convergence rates and accuracy as N (the number of basis functions) increases requires careful consideration of factors such as truncation tolerance ϵ and choice of collocation points xj.
Another limitation could arise from numerical stability issues when dealing with nonlocal operators like fractional Laplacians on unbounded domains. Artificial boundary layers may affect results if not handled properly during discretization processes.
Furthermore, adapting this method to handle time-dependent problems introduces additional complexities due to accumulating errors over time steps and maintaining stability in both spatial and temporal dimensions.

How can this novel approach be extended or adapted to solve other types of differential equations beyond those discussed in this content

This novel approach can be extended or adapted to solve various types of differential equations beyond those discussed in this context by modifying the choice of basis functions according to specific problem requirements. For instance:

Nonlinear Equations: By incorporating nonlinear terms into the equation formulation, one can explore how frame-based methods handle nonlinearity.
Variable Coefficients: Extending this approach to address equations with variable coefficients would involve adjusting the basis functions accordingly.
Higher Dimensions: Adapting this method for higher-dimensional problems involves selecting appropriate families of orthogonal polynomials suitable for multidimensional spaces.
Coupled Systems: Applying frame properties to coupled systems where multiple differential equations interact would require developing a framework that accounts for interdependencies among variables.
By customizing the selection and construction of frames based on different types of differential equations, one can enhance its applicability across diverse scientific fields requiring accurate numerical solutions for complex mathematical models.