Date of Award

Spring 2024

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical & Aerospace Engineering


Aerospace Engineering

Committee Director

Gene Hou

Committee Member

Brett Newman

Committee Member

Drew Landman

Committee Member

Hong Yang


Characterizing the behavior of dynamic systems requires the inclusion of initial conditions to propagate behavior forward in time. More realistic representations of system behavior quantify uncertainty about the initial conditions to assess sensitivity, reliability, and other stochastic response parameters. In many engineering applications, the uncertain initial conditions may be unknown given a desired response. This research applies the Fokker-Planck equation to reversible dynamic systems of select multi-dimensional nonlinear differential equations as a means for predicting the uncertainty about initial conditions. An alternating directions implicit numerical scheme is used to numerically solve the Fokker-Planck equation for both forward and reversed equations of motion. Initial conditions are predicted for a linear oscillating system, nonlinear trim solution, and atmospheric reentry equations. A use case is presented where the initial atmospheric entry conditions are predicted for a Mars reentry vehicle given a landing zone and parachute deployment response conditions. Monte Carlo simulations are also implemented to verify the outputs of the numerical scheme. Additional verification is conducted by comparing forward and reverse transient results of each problem set. It is shown that the initial conditions can be adequately predicted using the presented methodology. Computational resources quickly become a limitation as additional dimensions of variability are added to the problem.


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