Mathematical Sciences: Mathematical Problems in Various Applied Fields

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9401405 Keller Research will be conducted on a number of different problems in applied mathematics. The following problems will be studied: (1) the extension of a result deriving an exact nonreflecting boundary condition to electromagnetic waves governed by Maxwell's equations and to elastic waves governed by the linear theory of elasticity; (2) the development of a finite element formulation of scattering problems for the scalar wave equation incorporating the nonreflecting boundary condition; (3) the use of asymptotic methods to analyze the error at the nth step of an iterative numerical method; (4) the folding of elastic sheets and the coiling of ropes when they fall onto a horizontal surface, and (5) the study of resonantly interacting waves. Perturbation, asymptotic, and numerical methods will be developed and utilized to study a number of problems in applied mathematics. These problems will include the coiling of a rope falling on a horizontal plane, the folding of a sheet falling on a horizontal plane, the development of numerical methods employing exact nonreflecting boundary conditions which are nonlocal, the use of asymptotic methods to analyze the error in various iterative computational schemes, and the resonant interaction of nonlinear waves, such as gravity waves on the surface of a fluid.