Pseudospectral methods for solving an equation of hypergeometric type with a perturbation

✍ Scribed by H. Alıcı; H. Taşeli

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 430 KB
• Volume: 234
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2009.06.004

✦ Synopsis

Almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrödinger form

for a wide class of potentials V (x) may be transformed into the form

by means of intelligent transformations on both dependent and independent variables, where σ (ξ ) and τ (ξ ) are polynomials of degrees at most 2 and 1, respectively, and λ is a parameter. The last form is closely related to the equation of the hypergeometric type (EHT), in which Q (ξ ) is identically zero. It will be called here the equation of hypergeometric type with a perturbation (EHTP). The function Q (ξ ) may, therefore, be regarded as a perturbation. It is well known that the EHT has polynomial solutions of degree n for specific values of the parameter λ, i.e. λ := λ (0) n = -n[τ + 1 2 (n -1)σ ], which form a basis for the Hilbert space L 2 (a, b) of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of EHTP, and hence the energies E of the original Schrödinger equation. Specimen computations are performed to support the convergence numerically.