Applied Mathematics deals with the mathematical underpinnings of important real-world problem.

Applied Mathematics Research

Inverse Problems

Inverse problems are a major area of both theoretical and computational research in the department. The theoretical analysis of such problems requires a solid knowedge of such areas of mathematics as partial differential equations and Fourier analysis. Aircraft acoustics and crack detection are just two applications investigated by research groups in the department.

(Bukhgeym, Isakov, Sun)

Acoustical Holography

Nearfield Acoustical Holography is the inverse problem in acoustics of computing the normal velocities on the boundary of a region from measurements of the acoustical pressure on an interior surface. Methods for solving this problem are important for the reduction of noise levels in airplane cabins. A group of several faculty and graduate students have developed such methods with support from the National Science Foundation as well as a local aircraft company. The figure shows the reconstruction of normal velocity in comparison with the original normal velocity distribution.

(DeLillo, Isakov, Hrycak)

Fluid Dynamics

Fluid flows display an amazing range of phenomena, from regular patterns to turbulence. Differential equations are used to model these various phenomena mathematically, and numerical methods can be used to investigate the behavior of fluid flows in various situations. In recent research at WSU a numerical investigation has been carried out to determine steady, axisymmetric vortex flows past a spherical body. Various types of solutions were found including vortex rings and nearly spherical regions of vorticity. The figure shows the meridional cross-sections for a vortex ring and for a nearly spherical vortex behind the sphere.

(Elcrat, Miller)

Mathematical Physics

Various geometries are used in modern physics. In addition to the classic Euclidean geometry, other geometries have been used; first Bolyai-Lobachevskian, then Riemannian, then Kleinian, and more recently, various combinations of the last two. The graphaic shows one of these: a geodesic surface in the Heisenberg group with an indefinite metric of signature (+--).

(Parker)