Rational approximation to the fractional Laplacian operator in reaction-diffusion problems

TL;DR

This paper introduces a novel numerical method for efficiently approximating the fractional Laplacian in reaction-diffusion equations, transforming dense matrices into sparse banded matrices to enable faster computations on bounded domains.

Contribution

The paper presents a new matrix approximation technique that simplifies the fractional Laplacian operator, making numerical solutions more computationally efficient.

Findings

Effective approximation of the fractional Laplacian with banded matrices

Reduced computational complexity due to matrix sparsity

Numerical results confirm the method's accuracy and efficiency

Abstract

This paper provides a new numerical strategy to solve fractional in space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transform method the fractional Laplacian operator is replaced by a matrix which, in general, is dense. The approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices. This leads to a semi-linear initial value problem in which the matrices involved are sparse. Numerical results are presented to verify the effectiveness of the proposed solution strategy.