scholarly journals Quasi hyperbolic function expansion method and tanh-function method for solving vibrating string equation and elastic rod equation

2019 ◽  
Vol 38 (3-4) ◽  
pp. 1455-1465 ◽  
Author(s):  
Yi Tian
Keyword(s):  
Function Method ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
String Equation ◽  
Elastic Rod ◽  
Function Expansion ◽  
Vibrating String ◽  
Rod Equation

Solitary wave solutions of (2+1)-dimensional Maccari system

2021 ◽  
pp. 2150391
Author(s):  
Ghazala Akram ◽  
Naila Sajid
Keyword(s):  
Nonlinear Optics ◽  
Solitary Wave ◽  
Plasma Physics ◽  
Exact Solutions ◽  
Function Method ◽  
Expansion Method ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions

In this article, three mathematical techniques have been operationalized to discover novel solitary wave solutions of (2+1)-dimensional Maccari system, which also known as soliton equation. This model equation is usually of applicative relevance in hydrodynamics, nonlinear optics and plasma physics. The [Formula: see text] function, the hyperbolic function and the [Formula: see text]-expansion techniques are used to obtain the novel exact solutions of the (2+1)-dimensional Maccari system (arising in nonlinear optics and in plasma physics). Many novel solutions such as periodic wave solutions by [Formula: see text] function method, singular, combined-singular and periodic solutions by hyperbolic function method, hyperbolic, rational and trigonometric solutions by [Formula: see text]-expansion method are obtained. The exact solutions are shown through 3D graphics which present the movement of the obtained solutions.

GENERALIZED RICCATI EQUATION EXPANSION METHOD AND ITS APPLICATION TO THE (2+1)-DIMENSIONAL BOUSSINESQ EQUATION

2003 ◽  
Vol 14 (04) ◽  
pp. 471-482 ◽  
Author(s):  
YONG CHEN ◽  
BIAO LI ◽  
HONGQING ZHANG
Keyword(s):  
Riccati Equation ◽  
Function Method ◽  
Boussinesq Equation ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Generalized Riccati Equation ◽  
Hyperbolic Function Method ◽  
Formal Solutions ◽  
Singular Soliton

Based on the computerized symbolic system Maple and a Riccati equation, a new Riccati equation expansion method for constructing nontraveling wave and coefficient functions' soliton-like solutions is presented by a new general ansätz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method, and generalized hyperbolic-function method. By using the method, we not only successfully recovered the previously known formal solutions but could also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the (2+1)-dimensional Boussinesq equation and obtain rich new families of the exact solutions, including the nontraveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, and triangular functions solutions.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

A Class of Exact Solutions of (3+1)-Dimensional Generalized B-Type Kadomtsev–Petviashvili Equation

2017 ◽  
Vol 18 (2) ◽  
pp. 137-143 ◽  
Author(s):  
Shuang Liu ◽  
Yao Ding ◽  
Jian-Guo Liu
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Traveling Wave ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
New Exact Solutions ◽  
Petviashvili Equation

AbstractBy employing the generalized$(G'/G)$-expansion method and symbolic computation, we obtain new exact solutions of the (3 + 1) dimensional generalized B-type Kadomtsev–Petviashvili equation, which include the traveling wave exact solutions and the non-traveling wave exact solutions showed by the hyperbolic function and the trigonometric function. Meanwhile, some interesting physics structure are discussed.

SPATIAL XPM-PAIRED SOLITONS IN NONLINEAR METAMATERIALS

2013 ◽  
Vol 22 (01) ◽  
pp. 1350009
Author(s):  
ANLE FANG ◽  
YUANJIANG XIANG ◽  
BINXIAN ZHUANG ◽  
LEYONG JIANG ◽  
XIAOYU DAI ◽  
...  
Keyword(s):  
Refractive Index ◽  
Optical Fibers ◽  
Function Method ◽  
Hyperbolic Function ◽  
Nonlinear Polarization ◽  
Dark Solitons ◽  
Nonlinear Metamaterials ◽  
Theoretical Predictions ◽  
Hyperbolic Function Method ◽  
Propagation Properties

We investigate spatial XPM-paired solitons in nonlinear metamaterials (MMs) based on the (1 + 1)-dimensional coupled nonlinear Schrodinger equation (NLSE) describing the co-propagation of two optical beams of different frequencies in the MM with a Kerr-type nonlinear polarization. Three types of XPM-paired solitons including bright-bright, bright-dark and dark-dark solitons for different combination of the signs of refractive index experienced by the two beams, respectively, are obtained by using a generalized hyperbolic function method, which makes the temporal XPM-paired solitons in optical fibers find their spatial counterparts in MMs. Numerical simulations are performed to confirm the theoretical predictions and further identify the propagation properties of the spatial XPM-paired solitons in MMs described by Drude model.

New optical soliton solutions via two distinctive schemes for the DNA Peyrard–Bishop equation in fractal order

2021 ◽  
pp. 2150444
Author(s):  
Loubna Ouahid ◽  
M. A. Abdou ◽  
S. Owyed ◽  
Sachin Kumar
Keyword(s):  
Evolution Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Optical Soliton ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Closed Form Solutions ◽  
Special Solutions ◽  
Engineering Physics ◽  
Exact Soliton Solutions

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

Analytical Solutions of Two Space-Time Fractional Nonlinear Models Using Jacobi Elliptic Function Expansion Method

2021 ◽  
pp. 173-188
Author(s):  
Zillur Rahman ◽  
M. Zulfikar Ali ◽  
Harun-Or-Roshid ◽  
Mohammad Safi Ullah
Keyword(s):  
Elliptic Function ◽  
Nonlinear Models ◽  
Applied Mathematics ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Space Time ◽  
Jacobi Elliptic Function ◽  
Function Expansion ◽  
Partial Models ◽  
Jacobi Elliptic Function Expansion

In this manuscript, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM) models have been investigated which are frequently arises in nonlinear optics, solid states, fluid mechanics and shallow water. Jacobi elliptic function expansion integral technique has been used to build more innovative exact solutions of the s-tfEW and s-tfWBBM nonlinear partial models. In this research, fractional beta-derivatives are applied to convert the partial models to ordinary models. Several types of solutions have been derived for the models and performed some new solitary wave phenomena. The derived solutions have been presented in the form of Jacobi elliptic functions initially. Persevering different conditions on a parameter, we have achieved hyperbolic and trigonometric functions solutions from the Jacobi elliptic function solutions. Besides the scientific derivation of the analytical findings, the results have been illustrated graphically for clear identification of the dynamical properties. It is noticeable that the integral scheme is simplest, conventional and convenient in handling many nonlinear models arising in applied mathematics and the applied physics to derive diverse structural precise solutions.

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