An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method

Başhan, Ali; Yağmurlu, Nuri Murat; Uçar, Yusuf; Esen, Alaattin

doi:10.1016/j.chaos.2017.04.038

Cited by 43 publications

(11 citation statements)

References 27 publications

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“…It is seen that Bellman et al 41 have first proposed the differential quadrature method as the numerical approach method for the partial differential equations. Then, in time, many problems are solved by this proposed method arising in the mathematics, physics and engineering 41–48 …”

Section: Introductionmentioning

confidence: 99%

A new perspective for the numerical solution of the Modified Equal Width wave equation

Başhan

Yağmurlu

Uçar

et al. 2021

Math Methods in App Sciences

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Finding the approximate solutions to natural systems in the branch of mathematical modelling has become increasingly important and for this end various methods have been proposed. The purpose of the present paper is to obtain and analyze the numerical solutions of Modified Equal Width equation (MEW). This equation is one of those equations used to model nonlinear phenomena which has a significant role in several branches of science such as plasma physics, fluid mechanics, optics and kinetics. Firstly, for the discretization of spatial derivatives, a fifth‐order quintic B‐spline based scheme is directly implemented. Secondly, a forward finite difference formula is applied for the temporal discretization of derivatives with respect to time. Simulation results establish the validity and applicability of the suggested technique for a wide range of nonlinear equations. Then, the newly obtained theoretical consequences are numerically justified by the simulations and test problems. These illustrative test problems are presented verifying the superiority of the newly presented scheme compared to other existing schemes and techniques. The suggested method with symbolic computational software such as, Matlab, is proven as an effective way to obtain the soliton solutions of several nonlinear partial differential equations (PDEs). Finally, the newly obtained results are presented graphically to justify the approximate findings.

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Section: Introductionmentioning

confidence: 99%

A new perspective for the numerical solution of the Modified Equal Width wave equation

Başhan

Yağmurlu

Uçar

et al. 2021

Math Methods in App Sciences

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“…Its coefficient d plays an important role in the form of the rNLSE, as it determines solutions with different behavior. Many studies on Schrödinger and rNLSE are made by various scientists [19][20][21][22][23][24]. For instance, Williams et al [17] argued the stability and dynamical properties of soliton waves in rLNSE.…”

Section: Introductionmentioning

confidence: 99%

Generalized Sub-Equation Method for the (1+1)-Dimensional Resonant Nonlinear Schrodinger’s Equation

Taşbozan

Tozar

Kurt

2021

Bilecik Şeyh Edebali Üniversitesi Fen Bilimleri Dergisi

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Interest in studying nonlinear models has been increasing in recent years. Dynamical systems, in which the state of the system changes continuously over time, have nonlinear interactions. The use of unique nonlinear differential equations is inescapable in the evaluation of such systems. In mathematical point of view, for obtaining analytical solutions of nonlinear differential equations, it must be fully integrable. Consequently, the importance of fully integrable nonlinear differential equations for nonlinear science has become indisputable. Among these equations, one of the most studied by physicists and mathematicians is the nonlinear Schrödinger equation. This equation has undergone many modifications to evaluate different phenomena. In this study, the resonant nonlinear Schrödinger equation, which is the most important of these physical equations in terms of explaining many physical phenomena, is solved analytically with the generalized sub-equation method.

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“…Bellman et al [22] firstly supposed the DQM for the numerical solutions of the differential equations. Then, many problems are solved commonly used in the scientific area [22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

Başhan

Yağmurlu

Uçar

et al. 2020

Numerical Methods Partial

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The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B-spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.

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An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method

Cited by 43 publications

References 27 publications

A new perspective for the numerical solution of the Modified Equal Width wave equation

A new perspective for the numerical solution of the Modified Equal Width wave equation

Generalized Sub-Equation Method for the (1+1)-Dimensional Resonant Nonlinear Schrodinger’s Equation

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

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