Corrigendum to “A new hyperbolic auxiliary function method and exact solutions of the mBBM equation” [Commun Nonlinear Sci Numer Simul 2010;15:135–138]

2010 ◽  
Vol 15 (9) ◽  
pp. 2734
Author(s):  
Olawanle P. Layeni ◽  
Ade P. Akinola
Keyword(s):  
Exact Solutions ◽  
Function Method ◽  
Auxiliary Function

THE EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS IN THE GARDNER AND GARDNER–KP EQUATIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250126 ◽  
Author(s):  
FANG YAN ◽  
CUNCAI HUA ◽  
HAIHONG LIU ◽  
ZENGRONG LIU
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Traveling Wave Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Kp Equation ◽  
Gardner Equation ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kp Equations

By using the method of dynamical systems, this paper studies the exact traveling wave solutions and their bifurcations in the Gardner equation. Exact parametric representations of all wave solutions as well as the explicit analytic solutions are given. Moreover, several series of exact traveling wave solutions of the Gardner–KP equation are obtained via an auxiliary function method.

New Exact Solutions for Two Nonlinear Equations

2006 ◽  
Vol 61 (1-2) ◽  
pp. 1-6 ◽  
Author(s):  
Zonghang Yang
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Exact Solutions ◽  
Water Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
New Exact Solutions ◽  
Complex Phenomena ◽  
Sivashinsky Equation

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutions for the second were just N-soliton solutions. In this paper we present kinds of new exact solutions by using the extended tanh-function method.

Study on soliton solutions of the longitudinal wave equation and magneto-electro-elastic circular rod dynamical model

2021 ◽  
Vol 35 (13) ◽  
pp. 2150168
Author(s):  
Adel Darwish ◽  
Aly R. Seadawy ◽  
Hamdy M. Ahmed ◽  
A. L. Elbably ◽  
Mohammed F. Shehab ◽  
...  
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Longitudinal Wave ◽  
Physical Interpretation ◽  
Elliptic Function ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Jacobi Elliptic Function ◽  
Two Dimensional

In this paper, we use the improved modified extended tanh-function method to obtain exact solutions for the nonlinear longitudinal wave equation in magneto-electro-elastic circular rod. With the aid of this method, we get many exact solutions like bright and singular solitons, rational, singular periodic, hyperbolic, Jacobi elliptic function and exponential solutions. Moreover, the two-dimensional and the three-dimensional graphs of some solutions are plotted for knowing the physical interpretation.

scholarly journals Exact solution of fractional Nizhnik-Novikov-Veselov equation

2014 ◽  
Vol 18 (5) ◽  
pp. 1716-1717 ◽  
Author(s):  
Sui-Min Jia ◽  
Ming-Sheng Hu ◽  
Qiao-Ling Chen ◽  
Zhi-Juan Jia
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Function Method

The fractional Nizhnik-Novikov-Veselov equation is converted to its differential partner, and its exact solutions are successfully established by the exp-function method.

scholarly journals New Exact Solutions for a Generalized Double Sinh-Gordon Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Gabriel Magalakwe ◽  
Chaudry Masood Khalique
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Fluid Dynamics ◽  
Exact Solutions ◽  
Function Method ◽  
Gordon Equation ◽  
Quantum Field ◽  
Integrable Quantum ◽  
New Exact Solutions ◽  
Integrable Quantum Field Theory

We study a generalized double sinh-Gordon equation, which has applications in various fields, such as fluid dynamics, integrable quantum field theory, and kink dynamics. We employ the Exp-function method to obtain new exact solutions for this generalized double sinh-Gordon equation. This method is important as it gives us new solutions of the generalized double sinh-Gordon 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.

Abundant exact solutions to the strain wave equation in micro-structured solids

2021 ◽  
pp. 2150439
Author(s):  
Karmina K. Ali ◽  
R. Yilmazer ◽  
H. Bulut ◽  
Tolga Aktürk ◽  
M. S. Osman
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Exact Solutions ◽  
Rational Function ◽  
Physical Interpretation ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Singular Solutions ◽  
Strain Wave ◽  
Important Place

In this study, the strain wave equation in micro-structured solids which take an important place in solid physics is presented for consideration. The generalized exponential rational function method is used for this purpose which is one of the most powerful methods of constructing abundantly distinct, exact solutions of nonlinear partial differential equations. In micro-structured solids, wave propagation is based on the structure of the strain wave equation. As a consequence, we successfully received many different exact solutions, including non-topological solutions, periodic singular solutions, topological solutions, singular solutions, like periodic lump solutions. Furthermore, in order to better understand their physical interpretation, 2D, 3D, and counter plots are graphed for each of the solutions acquired.

scholarly journals Analytical Solution of System of Volterra Integral Equations Using OHAM

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Muhammad Akbar ◽  
Rashid Nawaz ◽  
Sumbal Ahsan ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Numerical Methods ◽  
Analytical Solution ◽  
Integral Equations ◽  
Function Method ◽  
Auxiliary Function ◽  
Volterra Integral Equations ◽  
Large Parameter ◽  
Reliable Technique ◽  
Present Technique ◽  
Optimal Homotopy Asymptotic Method

In this work, a reliable technique is used for the solution of a system of Volterra integral equations (VIEs), called optimal homotopy asymptotic method (OHAM). The proposed technique is successfully applied for the solution of different problems, and comparison is made with the relaxed Monto Carlo method (RMCM) and hat basis function method (HBFM). The comparisons show that the present technique is more suitable and reliable for the solution of a system of VIEs. The presented technique uses auxiliary function containing auxiliary constants, which control the convergence. Moreover, OHAM does not require discretization like other numerical methods and is also free from small or large parameter.

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