Solvability of Nonlinear Singular Problems for Ordinary Differential Equations

Two very accurate and efficient methods for computing eigenvalues of Sturm–Liouville problems

Sturm-Liouville problems are abundant in the numerical treatment of scientific and engineering problems. In the present contribution, we present an efficient and highly accurate method for computing eigenvalues of singular Sturm-Liouville boundary value problems. The proposed method uses the double exponential formula coupled with sinc collocation method. This method produces a symmetric positive-definite generalized eigenvalue system and has exponential convergence rate. Numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula.

The double exponential sinc collocation method for singular Sturm-Liouville problems

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Analysis of Singular Initial and Terminal Value Problems

We provide an effective finite element method to solve the Schrodinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from the interior of the domain to its boundary. Then by using orthogonal properties of spherical harmonic functions and variable separation technique we transform the original problem into a series of one-dimensional eigenvalue problems. We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. Combining with the spectral theory of Babuska and Osborn for self-adjoint positive definite eigenvalue problems, we obtain error estimates of approximation eigenvalues and eigenvectors. Finally, we provide some numerical examples to show the efficiency and accuracy of the algorithm.

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An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain

A modification of the central-difference method is given which greatly improves the convergence when applied to a certain class of singular eigenvalue problems, including the Klein-Gordon equation. The singularity given special treatment is at the finite end.

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G. W. Reddien

Papers

Finite-difference approximations to singular Sturm-Liouville eigenvalue problems