Optical soliton solutions for a space-time fractional perturbed nonlinear Schrödinger equation arising in quantum physics

Abdou, M. A.; Owyed, Saud; Abdel‐Aty, Abdel‐Haleem; Raffah, Bahaaudin M.; Abdel‐Khalek, S.

doi:10.1016/j.rinp.2019.102895

Cited by 51 publications

(14 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the analytical solution of the nonlinear fractional Schrödinger equation, one can refer to the residual power series method [29], double Laplace transform [30], homotopy analysis transform method [31], generalized Kudryshov method [32], adomian decomposition method [33], generalized Riccati equation mapping method and the modified Kudryashov method [34], and the fractional Riccati expansion method [35].…”

Section: Introductionmentioning

confidence: 99%

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

2022

View full text Add to dashboard Cite

This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The L1-approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms L2 and L∞ are also calculated at different values of N and t to evaluate the performance of the suggested method.

show abstract

Section: Introductionmentioning

confidence: 99%

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

2022

View full text Add to dashboard Cite

show abstract

“…In order to overcome these complications and model the nonlocal systems effectively, it is better to use Atangana-Baleanu (AB) derivative. In the literature, a large amount of work is available for the solution of differential or integrodifferential equations with AB derivative, see [15,20,[48][49][50][51] and references therein. In this article, we aim to approximate the solution of integrodifferential equations of Fredholm type with AB derivative of the form e left Riemann-Liouville (RL) integral for α > 0 is defined as [13]…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

Wang

Kamran

Jamal

et al. 2021

Mathematical Problems in Engineering

View full text Add to dashboard Cite

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

show abstract

“…Caputo and Fabrizio (CF) have given a mathematical operator by taking kernel as exponential decay law (Caputo and Fabrizio, 2015). The scientific assumptions of these arbitrary order differential operators can be seen in Atangana and Baleanu (2017); Jothimani and Valliammal (2018); Abdou et al (2020); Abdel-Aty et al (2020a); Abdel-Aty et al (2020b, 2020c); Arqub et al (2020); Raza et al (2020a, 2020b); and Raza and Rafiq (2020).…”

Section: Introductionmentioning

confidence: 99%

A study of a modified nonlinear dynamical system with fractal-fractional derivative

Kumar

Chauhan

Momani

et al. 2021

HFF

View full text Add to dashboard Cite

Purpose This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control. Design/methodology/approach The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems. Findings Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved. Originality/value This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.

show abstract

Optical soliton solutions for a space-time fractional perturbed nonlinear Schrödinger equation arising in quantum physics

Cited by 51 publications

References 45 publications

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

A study of a modified nonlinear dynamical system with fractal-fractional derivative

Contact Info

Product

Resources

About