In this project the principal investigator will work with the postdoctoral associate Patricia Lewis on the asymptotic analysis of certain classes of two-dimensional integral equations that contain a large parameter. The main thrust of the research will be to develop new methods for finding asymptotically correct approximate solutions of the equations as the large parameter approaches infinity. Such problems arise in investigations of the behavior of waves propagating in two dimensions in the limit of large wavenumber. This research will be especially valuable because numerical solutions of the integral equations are not reliable as the parameter becomes large. Asymptotic methods work because the largeness or smallness of a parameter in an equation can be used to simplify the equation to the point where an approximate solution, which is still meaningful, can be found more easily. In this project the principal investigator and a postdoctoral associate will use asymptotic methods to find accurate approximate solutions of integral equations that model the behavior of waves propagating in the limit of large wavenumber.
