Mathematical Sciences: Some Nonlinear Problems In Differential and Integral Equations

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In this project the principal investigators will analyze four problems involving nonlinear differential and integral equations. The first problem is an integro-differential equation that describes the entry of a wedge into a bath of water. The second problem involves the use of homotopy methods to study the existence of travelling wave solutions of integral and differential equations that arise in applications. The third problem involves the use of shooting methods to study the behavior of chaotic orbits. And finally the fourth problem concerns finding similarity solutions of some two-dimensional versions of the Navier-Stokes equations. Many of the mathematical models that describe the behavior of everyday phenomena involve integral equations or differential equations or combinations of these two types of equations. Very often it is easiest to study the solutions of such equations qualitatively, that is, instead of looking for exact analytical or numerical solutions that can be quite complicated, one looks for approximate representations of the solutions that contain, more or less, the essence of the exact solution. In this project the principal investigators will use qualitative methods to study the solutions of integral and differential equations that arise in fluid dynamics, biology and mechanics.