Comparative analysis of spectral methods in half-bounded domains; implementation of predefined exponential and Laguerre basis sets to study planar dielectric and plasmonic waveguides

Alharbi, Fahhad H.

doi:10.1007/s11082-010-9388-9

Cited by 12 publications

(10 citation statements)

References 15 publications

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“…A review paper by Shen and Wang discusses this problem in further detail [26]. Recently, a nonorthogonal predefined exponential basis set for eignevalue problems involving half bounded domains was reintroduced [27,28]. Similar sets were introduced in the 1970s by Raffenetti, Bardo, and Ruedenberg [33][34][35] for self-consistent field wave functions.…”

Section: Spectral Methods and The Exponential Basis Setsmentioning

confidence: 99%

“…Many techniques were introduced to overcome this challenge, such as using exponentially decaying functions as basis sets, the truncation of the computational windows, and applying size scaling. Recently a nonorthogonal predefined exponential basis set for eignevalue problems involving half bounded domains was introduced and used [27,28]. The set is easy to use and allows generally finding a wide range of eigenvalues simultaneously.…”

Section: Introductionmentioning

confidence: 99%

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Quantum criticality analysis by finite-size scaling and exponential basis sets

Alharbi

Kais

2013

Phys. Rev. E

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We combine the finite-size scaling method with the mesh-free spectral method to calculate quantum critical parameters for a given Hamiltonian. The basic idea is to expand the exact wave function in a finite exponential basis set and extrapolate the information about system criticality from a finite basis to the infinite basis set limit. The used exponential basis set, though chosen intuitively, allows handling a very wide range of exponential decay rates and calculating multiple eigenvalues simultaneously. As a benchmark system to illustrate the combined approach, we choose the Hulthen potential. The results show that the method is very accurate and converges faster when compared with other basis functions. The approach is general and can be extended to examine near-threshold phenomena for atomic and molecular systems based on even-tempered exponential and Gaussian basis functions.

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Section: Spectral Methods and The Exponential Basis Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum criticality analysis by finite-size scaling and exponential basis sets

Alharbi

Kais

2013

Phys. Rev. E

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“…One of the complexities in applying spectral methods (SM) to the TF equation is the fact that it is defined on a semiinfinite domain. Significant research has been conducted on applying SM on infinite and semi-infinite domains [7][8][9][10][11][12][13][14][15]. This has been achieved by implementing a wide range of approaches varying from using suitable basis sets and truncating the numerical window to forcing size scaling.…”

Section: Introductionmentioning

confidence: 99%

“…In our approach to solve the TF equation, we apply the spectral method based on the exponential basis set. This basis set and its polynomial version have been recently used for solving several differential equations on semiinfinite domains [13][14][15]. The use of a similar basis set was initially presented in the 1970s by Raffenetti, Bardo, and Ruedenberg [25][26][27] for self-consistent field wave functions.…”

Section: Introductionmentioning

confidence: 99%

Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

Jovanović

Kais

Alharbi

2014

Journal of Applied Mathematics

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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.

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“…This is due to the properties of the used basis function set. Previously, this type of methods has been also used on infinite and semi-infinite domains [18,25,26,27]. This has been achieved by adopting various numerical techniques such as using suitable basis sets, truncating the numerical window, and forcing size scaling.…”

Section: Introductionmentioning

confidence: 99%

Efficient method for localised functions using domain transformation and Fourier sine series

2013

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An efficient approach to handle localized states by using spectral methods (SM) in one and three dimensions is presented. The method consists of transformation of the infinite domain to the bounded domain in (0, π) and using the Fourier sine series as a set of basis functions for the SM. It is shown that with an appropriate choice of transformation functions, this method manages to preserve the good properties of original SMs; more precisely, superb computational efficiency when high level of accuracy is necessary. This is made possible by analytically exploiting the properties of the transformation function and the Fourier sine series. An especially important property of this approach is the possibility of calculating the Hartree energy very efficiently. This is done by exploiting the positive properties of the sine series as a basis set and conducting an extinctive part of the calculations analytically. We illustrate the efficiency of this method and implement it to solve the Poisson's and Helmholtz equations in both one and three dimensions. The efficiency of the method is verified through a comparison to recently published results for both one and three dimensional problems. * *

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Comparative analysis of spectral methods in half-bounded domains; implementation of predefined exponential and Laguerre basis sets to study planar dielectric and plasmonic waveguides

Cited by 12 publications

References 15 publications

Quantum criticality analysis by finite-size scaling and exponential basis sets

Quantum criticality analysis by finite-size scaling and exponential basis sets

Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

Efficient method for localised functions using domain transformation and Fourier sine series

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