Dr. Alan Whitman
Research
Over a number of years I have contributed to work that studies how a scalar field propagates in space filled with a medium whose propagation speed is randomly varying spatially. Applications of this work occur in Underwater Acoustics and in Optics when light propagates through a turbulent atmosphere. More recently, on accounting for temporal as well as spatial fluctuations, backscattering has been included and its effect on the propagation was elicited. At present we are trying to extend the work to include the vector, electromagnetic field. The physical application here is to thermal radiation which occurs in the atmospheric warming problem.
Asymptotic approximations to differential equations have a long and rich history going back to Prandtl's analysis of the fluid mechanical boundary layer and Poincare's studies of celestial mechanics. Since these early works there have been myriad applications of the techniques. Over the past several years I have been working on a number of problems using these methods.
An infectious disease process in a human body can be modeled as a predator-prey system between a pathogen population and an immune system. We analyzed a particular system that has been proposed, and obtained a solution that could be used to generate insights about the system behavior, organize data, and suggest new measurements.
An inverted pendulum can be stabilized by moving its base on the arc of a circle in a horizontal plane according to the dictates of a particular sliding mode control law. We analyzed this nonlinear system and obtained a solution that provided insight on how to choose the control parameters for optimal performance.
The equation of time quantifies the difference between time as measured by a sundial and time as measured by a mechanical clock. That this difference exists has been known since astronomers began making these measurements and comparing them. Moreover, since Newton showed how planetary orbits could be analyzed mathematically, it has been possible to calculate the difference as well. However, the calculation is complex and can not be represented by a simple explicit analytical expression. This work presents some asymptotic approximations to the known basic relations that produce a simple explicit formula for the equation of time that is accurate enough to be useful for correcting sundials.The entire paper is available here as a PDF file. A shorter version can be found in the Mathematical Details section of the Wikipedia article on the Equation of Time.
Copyright 1999 Villanova University
Department of Mechanical Engineering
Last updated 3/16/2016