Mathematical Sciences: Research in Differential Equations

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9302737 Hastings This project concerns investigations into mathematical problems of nonlinear ordinary differential equations. There are six areas under investigation. They include asymptotics and painleve transcendents involving solutions obtained by inverse scattering, shooting methods applied to the study of chaos, specially in the analysis of Lorenz equations, and self-similar solutions of Barenblatt's fourth order model for the decay of turbulent bursts. Work will also be done expanding recent studies of stability in modeling solid state diffusion in semiconductors, asymptotics beyond all orders in fifth order equations and on the smoothing of Stokes discontinuities in the asymptotic expansions of analytic functions. Differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***