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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 18, Issue 4

Issues

Symbolic Algorithm of the Functional-Discrete Method for a Sturm–Liouville Problem with a Polynomial Potential

Volodymyr MakarovORCID iD: http://orcid.org/0000-0002-4883-6574 / Nataliia RomaniukORCID iD: http://orcid.org/0000-0002-3497-7077
Published Online: 2017-10-13 | DOI: https://doi.org/10.1515/cmam-2017-0040

Abstract

A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete method is developed and justified for the Sturm–Liouville problem on a finite interval for the Schrödinger equation with a polynomial potential and the boundary conditions of Dirichlet type. The algorithm of the general scheme of our method is developed when the potential function is approximated by the piecewise-constant function. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential polynomial and on the correction number. Our method uses the algebraic operations only and does not need solutions of any boundary value problems and computations of any integrals unlike the previous version. A numerical example illustrates the theoretical results.

Keywords: Eigenvalue Problem; Sturm–Liouville Problem; Polynomial Potential; Functional-Discrete Method; Symbolic Algorithm; Super-Exponentially Convergence Rate

MSC 2010: 65L15; 65L20; 65L70; 34B09; 34B24; 34L16; 34L20

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About the article

Received: 2017-03-18

Revised: 2017-08-20

Accepted: 2017-08-24

Published Online: 2017-10-13

Published in Print: 2018-10-01


Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 4, Pages 703–715, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0040.

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