Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

Boundary methods for mixed boundary problems of Laplace׳s equation in elliptic domains with elliptic holes☆

Abstract: The particular solutions (PS) and fundamental solutions (FS) in polar coordinates can be found in many textbooks, but with much less coverage in elliptic coordinates (Chen et al., 2010 [5] , Chen et al., 2012 [6] , Morse and Feshbach, 1953 [20] , Li et al., 2015 [18] ). Since the elliptic domains with elliptic holes may be found in some engineering problems, the PS and the FS expansions in elliptic coordinates are essential for numerical computations. For Dirichlet problems of Laplace׳s equation in elliptic domains, the null field method (NFM), the interior field method (IFM) and the collocation Trefftz method (CTM) are reported in [18] . There seems to exist few reports for mixed problems, where the Dirichlet and Neumann conditions are assigned on the exterior and the interior boundaries, simultaneously. This paper is devoted to such mixed problems by the NFM and the IFM, and the explicit algebraic equations are derived for elliptic domains. Besides, other effective particular solutions (PS) are sought, and the collocation Trefftz method (CTM) [16] is employed. The CTM may be used for Robin problems in elliptic domains. The effective algorithms for the mixed problems of Laplace׳s equation on elliptic domains are the main goal of this paper. The techniques of the mixed techniques in this paper can be applied to Dirichlet problems, the dual techniques are called in Chen and Hong (1999 [4] ), Hong and Chen (1988 [8] ), and Portela et al. (1992 [21] ). A preliminary study for the dual techniques is one goal of this paper.

Chapter 6 – Coordinate Transformations

Abstract: This chapter presents the mathematical methodology of using tensors to transform a Cartesian coordinate system to other types of coordinate systems. The emphasis is on the development of generalized vertical coordinate systems including those based on pressure, potential temperature, and terrain-following. Specific examples of how the coordinate system can facilitate improved understanding of atmospheric flows is presented.

Algebro-Geometric Solutions to Fokas-Lenells Equation

Abstract: The Fokas-Lenells equation is given by the Lenard gradient sequence for a given spectral problemThen,this equation is decomposed into solvable ordinary differential equationsBased on the finite-order expansion of the Lax matrix,elliptic coordinates are introducedSo,the flow can be straighten out by the Abel-Jacobi coordinatesAt the end, algebro-geometric solutions of the Fokas-Lenells equation are presented by means of the Riemannθfunction

Natural Coordinate System in Curved Space-time

Abstract: In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be preseted according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it "natural coordinate system". The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In NCS we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein's equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is a nice coordinate system to describe the realistic curved space-time, and is helpful to understand the nature of space-time.