Solution of the Pontriagin–Vitt equation for the moments of time to first passage of the randomly accelerated particle by the finite element method

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Abstract
The Pontriagin–Vitt equation governing the mean of the time of first passage of a randomly accelerated particles has been studied extensively by Franklin and Rodemich.1 In their paper is presented the analytic solution for the two‐sided barrier problem and solutions by several finite difference procedures.This note demonstrates solution of the problem by a Petrov–Galerkin finite element method using upstream weighting functions,2 shown to give rapidly convergent results. In addition, the equation is generalized to include higher statistical moments, and solutions for the first few ordinary moments are reported.

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