A fractional spectral method with applications to some singular problems
type: Item
instance of: scholarly article
title: A fractional spectral method with applications to some singular problems
zbMATH Open document ID: 1382.65467
author: Dianming Hou
author: Chuanju Xu
published in: Advances in Computational Mathematics
publication date: 2017-12-06T00:00:00Z
review text: The authors introduce a novel type of projection operators that map into a class of generalized polynomials with non-integer exponents. Working in certain weighted Sobolev spaces, the mapping properties and -- in particular -- the approximation rates of these projections are investigated. These operators are then used as building blocks for solvers for fractional order differential, integral and integro-differential equations. It turns out that they can, for suitably chosen parameters, give rise to exponential convergence rates even in cases where the exact solutions exhibit the types of nonsmooth behaviour at the initial point that one usually finds in differential equations of fractional order. Moreover, it is shown that it is possible to employ such projection operators to more standard problems like a classical elliptic differential equation. In this case it may also happen that the given data lead to nonsmooth behaviour of the solution that can be compensated by the authors' approach in a way that drastically improves the convergence rate.
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65N35
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 34A08
zbMATH DE Number: 6816259
zbMATH Keywords: integro-differential equations
zbMATH Keywords: fractional differential equations
zbMATH Keywords: singularity
zbMATH Keywords: Müntz polynomials
zbMATH Keywords: fractional Jacobi polynomials
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: spectral accuracy
zbMATH Keywords: convergence
zbMATH Keywords: fractional spectral method
describes a project that uses: OPQ
MaRDI profile type: Publication
full work available at URL: s10444-016-9511-y
OpenAlex ID: W2571966495
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DOI: 10.1007/S10444-016-9511-Y
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