Petrov-Galerkin finite element solution for the first passage probability and moments of first passage time of the randomly accelerated free particle

A numerical solution to an initial boundary value problem governing the probability of failure of a randomly accelerated free particle is obtained using a Petrov-Galerkin finite element method. This direct solution is the first successful one, and no others have been reported in the literature. A solution of the Pontriagin-Vitt equation for the time to first passage of the particle is obtained first: in this case an analytical solution is available and used to evaluate the numerical algorithm. Extensions to the solution of other stochastic differential equations, in particular those governing the probability of failure of the linear oscillator, and applications to structural dynamics are discussed.