The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains
type: Item
instance of: scholarly article
title: The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains
zbMATH Open document ID: 1447.65103
author: Yanyan Li
published in: Numerical Algorithms
publication date: 2019-11-27T00:00:00Z
review text: The paper studies the numerical solution of the following time-fractional sub-diffusion equations (FSDEs) on unbounded domains: \[ _0^C\partial_t^{\alpha}u(x,t)-\partial_x^{2}u(x,t)+u(x,t)=f(x,t),\qquad x \in \mathbb{R},\, t\in \Lambda, \] subject to the following conditions: \[ \lim\limits_{|x|\to \infty} u(x,t) = 0,\quad u(x,0)=\phi(x), \] where \(0 < \alpha < 1\) and \(\Lambda = (0,+\infty)\). The authors use the Laguerre-Hermite spectral Galerkin and collocation methods to solve the above problem. The generalized associated Laguerre functions of the first kind (GALFs-1) and Hermite functions (HFs) are introduced as basis functions in time and spatial space, respectively. Some basic properties and definitions of the confluent hypergeometric functions, generalized associated Laguerre functions, Hermite functions, fractional calculus, and Laplace transform are given. In the time domain, the fractional derivative of the basis functions has derived in the Galerkin method by using the Laplace transformation. The collocation method expands the solution in terms of suitable global basis functions. The formulation of the Laguerre-Galerkin method and the spectral collocation methods based on GALFs-1 are given and the convergence behavior of numerical solutions is tested by some simple test problems. The error analysis is carried out separately on time and space domains, and also the time-space error is estimated. Lastly, some numerical results are given to validate the methods.
reviewed by: Srinivasan Natesan
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 7137009
zbMATH Keywords: spectral Galerkin method
zbMATH Keywords: spectral collocation method
zbMATH Keywords: generalized associated Laguerre functions
zbMATH Keywords: Hermite function
zbMATH Keywords: time-fractional sub-diffusion equation
zbMATH Keywords: unbounded domain
MaRDI profile type: Publication
full work available at URL: s11075-018-00652-z
OpenAlex ID: W2908039797
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The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains (DOI)
DOI: 10.1007/S11075-018-00652-Z

The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains (describes a project that uses)
describes a project that uses: Digital Library of Mathematical Functions

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