An analytical–numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations E. Mamontov a,b, * , A. Naess b,1 a Department of Physics, Faculty of Science, University of Gothenburg, SE 412 96 Gothenburg, Sweden b Centre for Ships and Ocean Structures and Department of Mathematical Sciences, The Norwegian University of Science and Technology, NO 7491 Trondheim, Norway article info Article history: Received 28 December 2007 Received in revised form 28 January 2008 Accepted 5 August 2008 Available online 24 September 2008 Communicated by K.R. Rajagopal Keywords: Itô’s stochastic differential equation Transition probability density Analytical–numerical method Damping matrix Motion of ship in stochastic sea Stochastic rolling of ship abstract Probability densities for solutions of nonlinear Itô’s stochastic differential equations are described by the corresponding Kolmogorov-forward/Fokker–Planck equations. The densi- ties 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 compli- cated damping matrix picture. The work suggests a few directions for future research. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Many phenomena analyzed in the natural sciences and engineering are described by solutions of nonlinear Itô’s stochastic differential equations (ISDEs) (e.g., [1–3]). In particular, may be specific applications refer to nonlinear behavior of ships and ocean structures under the stochastic-load effect due to sea waves or wind (e.g., [4–8]). A fairly general stochastic treatment describing sea waves with a random field can be found in [3, Chapter 8]. Minoura and Naito [9,10] present the results on the ISDE-based modelling of see waves which are less general but better suited to practical analysis. The nonlinear behavior of floating structures in the stochastic sea waves or wind is associated with stochastic systems which include components with both lumped and distributed characteristics. This area is one of many examples of engineering 0020-7225/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2008.08.001 * Corresponding author. Address: Department of Physics, Faculty of Science, University of Gothenburg, SE 412 96 Gothenburg, Sweden. Tel.: +46 (0) 31 7723489. E-mail addresses: eugen.mamontov@physics.gu.se, eugen.mamontov@ntnu.no (E. Mamontov). 1 Tel.: +47 73 597053. International Journal of Engineering Science 47 (2009) 116–130 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci