Strong 1.5 order scheme for fractional Langevin equation based on spectral approximation of white noise

A full-discrete scheme is presented for a Langevin equation involving Caputo fractional derivative and additive white noise. Based on a spectral truncation of white noise, the fractional Langevin equation is converted to an approximate equation with random parameters and a finite difference scheme i...

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Bibliographic Details
Published in:Numerical algorithms 2024, Vol.95 (1), p.423-450
Main Authors: Wang, Yibo, Cao, Wanrong
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Computer Science
Numeric Computing
Numerical Analysis
Original Paper
Theory of Computation
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Summary:A full-discrete scheme is presented for a Langevin equation involving Caputo fractional derivative and additive white noise. Based on a spectral truncation of white noise, the fractional Langevin equation is converted to an approximate equation with random parameters and a finite difference scheme is constructed. Consistency of the approximate equation as well as error estimate of the finite difference scheme are obtained. It is proved that when spectral truncation level and the step size are inversely proportional, the convergence order of the finite difference scheme is 1.5 in mean-square sense and independent of the value of fractional derivative order. Numerical examples verify the theoretical analysis. Moreover, to fulfill long-time simulation, a scheme based on piecewise spectral approximation of white noise is further developed.
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-023-01576-z