We review the connection between inflationary models and observations and concentrate to describe models based on softly broken supersymmetry, in particular running mass models, and their predictions. We then present a fit of the spectral index of the curvature perturbation, assuming a flat ΛCDM cosmology.

A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as a set of non-overlapping cubic regions, plus a set of maps to identify the faces of adjoining regions. The differential structure on these manifolds is fixe...

In the past few years we have been involved in the development and implementation of parallel finite element and pseudo-spectral methods for porous media flow. In this paper we present a brief review of our methods and show several examples of highly nonlinear simulations in two and three-dimensions, on different current high performance computers. New issues on finite element techniques and ps...

An orthogonal system of rational functions is introduced. Some results on rational approximations based on various orthogonal projections and interpolations are established. These results form the mathematical foundation of the related spectral method and pseudospectral method for solving differential equations on the half line. The error estimates of the rational spectral method and rational p...

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multiorder fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is ve...

We introduce a novel spectral element method based on the ultraspherical and hierarchical Poincar\'{e}-Steklov scheme for solving second-order linear partial differential equations polygonal domains with unstructured quadrilateral or triangular meshes. Properties of lead to almost banded systems, allowing be competitive in high-polynomial regime ($p > 5$). The enables precomputed solution opera...

We study effects of finite height and surrounding material on photonic crystal slabs of oneand two-dimensional photonic crystals with a pseudo-spectral method and finite difference time domain simulation methods. The band gap is shown to be strongly modified by the boundary material. As an application we suggest reflection and guiding of light by patterning the material on top/below the slab.

We propose and analyze the spectral collocation approximation for the partial integrodifferential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomia...