A fast \(O(N\log N)\) finite difference method for the one-dimensional space-fractional diffusion equation
type: Item
instance of: scholarly article
title: A fast \(O(N\log N)\) finite difference method for the one-dimensional space-fractional diffusion equation
zbMATH Open document ID: 1337.65097
published in: Mathematics
publication date: 2016-01-12T00:00:00Z
review text: Summary: This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of \(O(N^2)\) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of \(O(N^3)\) per time step, where \(N\) is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of \(O(N)\) and a computational cost of \(O(N\log N)\) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 15B05
Mathematics Subject Classification ID: 65T50
zbMATH DE Number: 6528860
zbMATH Keywords: circulant and Toeplitz matrices
zbMATH Keywords: fast finite difference methods
zbMATH Keywords: fast Fourier transform
zbMATH Keywords: fractional diffusion equations
author: Treena S. Basu
MaRDI profile type: Publication
full work available at URL: math3041032
OpenAlex ID: W2137811217
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DOI: 10.3390/MATH3041032
description: scientific article; zbMATH DE number 6528860