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An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations

Artikel i vetenskaplig tidskrift
Författare Eugen Mamontov
Arvid Naess
Publicerad i International Journal of Engineering Science
Volym 47
Nummer/häfte 1
Sidor 116-130
ISSN 0020-7225
Publiceringsår 2009
Publicerad vid Institutionen för fysik (GU)
Sidor 116-130
Språk en
Länkar dx.doi.org/10.1016/j.ijengsci.2008....
Ämnesord Itô’s stochastic differential equation, transition probability density, analytical-numerical method, damping matrix, motion of ship in stochastic sea, stochastic rolling of ship
Ämneskategorier Matematisk analys, Matematisk fysik, Beräkningsfysik, Icke-linjär dynamik, kaos, Strömningsmekanik, Övrig teknisk mekanik, Övrig teknisk fysik

Sammanfattning

Probability densities for solutions of nonlinear Itô’s stochastic differential equations are described by the corresponding Kolmogorov-forward/Fokker-Planck equations. The densities provide the most complete information on the related probability distributions. This is an advantage crucial in many applications such as modelling floating structures under the stochastic load due to wind or sea waves. Practical methods for numerical solution of the probability-density equations are combined, analytical-numerical techniques. The present work develops a new analytical-numerical approach, the successive-transition (ST) method, which is a version of the path-integration (PI) method. The ST technique is based on an analytical approximation for the transition-probability density. It enables the PI approach to explicitly allow for the damping matrix in the approximation. This is achieved by extending another method, introduced earlier for bistable nonlinear reaction-diffusion equations, to the probability-density equations. The ST method also includes a control for the size of the time step. The overall accuracy of the ST method can be tested on various nonlinear examples. One such example is proposed. It is one-dimensional nonlinear Itô’s equation describing the velocity of a ship maneuvering along a straight line under the action of the stochastic drag due to wind or sea waves. Another problem in marine engineering, the rolling of a ship up to its possible capsizing is also discussed in connection with the complicated damping-matrix picture. The work suggests a few directions for future research.

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