New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity

Bhrawy, A. H.; Mallawi, Fouad; Abdelkawy, Mohamed

doi:10.1186/s13662-016-0752-3

Cited by 3 publications

(2 citation statements)

References 29 publications

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“…The interested reader can be referred, for example, to refs. − . Collocation has also been used to solve other types of differential equations such as the equations of classical density-functional theory, where collocation is convenient as it allows to modulate the concentration of collocation points which is higher in regions where the density exhibits rapid variations. − While those results have conceptual utility for understanding the potential of application of collocation to the electronic (with rapidly varying potentials) Schrödinger equation for real-life systems, we focus here on the use of collocation to solve the electronic Schrödinger equation for real atomic systems in three dimensions.…”

Section: Applications Of Collocationmentioning

confidence: 99%

Using Collocation to Solve the Schrödinger Equation

Manzhos

Ihara

Carrington

2023

J. Chem. Theory Comput.

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We review the collocation approach to the solution of the Schrodinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule−surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.

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Section: Applications Of Collocationmentioning

confidence: 99%

Using Collocation to Solve the Schrödinger Equation

Manzhos

Ihara

Carrington

2023

J. Chem. Theory Comput.

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“…2006; Bhrawy and Alofi 2013; Gu and Chen 2014; Bhrawy and Abdelkawy 2015; Bhrawy 2016a) is a specific type of spectral methods, that is more applicable and widely used to solve almost types of differential (Bhrawy et al. 2016b; Tatari and Haghighi 2014), integral (Bhrawy et al. 2016c; Rahmoune 2013), integro-differential (Jiang and Ma 2013; Ma and Huang 2014) and delay differential (Bhrawy et al.…”

Section: Introductionmentioning

confidence: 99%

A space–time spectral collocation algorithm for the variable order fractional wave equation

et al. 2016

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The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space–time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi–Gauss–Lobatto collocation scheme for the spatial discretization and the shifted Jacobi–Gauss–Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.

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