Singular Sturm-Liouville Equation Research Articles

A Spectral Method for Polar Coordinates

A new set of polynomial functions that can be used in spectral expansions of C∞ functions in polar coordinates (r, φ) is defined by a singular Sturm-Liouville equation. With the use of the basis functions, the spectral representations remain analytic at the pole despite the coordinate singularity because the pole condition is exactly satisfied at the origin for ag azimuthal modes, not just a few of the gravest modes (which is the usual case). Based on recurrence relations, fast and stable numerical operators for 1/r, r(d/dr), the Laplacian and Helmholtz operators and their inverses are developed. Although the spacings in the azimuthal direction of the collocation points near the origin are small (i.e., α 1/M2, where M is the number of radial modes), the explicit numerical method for Euler's equation is not stiff at the origin. Namely, the CFL number σ is O(1) where the grid size in is defined as π/M (i.e., the maximum allowable timestep is proportional to 1/M, not 1/M2).

Sturm-Liouville operators with an indefinite weight function

SynopsisSpectral properties of the singular Sturm-Liouville equation –(p−1y′)′ + qy = λry with an indefinite weight function r are studied in . The main tool is the theory of definitisable operators in spaces with an indefinite scalar product.

Existence and Uniqueness of Solutions to a Fourth Order Nonlinear Eigenvalue Problem

The Euler equations for an isoperimetric eigenvalue problem consist of a singular Sturm–Liouville equation coupled to a second order nonlinear ordinary differential equation. An existence proof for solutions to the isoperimetric problem is given, thereby proving the existence of solutions to the Euler equations. Next, it is proved that for each eigenvalue any solution to these equations provides a weak relative minimum of the functional for the isoperimetric problem. The proof that this solution is unique follows from the strict convexity of this functional with respect to all admissible functions.

On a Problem of Weyl in the Theory of Singular Sturm-Liouville Equations

On a Problem of Weyl in the Theory of Singular Sturm-Liouville Equations

On the properties of a singular Sturm-Liouville equation determined by its spectral functions.

On the properties of a singular Sturm-Liouville equation determined by its spectral functions.