Sentence examples for the elliptic coordinates from inspiring English sources

The elliptic coordinates.

To analytically study this eigenproblem, the elliptic coordinates and Mathieu functions are adopted.

The fundamental solution is expanded into the separable kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion.

By fully employing the elliptical geometry, fundamental solutions were expanded into the degenerate kernel by using an addition theorem in terms of the elliptic coordinates and boundary densities are described by using the eigenfunction expansion.

In the exterior domain, the 2D scattering problem is solved in the elliptic coordinate system using the Mathieu functions.

Potential function and complex function in the elliptic coordinate system are employed to solve the problem of scattering harmonic plane waves by multiple elliptic cavities in water saturated soil medium.

Then we obtain a new differential-difference integrable system with three-potential functions, whose algebraic-geometric solution is derived from the theory of algebraic curves, where we construct the new elliptic coordinates to straighten out the continuous and discrete flows by introducing the Abel maps as well as the Riemann-Jacobi inversion theorem.

The formulation in elliptic coordinates leads to an exact series solution involving Mathieu functions.

(14) Equation (14) is now in the form of the 2-dimensional wave equation in elliptic coordinates, and hence, the solutions (W_{n}) are in the form of the Mathieu and modified Mathieu equations as we mention in Section 2.2 for (q=-is_{n}=-inomegarho c^{2}{ 4mu}^{-1}).

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.

Analytical expressions for the single scatterings are obtained in the spacetime domain using elliptic coordinates.