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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:234

Article; Proceedings Paper

Alici, H.¹  Taseli, H.¹ 

[1] Middle E Tech Univ, Dept Math, TR-06531 Ankara, Turkey

关键词: Schrodinger operator;    Regular and singular Sturm-Liouville eigenvalue problems;    Pseudospectral methods;    Equation of hypergeometric type;    Classical orthogonal polynomials;   

DOI  :  10.1016/j.cam.2009.06.004

来源: Elsevier

【 摘 要 】

Almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrodinger form -Psi '' (x) + V(x) Psi (x) = E Psi (x). x is an element of ((a) over bar, (b) over bar) subset of R, Psi(x) is an element of L-2 ((a) over bar, (b) over bar) for a wide class of potentials V(x) may be transformed into the form s(xi)y '' + tau(xi) y' + Q (xi)y = -lambda y. xi is an element of (a, b) subset of R by means of intelligent transformations on both dependent and independent variables, where sigma(xi) and tau(xi) are polynomials of degrees at most 2 and 1, respectively, and lambda is a parameter. The last form is closely related to the equation of the hypergeometric type (EHT), in which Q(xi) is identically zero. It will be called here the equation of hypergeometric type with a perturbation (EHTP). The function Q(xi) 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 lambda, i.e. lambda : = lambda((0))(n) = -n vertical bar tau' + 1/2 (n - 1) sigma ''vertical bar, 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 Schrodinger equation. Specimen computations are performed to support the convergence numerically. (C) 2009 Elsevier B.V. All rights reserved.

【 授权许可】

Free