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 -