Details
| Original language | English |
|---|---|
| Article number | 100436 |
| Journal | Theoretical and Applied Mechanics Letters |
| Volume | 13 |
| Issue number | 3 |
| Early online date | 24 Feb 2023 |
| Publication status | Published - May 2023 |
Abstract
Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.
Keywords
- Analytical intrinsic drift coefficient, Dimension reduction, Globally-evolving-based generalized density evolution equation (GE-GDEE), Linear fractional differential system, Non-Markovian system
ASJC Scopus subject areas
- Engineering(all)
- Computational Mechanics
- Environmental Science(all)
- Environmental Engineering
- Engineering(all)
- Civil and Structural Engineering
- Engineering(all)
- Biomedical Engineering
- Engineering(all)
- Aerospace Engineering
- Engineering(all)
- Ocean Engineering
- Engineering(all)
- Mechanics of Materials
- Engineering(all)
- Mechanical Engineering
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In: Theoretical and Applied Mechanics Letters, Vol. 13, No. 3, 100436, 05.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise
AU - Luo, Yi
AU - Lyu, Meng Ze
AU - Chen, Jian Bing
AU - Spanos, Pol D.
N1 - Funding Information: The supports of the National Natural Science Foundation of China (Grant Nos. 51725804 and U1711264 ), the Research Fund for State Key Laboratories of Ministry of Science and Technology of China (SLDRCE19-B-23), and the Shanghai Post-Doctoral Excellence Program (2022558) are highly appreciated.
PY - 2023/5
Y1 - 2023/5
N2 - Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.
AB - Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.
KW - Analytical intrinsic drift coefficient
KW - Dimension reduction
KW - Globally-evolving-based generalized density evolution equation (GE-GDEE)
KW - Linear fractional differential system
KW - Non-Markovian system
UR - http://www.scopus.com/inward/record.url?scp=85150066655&partnerID=8YFLogxK
U2 - 10.1016/j.taml.2023.100436
DO - 10.1016/j.taml.2023.100436
M3 - Article
AN - SCOPUS:85150066655
VL - 13
JO - Theoretical and Applied Mechanics Letters
JF - Theoretical and Applied Mechanics Letters
IS - 3
M1 - 100436
ER -