Painlevé analysis and traveling wave solutions of the sixth order differential equation with non-local nonlinearity

Optik ◽  
2021 ◽  
pp. 167586
Author(s):  
Nikolay A. Kudryashov ◽  
Dariya V. Safonova
Keyword(s):  
Differential Equation ◽  
Traveling Wave ◽  
Order Differential Equation ◽  
Traveling Wave Solutions ◽  
Sixth Order ◽  
Painlevé Analysis ◽  
Wave Solutions ◽  
Non Local ◽  
Local Nonlinearity ◽  
Painleve Analysis

scholarly journals Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation

2000 ◽  
Vol 24 (6) ◽  
pp. 371-377 ◽  
Author(s):  
Kenneth L. Jones ◽  
Xiaogui He ◽  
Yunkai Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equations ◽  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
De Vries ◽  
Periodic Traveling Wave

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries equation in the form of(uη+u/(2η)+[f(u)]ξ+uξξξ)ξ+uθθ/η2=h0. The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili equation,(ut+[f(u)]x+uxxx)x+uyy=h0, and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schauder's fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

scholarly journals New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah ◽  
M. Ali Akbar
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solutions ◽  
Traveling Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equation ◽  
Traveling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions ◽  
Higher Dimensional

We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

A bounded and monotone finite-difference solution of a hyperbolic Burgers–Fisher equation

2018 ◽  
Vol 29 (12) ◽  
pp. 1850122
Author(s):  
L. A. Flores-Oropeza ◽  
A. Román-Loera ◽  
Ahmed S. Hendy
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Traveling Wave ◽  
Boundary Data ◽  
Traveling Wave Solutions ◽  
Initial Boundary ◽  
Fisher Equation ◽  
Wave Solutions ◽  
Nonlinear Advection ◽  
Finite Difference Solution

In this work, a nonlinear finite-difference scheme is provided to approximate the solutions of a hyperbolic generalization of the Burgers–Fisher equation from population dynamics. The model under study is a partial differential equation with nonlinear advection, reaction and damping terms. The existence of some traveling-wave solutions for this model has been established in the literature. In the present manuscript, we investigate the capability of our technique to preserve some of the most important features of those solutions, namely, the positivity, the boundedness and the monotonicity. The finite-difference approach followed in this work employs the exact solutions to prescribe the initial-boundary data. In addition to providing good approximations to the analytical solutions, our simulations suggest that the method is also capable of preserving the mathematical features of interest.

scholarly journals Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1999
Author(s):  
Hongpeng Guo ◽  
Zhiming Guo
Keyword(s):  
Diffusion Equation ◽  
Traveling Wave ◽  
Local Reaction ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Birth Function ◽  
Wave Solutions ◽  
Monotonicity Assumption ◽  
Non Local

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed c>c*, where c*>0 is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions.

Algebraic Traveling Wave Solutions of a Non-local Hydrodynamic-type Model

2014 ◽  
Vol 17 (3-4) ◽  
pp. 465-482 ◽  
Author(s):  
Aiyong Chen ◽  
Wenjing Zhu ◽  
Zhijun Qiao ◽  
Wentao Huang
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Hydrodynamic Type ◽  
Type Model ◽  
Wave Solutions ◽  
Non Local

EXISTENCE OF EXACT FAMILIES OF TRAVELING WAVE SOLUTIONS FOR THE SIXTH-ORDER RAMANI EQUATION AND A COUPLED RAMANI EQUATION

2012 ◽  
Vol 22 (01) ◽  
pp. 1250002 ◽  
Author(s):  
JIBIN LI
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Sixth Order ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

By using the method of dynamical systems and the results in [Li & Zhang, 2011] to the sixth-order Ramani equation and a coupled Ramani equation, the families of exact traveling wave solutions can be obtained.

scholarly journals Explicit Exact Solutions of the (2+1)-Dimensional Integro-Differential Jaulent–Miodek Evolution Equation Using the Reliable Methods

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Supaporn Kaewta ◽  
Sekson Sirisubtawee ◽  
Nattawut Khansai
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Transformation ◽  
Jacobi Elliptic Function ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Wave Solutions

In this article, we utilize the G′/G2-expansion method and the Jacobi elliptic equation method to analytically solve the (2 + 1)-dimensional integro-differential Jaulent–Miodek equation for exact solutions. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials (Jaulent and Miodek, 1976; Jaulent, 1976). The equation is converted into a fourth order partial differential equation using a transformation. After applying a traveling wave transformation to the resulting partial differential equation, we obtain an ordinary differential equation which is the main equation to which the both schemes are applied. As a first step, the two methods give us distinguish systems of algebraic equations. The first method provides exact traveling wave solutions including the logarithmic function solutions of trigonometric functions, hyperbolic functions, and polynomial functions. The second approach provides the Jacobi elliptic function solutions depending upon their modulus values. Some of the obtained solutions are graphically characterized by the distinct physical structures such as singular periodic traveling wave solutions and peakons. A comparison between our results and the ones obtained from the previous literature is given. Obtaining the exact solutions of the equation shows the simplicity, efficiency, and reliability of the used methods, which can be applied to other nonlinear partial differential equations taking place in mathematical physics.

scholarly journals Exact traveling wave solutions for a new nonlinear heat transfer equation

2017 ◽  
Vol 21 (4) ◽  
pp. 1833-1838 ◽  
Author(s):  
Feng Gao ◽  
Xiao-Jun Yang ◽  
Yu-Feng Zhang
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Linear Partial Differential Equation ◽  
Heat Transfer Equation ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Transfer Problems

In this paper, we propose a new non-linear partial differential equation to de-scribe the heat transfer problems at the extreme excess temperatures. Its exact traveling wave solutions are obtained by using Cornejo-Perez and Rosu method.

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