Laguerre Functions and Their Applications to Tempered Fractional Differential Equations on Infinite Intervals

Tempered fractional diffusion equations (TFDEs) involving tempered fractional derivatives on the whole space were first introduced in Sabzikar et al. (J Comput Phys 293:14–28, 2015 ), but only the finite-difference approximation to a truncated problem on a finite interval was proposed therein. In th...

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Bibliographic Details
Published in:Journal of scientific computing 2018-03, Vol.74 (3), p.1286-1313
Main Authors: Chen, Sheng, Shen, Jie, Wang, Li-Lian
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Basis functions
Computational Mathematics and Numerical Analysis
Derivatives
Differential equations
Fourier transforms
Fractional calculus
Integrals
Laguerre functions
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Polynomials
Theoretical
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Summary:Tempered fractional diffusion equations (TFDEs) involving tempered fractional derivatives on the whole space were first introduced in Sabzikar et al. (J Comput Phys 293:14–28, 2015 ), but only the finite-difference approximation to a truncated problem on a finite interval was proposed therein. In this paper, we rigorously show the well-posedness of the models in Sabzikar et al. ( 2015 ), and tackle them directly in infinite domains by using generalized Laguerre functions (GLFs) as basis functions. We define a family of GLFs and derive some useful formulas of tempered fractional integrals/derivatives. Moreover, we establish the related GLF-approximation results. In addition, we provide ample numerical evidences to demonstrate the efficiency and “tempered” effect of the underlying solutions of TFDEs.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-017-0495-7