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

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Abstract

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.

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Acknowledgements

The authors would like to thank Prof. Guofei Pang for his valuable comments and suggestions.

Funding

This work was supported by the National Natural Science Foundation of China (No. 12071073). The first author was also supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province, China (No. KYCX22_0235).

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Yibo Wang & Wanrong Cao

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  1. Yibo Wang
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  2. Wanrong Cao
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Wanrong Cao: Writing — reviewing and editing, funding acquisition, conceptualization, resources, supervision. Yibo Wang: writing — original draft, software, formal analysis.

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Correspondence to Wanrong Cao.

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Wang, Y., Cao, W. Strong 1.5 order scheme for fractional Langevin equation based on spectral approximation of white noise. Numer Algor 95, 423–450 (2024). https://doi.org/10.1007/s11075-023-01576-z

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