scholarly journals New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yusuf Pandir ◽  
Halime Ulusoy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Wave Equations ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
New Exact Solutions

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

scholarly journals A Family of Novel Exact Solutions to2+1-Dimensional KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhen Wang ◽  
Li Zou ◽  
Zhi Zong ◽  
Hongde Qin
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Lax Pair ◽  
Partial Differential ◽  
Independent Variables ◽  
New Exact Solutions

We introduce two subequations with different independent variables for constructing exact solutions to nonlinear partial differential equations. In order to illustrate the efficiency and usefulness, we apply this method to2+1-dimensional KdV equation, which was first derived by Boiti et al. (1986) using the idea of the weak Lax pair. As a result, we obtained many new exact solutions.

scholarly journals Classification of Exact Solutions for Some Nonlinear Partial Differential Equations with Generalized Evolution

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yusuf Pandir ◽  
Yusuf Gurefe ◽  
Ugur Kadak ◽  
Emine Misirli
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Polynomial Method ◽  
Partial Differential ◽  
Elliptic Solutions ◽  
The One

We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.

scholarly journals Application of the extended exp(-φ(ξ))-expansion method to the nonlinear conformable time-fractional partial differential equations

2019 ◽  
Vol 7 (2) ◽  
pp. 81
Author(s):  
Dipankar Kumar ◽  
Samir Chandra Ray
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Expansion Method ◽  
Point Of View ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Kawahara Equation ◽  
Partial Differential ◽  
New Exact Solutions

This paper investigates the new exact solutions of the three nonlinear time fractional partial differential equations namely the nonlinear time fractional Clannish Random Walker’s Parabolic (CRWP) equation, the nonlinear time fractional modified Kawahara equation, and the nonlinear time fractional BBM-Burger equation by utilizing an extended form of exp(-φ(ξ))-expansion method in the sense of conformable fractional derivative. As outcomes, some new exact solutions are obtained and signified by hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Some solutions have been plotted by MATLAB software to show the physical significance of our studied equations. In the point of view of our executed method and generated results, we may conclude that extended exp (-φ(ξ))-expansion method is more efficient than exp(-φ(ξ))-expansion method to extract the new exact solutions for solving any types of integer and fractional differential equations arising in mathematical physics.   

scholarly journals (Kink; Kink; Kink; Kink) and (Pulse; Pulse; Pulse; Pulse) Solutions of a Set of Four Equations Modeled in a Nonlinear Hybrid Electrical Line with Crosslink Capacitor

2019 ◽  
pp. 1-14
Author(s):  
Tiague Takongmo Guy ◽  
Jean Roger Bogning
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Solitary Waves ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Methods ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
Hybrid Parts ◽  
Nonlinear Hybrid

The physics system that helps us in the study of this paper is a nonlinear hybrid electrical line with crosslink capacitor. Meaning it is composed of two different nonlinear hybrid parts Linked by capacitors with identical constant capacitance. We apply Kirchhoff laws to the circuit of the line to obtain new set of four nonlinear partial differential equations which describe the simultaneous dynamics of four solitary waves. Furthermore, we apply efficient mathematical methods based on the identification of coefficients of basic hyperbolic functions to construct exact solutions of those set of four nonlinear partial differential equations. The obtained results have enabled us to discover that, one of the two nonlinear hybrid electrical line with crosslink capacitor that we have modeled accepts the simultaneous propagation of a set of four solitary waves of type (Pulse; Pulse; Pulse; Pulse), while the other accepts the simultaneous propagation of a set of four solitary waves of type (Kink; Kink; Kink; Kink) when certain conditions we have established are respected. We ameliorate the quality of the signals by changing the sinusoidal waves that are supposed to propagate in the hybrid electrical lines with crosslink capacitor to solitary waves which are propagating in the new nonlinear hybrid electrical lines; we therefore, facilitate the choice of the type of line relative to the type of signal that we want to transmit.

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