scholarly journals On the modified (G′G2)-expansion method for finding some analytical solutions of the traveling waves.

2021 ◽  
Author(s):  
S. Behera ◽  
N.H. Aljahdaly ◽  
J.P.S. Virdi
Keyword(s):  
Traveling Waves ◽  
Analytical Solutions ◽  
Expansion Method

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Searching for Analytical Solutions of the (2+1)-Dimensional KP Equation by Two Different Systematic Methods

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Yongyi Gu ◽  
Fanning Meng
Keyword(s):  
Nonlinear Systems ◽  
Exact Solutions ◽  
Rational Function ◽  
Analytical Solutions ◽  
Direct Methods ◽  
Expansion Method ◽  
Complex Method ◽  
Kp Equation ◽  
Extended Complex ◽  
Complex Nonlinear Systems

In this paper, we derive analytical solutions of the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation by two different systematic methods. Using the exp⁡(-ψ(z))-expansion method, exact solutions of the mentioned equation including hyperbolic, exponential, trigonometric, and rational function solutions have been obtained. Based on the work of Yuan et al., we proposed the extended complex method to seek exact solutions of the (2+1)-dimensional KP equation. The results demonstrate that the applied methods are efficient and direct methods to solve the complex nonlinear systems.

A STUDY ABOUT THE SOLUTIONS OF SCHRÖDINGER EQUATION WITH AN ASYMPTOTICALLY LINEAR POTENTIAL

1996 ◽  
Vol 11 (03) ◽  
pp. 207-209 ◽  
Author(s):  
ELSO DRIGO FILHO
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Expansion Method ◽  
Linear Potential ◽  
Asymptotically Linear

We determine the solutions of the Schrödinger equation for an asymptotically linear potential. Analytical solutions are obtained by superalgebra in quantum mechanics and we establish when these solutions are possible. Numerical solutions for the spectra are obtained by the shifted 1/N expansion method.

Optical soliton solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation

2019 ◽  
Vol 33 (32) ◽  
pp. 1950402 ◽  
Author(s):  
Behzad Ghanbari ◽  
J. F. Gómez-Aguilar
Keyword(s):  
Rational Function ◽  
Imaginary Part ◽  
Traveling Waves ◽  
Analytical Solutions ◽  
Efficient Method ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Complex Structures ◽  
Exponential Rational Function Method

In this paper, the generalized exponential rational function method is applied to obtain analytical solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation. We obtain novel soliton, traveling waves and kink-type solutions with complex structures. We also present the two- and three-dimensional shapes for the real and imaginary part of the solutions obtained. It is illustrated that generalized exponential rational function method (GERFM) is simple and efficient method to reach the various type of the soliton solutions.

scholarly journals Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ai-Min Yang ◽  
Xiao-Jun Yang ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Series Expansion ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Expansion Method ◽  
Cantor Sets ◽  
Diffusion Equations ◽  
And Diffusion ◽  
Series Expansion Method

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

scholarly journals On the explicit solutions of fractional Bagley-Torvik equation arises in engineering

2019 ◽  
Vol 9 (3) ◽  
pp. 52-58
Author(s):  
Zehra Pinar
Keyword(s):  
Analytical Solutions ◽  
Expansion Method ◽  
Equation Method ◽  
Explicit Solutions ◽  
Science And Engineering ◽  
Conformable Derivatives

In this work, Bagley-Torvik equation is considered with conformable derivatives. The analytical solutions will be obtained via Sine-Gordon expansion method and Bernouli equation method for the two cases of Bagley-Torvik equation. We will illustrate and discuss about the methodology and solutions therefore the proposed equation has meaning in different areas of science and engineering.

A Generalized Sub-Equation Expansion Method and Some Analytical Solutions to the Inhomogeneous Higher-Order Nonlinear Schrödinger Equation

2008 ◽  
Vol 63 (12) ◽  
pp. 763-777 ◽  
Author(s):  
Biao Li ◽  
Yong Chen ◽  
Yu-Qi Li
Keyword(s):  
Solitary Wave ◽  
Analytical Solutions ◽  
Elliptic Function ◽  
Schrodinger Equation ◽  
Expansion Method ◽  
High Order ◽  
Solitary Wave Solutions ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

On the basis of symbolic computation a generalized sub-equation expansion method is presented for constructing some exact analytical solutions of nonlinear partial differential equations. To illustrate the validity of the method, we investigate the exact analytical solutions of the inhomogeneous high-order nonlinear Schrödinger equation (IHNLSE) including not only the group velocity dispersion, self-phase-modulation, but also various high-order effects, such as the third-order dispersion, self-steepening and self-frequency shift. As a result, a broad class of exact analytical solutions of the IHNLSE are obtained. From our results, many previous solutions of some nonlinear Schrödinger-type equations can be recovered by means of suitable selections of the arbitrary functions and arbitrary constants. With the aid of computer simulation, the abundant structure of bright and dark solitary wave solutions, combined-type solitary wave solutions, dispersion-managed solitary wave solutions, Jacobi elliptic function solutions and Weierstrass elliptic function solutions are shown by some figures.

Analytical Solutions of the Two-Site Holstein Model by the Coherent State Expansion

2003 ◽  
Vol 17 (22n24) ◽  
pp. 4252-4259 ◽  
Author(s):  
Zijing Lin ◽  
Rongsheng Han ◽  
Wang Kelin
Keyword(s):  
Coherent State ◽  
Analytical Solutions ◽  
Order Approximation ◽  
Nonlinear Dynamical System ◽  
Expansion Method ◽  
Third Order ◽  
State Expansion ◽  
The Third ◽  
Holstein Model ◽  
Highly Nonlinear

The polaron problem presents a highly nonlinear dynamical system in which the charge and lattice deformations are intricately coupled together. The coherent state expansion method was applied to the two-site Holstein model. Analytical solutions are found for the first, the second and the third order approximations. Comparison is made among the exact numerical results, based upon solving a recursive relation for the expansion coefficients, and the analytical solutions up to the third order approximation for various e-ph coupling strengths and electron hopping parameters.

scholarly journals Analytical solitons for the space-time conformable differential equations using two efficient techniques

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmad Neirameh ◽  
Foroud Parvaneh
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Space Time ◽  
Nonlinear Phenomena ◽  
Approximate Analytical Solutions ◽  
Coupled Burgers Equation ◽  
Mathematics And Physics

AbstractExact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

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