A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: Strong solutions

Most available integration techniques for stochastically driven engineering dynamical systems are based on stochastic Taylor expansions of the response variables and thus require numerical modelling of multiple stochastic integrals (MSI-s). Since the latter is an extremely involved numerical task and becomes inaccurate for higher level MSI-s, these methods fail to achieve an accuracy beyond a limited order. Recently, the first author has proposed a locally transversal linearization (LTL) technique that completely avoids the use of Taylor-like expansions in the construction of the integration map [Proc. Roy. Soc. Ser. A, 457 (2001) 539; Int. J. Numer. Methods Eng. 61 (2004) 764]. A crucial step in the implementation of the LTL method is to arrive at a conditionally linearized solution which is then made to transversally intersect the non-linear solution manifold in the associated phase space. The present paper is the first part of an investigation consisting of two parts to considerably simplify and numerically expedite the generation of the conditionally linearized solution without affecting the local and global error orders of the original LTL method. In particular, the derivation of the conditionally linear form of the stochastic differential equations is done in such a way that the corresponding fundamental solution matrix (and consequently its inverse) remains unchanged during the entire integration process. In this part of the work, only strong (path wise) stochastic solutions are constructed. Through formal error estimates, it is verified that the present version of the LTL method has the same error orders as its older counterpart. Further, a host of numerical examples on stochastically driven non-linear oscillators are presented to illustrate its superior computational speed and ease of implementation. In a companion paper related to this work, a weak form of the alternative LTL approach will be derived.