scholarly journals Analysis of Optical Solitons for Nonlinear Schrödinger Equation with Detuning Term by Iterative Transform Method

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1850
Author(s):  
Nehad Ali Shah ◽  
Praveen Agarwal ◽  
Jae Dong Chung ◽  
Essam R. El-Zahar ◽  
Y. S. Hamed
Keyword(s):  
Fractional Order ◽  
Present Method ◽  
Nonlinear Partial Differential Equations ◽  
Optical Solitons ◽  
Optical Soliton ◽  
Transform Method ◽  
Computational Costs ◽  
Nonlinear Schrödinger ◽  
Linear And Nonlinear ◽  
Integral Order

In this article, the iteration transform method is used to evaluate the solution of a fractional-order dark optical soliton, bright optical soliton, and periodic solution of the nonlinear Schrödinger equations. The Caputo operator describes the fractional-order derivatives. The solutions of some illustrative examples are presented to show the validity of the proposed technique without using any polynomials. The proposed method provides the series form solutions, which converge to the exact results. Using the present methodology, the solutions of fractional-order problems as well as integral-order problems are calculated. The present method has less computational costs and a higher rate of convergence. Therefore, the suggested algorithm is constructive to solve other fractional-order linear and nonlinear partial differential equations.

Numerical simulation for coupled nonlinear Schrödinger–Korteweg–de Vries and Maccari systems of equations

2021 ◽  
pp. 2150339
Author(s):  
Lanre Akinyemi ◽  
Pundikala Veeresha ◽  
Samuel Oluwatosin Ajibola
Keyword(s):  
Numerical Simulation ◽  
Plasma Physics ◽  
Acoustic Waves ◽  
Electromagnetic Waves ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Langmuir Waves ◽  
Suggested Approach ◽  
Nonlinear Schrödinger ◽  
De Vries

The primary goal of this paper is to seek solutions to the coupled nonlinear partial differential equations (CNPDEs) by the use of q-homotopy analysis transform method (q-HATM). The CNPDEs considered are the coupled nonlinear Schrödinger–Korteweg–de Vries (CNLS-KdV) and the coupled nonlinear Maccari (CNLM) systems. As a basis for explaining the interactive wave propagation of electromagnetic waves in plasma physics, Langmuir waves and dust-acoustic waves, the CNLS-KdV model has emerged as a model for defining various types of wave phenomena in mathematical physics, and so forth. The CNLM model is a nonlinear system that explains the dynamics of isolated waves, restricted in a small part of space, in several fields like nonlinear optics, hydrodynamic and plasma physics. We construct the solutions (bright soliton) of these models through q-HATM and present the numerical simulation in form of plots and tables. The solutions obtained by the suggested approach are provided in a refined converging series. The outcomes confirm that the proposed solutions procedure is highly methodological, accurate and easy to study CNPDEs.

scholarly journals A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 949 ◽  
Author(s):  
Hassan Eltayeb ◽  
Said Mesloub ◽  
Yahya T. Abdalla ◽  
Adem Kılıçman
Keyword(s):  
Present Method ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Numerical Examples ◽  
One Dimensional ◽  
Linear And Nonlinear ◽  
Laplace Decomposition Method

The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to demonstrate the efficiency, high accuracy, and the simplicity of present method.

scholarly journals ANALYSIS OF TIME-FRACTIONAL BURGERS AND DIFFUSION EQUATIONS BY USING MODIFIED q-HATM

Fractals ◽  
2021 ◽  
pp. 2240012
Author(s):  
NEHAD ALI SHAH ◽  
PRAVEEN AGARWAL ◽  
JAE DONG CHUNG ◽  
SAAD ALTHOBAITI ◽  
SAMY SAYED ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Nonlinear Partial Differential Equations ◽  
Convergent Sequence ◽  
Exact Result ◽  
Diffusion Equations ◽  
Homotopy Analysis ◽  
Linear And Nonlinear ◽  
And Diffusion

In this paper, the q-homotopy analysis transform technique is implemented to analyze the solution of fractional-order Burgers and diffusion equations with the help of Caputo operator. The results of the proposed method are shown and analyzed with the help of figures. This approach is used to determine the solution in a convergent sequence and illustrate the q-homotopy analysis transform technique solutions convergence to the exact result. Several examples showed the reliability and simplicity of the technique and highlighted the significance of this work. Therefore, the proposed method is successful in investigating other fractional-order linear and nonlinear partial differential equations.

scholarly journals An Analytical Investigation of Fractional-Order Biological Model Using an Innovative Technique

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Hassan Khan ◽  
Adnan Khan ◽  
Maysaa Al Qurashi ◽  
Dumitru Baleanu ◽  
Rasool Shah
Keyword(s):  
Fractional Order ◽  
Present Method ◽  
Iterative Procedure ◽  
Laplace Transformation ◽  
Small Volume ◽  
Analytical Investigation ◽  
Form Solution ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Closed Contact

In this paper, a new so-called iterative Laplace transform method is implemented to investigate the solution of certain important population models of noninteger order. The iterative procedure is combined effectively with Laplace transformation to develop the suggested methodology. The Caputo operator is applied to express the noninteger derivative of fractional order. The series form solution is obtained having components of convergent behavior toward the exact solution. For justification and verification of the present method, some illustrative examples are discussed. The closed contact is observed between the obtained and exact solutions. Moreover, the suggested method has a small volume of calculations; therefore, it can be applied to handle the solutions of various problems with fractional-order derivatives.

Linear and nonlinear generalized Fourier transforms

2006 ◽  
Vol 364 (1849) ◽  
pp. 3231-3249 ◽  
Author(s):  
Beatrice Pelloni
Keyword(s):  
Fourier Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourier Transforms ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Linear And Nonlinear ◽  
Mathematical Techniques ◽  
The Fourier Transform

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.

scholarly journals A New Coupled Fractional Reduced Differential Transform Method for the Numerical Solutions of(2+1)-Dimensional Time Fractional Coupled Burger Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
S. Saha Ray
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Present Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Transform Method ◽  
Differential Transform ◽  
Fractional Differential ◽  
Linear And Nonlinear

A very new technique, coupled fractional reduced differential transform, has been implemented to obtain the numerical approximate solution of (2 + 1)-dimensional coupled time fractional burger equations. The fractional derivatives are described in the Caputo sense. By using the present method we can solve many linear and nonlinear coupled fractional differential equations. The obtained results are compared with the exact solutions. Numerical solutions are presented graphically to show the reliability and efficiency of the method.

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