Spectral method for solving the nonlinear thomas-fermi equation based on exponential functions

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work.

Original languageEnglish
Article number168568
JournalJournal of Applied Mathematics
Volume2014
DOIs
Publication statusPublished - 1 Jan 2014

Fingerprint

Exponential functions
Spectral Methods
Ordinary differential equations
Nonlinear systems
Atoms
Infinite Domain
Numerical integration
System of equations
Rate of Convergence
Ordinary differential equation
Robustness
Range of data

Cite this

@article{2325ed4e244a43d89f3af01e9b337c34,
title = "Spectral method for solving the nonlinear thomas-fermi equation based on exponential functions",
abstract = "We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work.",
author = "Raka Jovanovic and Sabre Kais and Fahhad Alharbi",
year = "2014",
month = "1",
day = "1",
doi = "10.1155/2014/168568",
language = "English",
volume = "2014",
journal = "Journal of Applied Mathematics",
issn = "1110-757X",
publisher = "Hindawi Publishing Corporation",

}

TY - JOUR

T1 - Spectral method for solving the nonlinear thomas-fermi equation based on exponential functions

AU - Jovanovic, Raka

AU - Kais, Sabre

AU - Alharbi, Fahhad

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work.

AB - We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work.

UR - http://www.scopus.com/inward/record.url?scp=84912116957&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84912116957&partnerID=8YFLogxK

U2 - 10.1155/2014/168568

DO - 10.1155/2014/168568

M3 - Article

VL - 2014

JO - Journal of Applied Mathematics

JF - Journal of Applied Mathematics

SN - 1110-757X

M1 - 168568

ER -