An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.

Virtual Engineering (VE), also known as Model-Based Systems Engineering (MBSE), is necessary in both current operational engineering qualifications and to help reduce the costs of future vertical lift design and analysis. As computational power continues to provide increasing capability to the rotorcraft engineering community to perform simulations in both real time and off line, it is imperative that the community develop verification and validation protocols and processes to certify these methods so that they can be reliably used to help reduce engineering cost and schedule. Computational Fluid Dynamics (CFD) has become a major Computational Science and Engineering (CSE) tool in the fixed wing and vertical lift communities, but it has not been developed to the point where it is accepted as a replacement for testing in certification of new or existing systems or vehicles. Since the rise of modern CFD in the 1980s, the promise of CFD’s capabilities has been met or exceeded, but its role in certification arguably remains less prominent than projected. The ability to implement transformative technologies further drives the need for CFD in design. To meet CFD’s role in certification, several goals must be met to provide a true “numerical experiment” from which accuracies (error estimates), sensitivities, and consistent application results can be extracted. This paper discusses the progress and direction towards developing CFD strategies for certification.