An exponential spline solution of nonlinear Schrödinger equations with constant and variable coefficients

Mohammadi, Robab

doi:10.1016/j.cpc.2013.12.015

Cited by 16 publications

(8 citation statements)

References 59 publications

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“…In this paper, we present an implicit non-polynomial spline functions based scheme for the numerical solution of the following one dimensional convection-diffusion equation, ( , (0, ) ( ), (1, ) ( ), 0 u t g t u t g t t t (2) and with the initial condition ( ,0) ( ), 0 1 u x f x x d d (3) where 0 H ! , is the phase speed and 0…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

Kanth

2016

MATEC Web of Conferences

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Abstract. In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.

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Section: Introductionmentioning

confidence: 99%

“…Ravi Kanth and Aruna [1] implemented the differential transform method for solving linear and nonlinear Klein-Gordon equation. Mohammadi [2] presented the exponential spline approach and Lin [3] constructed the parametric cubic spline method for the numerical solution of non-linear Schrodinger equation.…”

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confidence: 99%

Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

Kanth

2016

MATEC Web of Conferences

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“…The numerical solutions to the nonlinear sine-Gordon equation have received considerable attention in the literature (see [10][11][12][13][14][15][16]). Mohammadi [17]- [19] developed different spline schemes to find the numerical solution of different type of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

An Exponential Spline Approach to the Generalized Sine-Gordon Equation

Mohammadi¹

2015

IJAPM

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Abstract:The nonlinear sine-Gordon equation is used to model many nonlinear phenomena. Numerical simulation of the solution to the one-dimensional generalized sine-Gordon equation is considered here. Two implicit three time-level difference schemes are developed, by using the exponential spline function approximation. We consider both Dirichlet and Neumann boundary conditions. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of. In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.

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“…Numerical methods involved finite-difference approach [19,21,22], bi-k-Lagrange elements [23], spectral collocation methods with Chebyshev polynomials of the first and second kind [24] as well as basis set expansion technique [25]. Time-dependent equations were solved with implicit and semi-implicit Crank-Nicolson methods [26,27,18,28,29], Euler scheme [22], third and fourth-order adaptive Runge-Kutta methods [30], splitstep finite difference method [22] and time-splitting sine and Fourier pseudospectral methods [31,32]. In the latter case, space was discretized with second-and fourth-order finite differences, exponential splines [29] or with Chebyshev-Tau spectral discretization method [26].…”

Section: Introductionmentioning

confidence: 99%

Numerical modeling of exciton–polariton Bose–Einstein condensate in a microcavity

Voronych

Buraczewski

Matuszewski

et al. 2017

Computer Physics Communications

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A novel, optimized numerical method of modeling of an exciton-polariton superfluid in a semiconductor microcavity was proposed. Exciton-polaritons are spin-carrying quasiparticles formed from photons strongly coupled to excitons. They possess unique properties, interesting from the point of view of fundamental research as well as numerous potential applications. However, their numerical modeling is challenging due to the structure of nonlinear differential equations describing their evolution. In this paper, we propose to solve the equations with a modified Runge-Kutta method of 4th order, further optimized for efficient computations. The algorithms were implemented in form of C++ programs fitted for parallel environments and utilizing vector instructions. The programs form the EPCGP suite which have been used for theoretical investigation of exciton-polaritons.Keywords: exciton-polariton superfluid; Bose-Einstein condensate; microcavity; Gross-Pitaevskii equation; Runge-Kutta method Nature of problem: An exciton-polariton superfluid is a novel, interesting physical system allowing investigation of high temperature Bose-Einstein condensation of exciton-polaritons-quasiparticles carrying spin. They have brought a lot of attention due to their unique properties and potential applications in polariton-based optoelectronic integrated circuits. This is an out-of-equilibrium quantum system confined within a semiconductor microcavity. It is described by a set of nonlinear differential equations similar in spirit to the Gross-Pitaevskii (GP) equation, but their unique properties do not allow standard GP solving frameworks to be utilized. Finding an accurate and efficient numerical algorithm as well as development of optimized numerical software is necessary for effective theoretical investigation of exciton-polaritons. Program summarySolution method: A Runge-Kutta method of 4th order was employed to solve the set of differential equations describing exciton-polariton superfluids. The method was fitted for the exciton-polariton equations and further optimized. The C++ programs utilize OpenMP extensions and vector operations in order to fully utilize the computer hardware.

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An exponential spline solution of nonlinear Schrödinger equations with constant and variable coefficients

Cited by 16 publications

References 59 publications

Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

An Exponential Spline Approach to the Generalized Sine-Gordon Equation

Numerical modeling of exciton–polariton Bose–Einstein condensate in a microcavity

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