Pseudospectral Methods in Applied Mathematics

点击翻译按钮获取中文摘要

( ) Refs in CCV; [ ] Research Refs.***This proposal emphasizes the use of pseudospectral methods based on nonclassical quadrature grids [1] applied to the solution of important problems in statistical mechanics (4,9-11), quantum chemistry (1,7) and physics (6), mathematical biology and image reconstruction. A pseudospectral method refers to a numerical solution of PDEs and Integral eqs. (IE) on a grid [1]. The popular choices are the uniform Fourier grid or the nonuniform Chebyshev quadrature. The disadvantage of a Fourier grid is the Gibbs phenomena which is the non-spectral convergence of the Fourier series for functions at jump discontinuties. The Gibbs phenomenon contaminates images in all forms of tomography. A major ongoing objective is to develop a method of identifying the "edges" where the oscillations occur and to employ the previous Gibbs resolution techniques [23,24] to resolve the images. Likewise, the interpolation of data on a uniform grid leads to large oscillations at the ends of the interval; the Runge phenomenon (RP). The RP can be resolved with nonuniform nonclassical quadratures with spectral accuracy. ******The topics in mathematical biology include the advance of a gene in a population modeled with Fishers equation and spiral waves in cardiac tissue studied with the Fitzhugh-Nagumo equation. The approach to equilibrium for nonequilibrium systems are studied with solutions of the Fokker Planck equation (FPE) (4) and the Boltzmann equation (BE) (1,11). The BE is also used to model the the high altitude regions of the atmospheres of Earth and Mars in comparison with available satellite data from enhanced Polar Outflow Probe (ePOP) and MAVEN. The resolution of images for tomography and magnetic resonance images is considered based on previous work for the resolution of the Gibbs phenomenon, the nonspectral convergence of a Fourier series of piecewise smooth functions. The basis for the modeling these different phenomena are multidimensional partial differential (PDE) and integral (IE) equations. Fishers reaction-diffusion equation [27], which models the advance of a gene, has localized traveling waves and solutions that are difficult to resolve. Spiral waves in cardiac tissue that are suspected to give rise to cardiac arrhythmias [28] are studied with the two-dimensional Fitzhugh-Nagumo equation with different ionic membrane models. For plasmas and globular clusters, the BE is approximated with a FPE which with particular diffusion coefficients yields a steady Kappa distribution used extensively in space science. A Poisson solver [15] is required to define the drift and diffusion coefficients in the FPE for plasmas. Ffficient Poisson solvers for Poisson's equation are required in fluid dynamics, plasma physics and cosmology. Efficient and accurate numerical methods are required to solve the Schroedinger equation (SE) (6) to describe many chemical and physical phenomenon. *** *** ******