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.

Use of Exterior Differential Forms to Express the Time Dependent Diffusion Equation in Any Orthogonal Coordinate System

Abstract: This paper illustrates the applicability of exterior differential forms to reactor theory by using them to represent the time dependent diffusion equation suitable for neutronic calculations. This formalism allows one to express the equation straightforwardly and easily in any orthogonal coordinate system. Because of present interest in CTR designs an application is made to a toroidal geometry with a general elliptical cross section.

Quasi-periodic solutions for an extension of AKNS hierarchy and their reductions

Abstract: Quasi-periodic solutions of an extension of the AKNS hierarchy are derived. Based on finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the equations are separated into solvable ordinary differential equations. Then various flows are straightened out through the Abel–Jacobi coordinates. By the standard Jacobi inversion treatment, explicit quasi-periodic solutions of the evolution equations are constructed in terms of the Riemann theta functions. Furthermore, the solutions of a new generalized nonlinear Schrodinger equation, which are the reductions of the above system, are deduced.

The airy stress function in curvilinear coordinates with application to the uniform flexure of a naturally curved spiral beam

Abstract: In problems of plane elasticity, in the absence of body forces, the stresses are derivable from a scalar function known as the Airy stress function. By expressing this relation as a tensor equation, the use of the Airy function is generalized for any plane, orthogonal coordinate system. A general expression for the compatibility equation in terms of the stress function is given. It is found that considerable simplification results if isometric coordinates are used. The geometry of the isometric, curvilinear coordinate system is determined by a conformal mapping of the rectangular, Cartesian coordinate plane upon the curvilinear coordinate plane. For a given coordinate system the components of the metric tensor are expressible in terms of the mapping function. As an application of the preceding theory the uniform flexure of a beam whose edges are bounded by logarithmic spirals is discussed. It is found that all spiral beams may be specified by two dimensionless shape parameters. Stress distributions in a typical beam are exhibited. As limiting cases of the solution for a spiral beam, the uniform flexure of a wedge acted upon by a moment at the vertex and the uniform flexure of a sector of a circular ring are obtained.