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Автор: Grafiati
Опубліковано: 10 грудня 2022
Оновлено: 28 січня 2023

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1

Паасонен, Виктор Иванович, and Михаил Петрович Федорук. "On the efficiency of high-order difference schemes for the Schro¨dinger equation." Вычислительные технологии, no. 6 (December 16, 2021): 68–81. http://dx.doi.org/10.25743/ict.2021.26.6.006.

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Исследуется ряд двух- и трехслойных разностных схем, построенных на расширенных шаблонах, до восьмого порядка точности для уравнения Шрёдингера. Наряду с многоточечными схемами рассматривается метод коррекции Ричардсона в приложении к схеме четвертого порядка аппроксимации, повышающий порядок точности путем построения линейных комбинаций приближенных решений, полученных на различных вложенных сетках. Проведено сравнение методов по устойчивости, сложности реализации алгоритмов и объему вычислений, необходимых для достижения заданной точности. На основе теоретического анализа и численных экспериментов выявлены методы, наиболее эффективные для практического применения The efficiency of difference methods for solving problems of nonlinear wave optics is largely determined by the order of accuracy. Schemes up to the fourth order of accuracy have the traditional architecture of three-point stencils and standard conditions for the application of algorithms. However, a further increase in the order in the general case is associated with the need to expand the stencils using multipoint difference approximations of the derivatives. The use of such schemes forces formulating additional boundary conditions, which are not present in the differential problem, and leads to the need to invert the matrices of the strip structure, which are different from the traditional tridiagonal ones. An exception is the Richardson correction method, which is aimed at increasing the order of accuracy by constructing special linear combinations of approximate solutions obtained on various nested grids according to traditional structure schemes. This method does not require the formulation of additional boundary conditions and inversion of strip matrices. In this paper, we consider several explicit and implicit multipoint difference schemes up to the eighth order of accuracy for the Schr¨odinger equation. In addition, a simple and double Richardson correction method is also investigated in relation to the classical fourth-order scheme. A simple correction raises the order to sixth and a double correction to eighth. This large collection of schemes is theoretically compared in terms of their properties such as the order of approximation, stability, the complexity of the implementation of a numerical algorithm, and the amount of arithmetic operations required to achieve a given accuracy. The theoretical analysis is supplemented by numerical experiments on the selected test problem. The main conclusion drawn from the research results is that of all the considered schemes, the Richardson-corrected scheme is the most preferable in terms of the investigated properties
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2

Zlotnik, Alexander, and Olga Kireeva. "ON COMPACT 4TH ORDER FINITE-DIFFERENCE SCHEMES FOR THE WAVE EQUATION." Mathematical Modelling and Analysis 26, no. 3 (September 10, 2021): 479–502. http://dx.doi.org/10.3846/mma.2021.13770.

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We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.
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3

Petridis, Athanasios N., Lawrence P. Staunton, Jon Vermedahl, and Marshall Luban. "Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times." Journal of Modern Physics 01, no. 02 (2010): 124–36. http://dx.doi.org/10.4236/jmp.2010.12018.

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4

Fermo, L., Mee Van, and S. Seatzu. "Emerging problems in approximation theory for the numerical solution of the nonlinear Schrödinger equation." Publications de l'Institut Math?matique (Belgrade) 96, no. 110 (2014): 125–41. http://dx.doi.org/10.2298/pim1410125f.

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We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schr?dinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solution of Fredholm integral equations and the identification of coefficients and parameters of monomial-exponential sums. The aim of the paper is twofold: propose a method for solving the above mentioned problems under particular hypothesis; arise interest in the issues illustrated to achieve an effective method for solving the problem under more general assumptions.
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5

Kapoor, Mamta, Nehad Ali Shah, and Wajaree Weera. "Analytical solution of time-fractional Schr<i>ö</i>dinger equations via Shehu Adomian Decomposition Method." AIMS Mathematics 7, no. 10 (2022): 19562–96. http://dx.doi.org/10.3934/math.20221074.

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<abstract> <p>Present research deals with the time-fractional Schr<italic>ö</italic>dinger equations aiming for the analytical solution via Shehu Transform based Adomian Decomposition Method [STADM]. Three types of time-fractional Schr<italic>ö</italic>dinger equations are tackled in the present research. Shehu transform ADM is incorporated to solve the time-fractional PDE along with the fractional derivative in the Caputo sense. The developed technique is easy to implement for fetching an analytical solution. No discretization or numerical program development is demanded. The present scheme will surely help to find the analytical solution to some complex-natured fractional PDEs as well as integro-differential equations. Convergence of the proposed method is also mentioned.</p> </abstract>
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6

Ritchie, Burke, and Charles A. Weatherford. "Numerical solution of the time-dependent Schr�dinger equation for continuum states." International Journal of Quantum Chemistry 80, no. 4-5 (2000): 934–41. http://dx.doi.org/10.1002/1097-461x(2000)80:4/5<934::aid-qua42>3.0.co;2-o.

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7

Campoy, G., and A. Palma. "On the numerical solution of the Schr�dinger equation with a polynomial potential." International Journal of Quantum Chemistry 30, S20 (March 10, 1986): 33–43. http://dx.doi.org/10.1002/qua.560300706.

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8

Zeng Si-Liang, Zou Shi-Yang, Wang Jian-Guo, and Yan Jun. "Numerical solution of the three-dimensional time-dependent Schr?dinger equation and its application." Acta Physica Sinica 58, no. 12 (2009): 8180. http://dx.doi.org/10.7498/aps.58.8180.

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9

Vigo-Aguiar, Jes�s, and Higinio Ramos. "A variable-step Numerov method for the numerical solution of the Schr�dinger equation." Journal of Mathematical Chemistry 37, no. 3 (April 2005): 255–62. http://dx.doi.org/10.1007/s10910-004-1467-3.

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10

Anastassi, Z. A., and T. E. Simos. "Trigonometrically fitted Runge?Kutta methods for the numerical solution of the Schr�dinger equation." Journal of Mathematical Chemistry 37, no. 3 (April 2005): 281–93. http://dx.doi.org/10.1007/s10910-004-1470-8.

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11

Sakas, D. P., and T. E. Simos. "A family of multiderivative methods for the numerical solution of the Schr�dinger equation." Journal of Mathematical Chemistry 37, no. 3 (April 2005): 317–32. http://dx.doi.org/10.1007/s10910-004-1472-6.

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12

Liu, Xue-Shen, Xiao-Yan Liu, Zhong-Yuan Zhou, Pei-Zhu Ding, and Shou-Fu Pan. "Numerical solution of one-dimensional time-independent Schr�dinger equation by using symplectic schemes." International Journal of Quantum Chemistry 79, no. 6 (2000): 343–49. http://dx.doi.org/10.1002/1097-461x(2000)79:6<343::aid-qua2>3.0.co;2-o.

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13

Kalogiratou, Z., Th Monovasilis, and T. E. Simos. "Numerical solution of the two-dimensional time independent Schr�dinger equation with Numerov-type methods." Journal of Mathematical Chemistry 37, no. 3 (April 2005): 271–79. http://dx.doi.org/10.1007/s10910-004-1469-1.

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14

Liu, Xuan, Muhammad Ahsan, Masood Ahmad, Muhammad Nisar, Xiaoling Liu, Imtiaz Ahmad, and Hijaz Ahmad. "Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion." Energies 14, no. 23 (November 23, 2021): 7831. http://dx.doi.org/10.3390/en14237831.

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Анотація:
This article is concerned with the numerical solution of nonlinear hyperbolic Schro¨dinger equations (NHSEs) via an efficient Haar wavelet collocation method (HWCM). The time derivative is approximated in the governing equations by the central difference scheme, while the space derivatives are replaced by finite Haar series, which transform it to full algebraic form. The experimental rate of convergence follows the theoretical statements of convergence and the conservation laws of energy and mass are also presented, which strengthens the proposed method to be convergent and conservative. The Haar wavelets based on numerical results for solitary wave shape of |φ| are discussed in detail. The proposed approach provides a fast convergent approximation to the NHSEs. The reliability and efficiency of the method are illustrated by computing the maximum error norm and the experimental rate of convergence for different problems. Comparisons are performed with various existing methods in recent literature and better performance of the proposed method is shown in various tables and figures.
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15

刘, 学. "Four Types of Functions Solutions of the Novel Auxiliary Equation and Its Application on the Perturbed Nonlinear Schro¨dinger Equation." Advances in Applied Mathematics 04, no. 03 (2015): 217–23. http://dx.doi.org/10.12677/aam.2015.43027.

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16

Fatokun, Johnson Oladele. "A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrödinger Equation." American Journal of Computational Mathematics 04, no. 04 (2014): 271–79. http://dx.doi.org/10.4236/ajcm.2014.44023.

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17

Psihoyios, G., and T. E. Simos. "Sixth algebraic order trigonometrically fitted predictor?corrector methods for the numerical solution of the radial Schr�dinger equation." Journal of Mathematical Chemistry 37, no. 3 (April 2005): 295–316. http://dx.doi.org/10.1007/s10910-004-1471-7.

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18

García A., Carlos, Nicolas Dauchez, and Gautier Lefebvre. "Radiation of thin complex plates estimated with the landscape of localisation theory." Journal of the Acoustical Society of America 151, no. 4 (April 2022): A167. http://dx.doi.org/10.1121/10.0010996.

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The landscape of localisation is a practical tool that enables the prediction of the geographical localisation of localized modes and helps us to understand the transition between localized and delocalised states. Moreover, approximations based on the Rayleigh quotient and on a variant of Weyl’s law are employed to predict the eigenfrequencies for the Schro ¨ dinger operator in quantum mechanics, but they are also valid for the Laplace and biharmonic operators, which characterize the behaviour of most dynamical systems in acoustics and vibrations. When studying the acoustic radiation from a vibrating structure, three global parameters are key indicators: the mean squared velocity, the acoustic radiated power, and the radiation efficiency. The literature on this subject is very vast for the plate case, where for simple geometries, it is still possible to derive analytical solutions or, at least, very useful approximations. For more complex structures, numerical simulations seem to be appropriate for lack of a simpler solution. In this context, this work aims to give some light to create a direct relationship between these global parameters and the landscape of localisation function, based on the multipolar radiation behaviour presented by localized modes and estimated by geometrical means.
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19

Chen, Yu, Iman Tavakkolnia, Alex Alvarado, and Majid Safari. "On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers." Entropy 22, no. 8 (August 15, 2020): 899. http://dx.doi.org/10.3390/e22080899.

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The capacity limits of fiber-optic communication systems in the nonlinear regime are not yet well understood. In this paper, we study the capacity of amplitude modulated first-order soliton transmission, defined as the maximum of the so-called time-scaled mutual information. Such definition allows us to directly incorporate the dependence of soliton pulse width to its amplitude into capacity formulation. The commonly used memoryless channel model based on noncentral chi-squared distribution is initially considered. Applying a variance normalizing transform, this channel is approximated by a unit-variance additive white Gaussian noise (AWGN) model. Based on a numerical capacity analysis of the approximated AWGN channel, a general form of capacity-approaching input distributions is determined. These optimal distributions are discrete comprising a mass point at zero (off symbol) and a finite number of mass points almost uniformly distributed away from zero. Using this general form of input distributions, a novel closed-form approximation of the capacity is determined showing a good match to numerical results. Finally, mismatch capacity bounds are developed based on split-step simulations of the nonlinear Schro¨dinger equation considering both single soliton and soliton sequence transmissions. This relaxes the initial assumption of memoryless channel to show the impact of both inter-soliton interaction and Gordon–Haus effects. Our results show that the inter-soliton interaction effect becomes increasingly significant at higher soliton amplitudes and would be the dominant impairment compared to the timing jitter induced by the Gordon–Haus effect.
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20

Sous, A. J., and M. I. El-Kawni. "Numerical Solution of the Schrödinger Equation for a Short-Range Potential with Any <i>l</i> Angular Momentum." Journal of Applied Mathematics and Physics 06, no. 04 (2018): 901–9. http://dx.doi.org/10.4236/jamp.2018.64077.

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21

Katrakhov, Valeriy V., and Sergey M. Sitnik. "The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations." Contemporary Mathematics. Fundamental Directions 64, no. 2 (December 15, 2018): 211–426. http://dx.doi.org/10.22363/2413-3639-2018-64-2-211-426.

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The main content of this book is composed from two doctoral theses: by V. V. Katrakhov (1989) and by S. M. Sitnik (2016). In our work, for the first time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to differential equations with singularities in coefficients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Significant part of these results is published with detailed proofs for the first time. In the first chapter, we give historical background, necessary notation, definitions, and auxiliary facts. In the second chapter, we give the detailed theory of Sonin and Poisson transmutations. In the third chapter, we describe an important special class of the Buschman-Erde´lyi transmutations and their applications. In the fourth chapter, we consider new weighted boundary-value problems with Sonin and Poisson transmutations. In the fifth chapter, we consider applications of the Buschman-Erde´lyi transmutations of special form to new boundary-value problems for elliptic equations with significant singularities of solutions. In the sixth chapter, we describe a universal compositional method for construction of transmutations and its applications. In the concluding seventh chapter, we consider applications of the theory of transmutations to differential equations with variable coefficients: namely, to the problem of construction of a new class of transmutations with sharp estimates of kernels for perturbed differential equations with the Bessel operator, and to special cases of the well-known Landis problem on exponential estimates of the rate of growth for solutions of the stationary Schro¨dinger equation. The book is concluded with a brief biographic essay about Valeriy V. Katrakhov, as well as detailed bibliography containing 648 references.
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22

Zhong, Yu, H. Triki, and Qin Zhou. "Analytical and numerical study of chirped optical solitons in a spatially inhomogeneous polynomial law fiber with parity-time symmetry potential." Communications in Theoretical Physics, November 23, 2022. http://dx.doi.org/10.1088/1572-9494/aca51c.

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Abstract This work studies the dynamical transmission of chirped optical solitons in a spatially inhomogeneous nonlinear fiber with cubic-quintic-septic nonlinearity, weak nonlocal nonlinearity, self-frequency shift and parity-time ($\mathcal{PT}$) symmetry potential. A generalized variable-coefficient nonlinear Schr"{o}dinger equation that models the dynamical evolution of solitons has been investigated by the analytical method of similarity transformation and the numerical mixed method of split-step Fourier method and Runge-Kutta method. The analytical self-similar bright and kink solitons, as well as their associated frequency chirps, are derived for the first time. We found that the amplitude of the bright and kink solitons can be controlled by adjusting the imaginary part of $\mathcal{% PT}$-symmetric potential. Moreover, the influence of initial chirp parameter on the soliton pulse widths is quantitatively analyzed. It is worth emphasizing that we could control the chirp whether it is linear or nonlinear by adjusting optical fiber parameters. The simulation results of bright and kink solitons fit perfectly with the analytical ones, and the stabilities of these soliton solutions against noises are checked by numerical simulation.
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23

"Statistical Approach to Nonlinear Schro ̈ dinger Equation. Quantum Case." Applied Mathematics & Information Sciences 16, no. 4 (July 1, 2022): 617–21. http://dx.doi.org/10.18576/amis/160415.

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24

Bilal, Muhammad, Muhammad Younis, Shafqat-Ur-Rehman, Jamshad Ahmad, and Usman Younas. "Investigation of new solitons and other solutions to the modified nonlinear Schro¨dinger equation in ocean engeneering." Journal of Ocean Engineering and Science, April 2022. http://dx.doi.org/10.1016/j.joes.2022.04.031.

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25

Younas, Usman, T. A. Sulaiman, Jingli Ren, and A. Yusuf. "Investigation of optical solitons and other solutions in optic fibers modelled by the improved perturbed nonlinear Schro¨dinger equation." Journal of Ocean Engineering and Science, June 2022. http://dx.doi.org/10.1016/j.joes.2022.06.038.

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