Mathematical Sciences: Asymptotic and Singular Perturbation Methods for Bifurcation and Moving Boundary Problems with Applications

点击翻译按钮获取中文摘要

With this award the principal investigator will continue his application of asymptotic methods of singular perturbation type to the study of bifurcation problems and moving boundary problems that arise from reaction-diffusion systems. In particular, he will investigate so-called singular bifurcation problems, wherein small changes in the bifurcation parameter can cause large responses in the system, laser instabilities, and the changing time history of the moving interface in certain pharmaceutical problems involving the time-release of medication. This proposal is concerned with bifurcation phenomena. As an illustration, stand a plastic ruler on its end and push on the top end. If you do not push very hard nothing happens. But if you push sufficiently hard the first thing that happens is that the ruler bows, either to the left or to the right. We call this a bifurcation: the configuration of the straight ruler has split (bifurcated) into one of two possible shapes. Applying still more force to the ruler results in an even number of bowed shapes having more and more curves. The principal investigator will study the number of solutions that bifurcate from a given solution of systems of equations that govern lasers and release of chemicals in the body.