scholarly journals Traveling wave solutions for (3 + 1) dimensional equations arising in fluid mechanics

2014 ◽  
Vol 3 (4) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Hassam Khan
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Traveling Wave ◽  
Couple Stress ◽  
Traveling Wave Solutions ◽  
Wave Parameter ◽  
Fluid Models ◽  
Restrictive Assumption ◽  
Wave Solutions

AbstractIn this note, traveling wave solutions for (3 + 1) dimensional fluid models of incompressible flow are considered. The governing partial differential equations of two models are reduced to ordinary differential equation by employing wave parameter and exact solutions are obtained. It is shown that these fluid models allow 3 + 1 dimensional solutions amongst each other. The methodology used in this work is independent of symmetric consideration and other restrictive assumption. Finally, a set of example of boundary condition is discussed for the couple stress fluid. It is observed that velocity profile strongly depends upon couple stress parameter.

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.

Traveling Wave Solutions for MHD Aligned Flow of a Second Grade Fluid

2010 ◽  
Vol 8 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Amir Mahmood ◽  
Muhammad Jamil ◽  
Nasir-Uddin Khan
Keyword(s):  
Differential Equations ◽  
Wave Phenomenon ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Parameter ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Restrictive Assumption ◽  
Wave Solutions ◽  
Grade Fluid

In this work, an approach based on traveling wave phenomenon is implemented for finding exact solutions of MHD aligned flow of an incompressible second grade fluid. The partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) by using wave parameter. The methodology used in this work is independent of symmetry consideration and other restrictive assumption. Comparison is made with the results obtained previously.

scholarly journals Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation

1999 ◽  
Vol 22 (3) ◽  
pp. 643-648 ◽  
Author(s):  
Kenneth L. Jones ◽  
Yunkai Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equation ◽  
Traveling Wave ◽  
Boussinesq Equation ◽  
Traveling Wave Solutions ◽  
Nonlinear Ordinary Differential Equation ◽  
Wave Solutions ◽  
Type Integral ◽  
Periodic Traveling Wave

The generalized forced Boussinesq equation,utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transformz=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous periodic functions with a given period2T. The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant periodic traveling wave solutions to the generalized forced Boussinesq equation is established.

Numerical Study of Periodic Traveling Wave Solutions for the Predator–Prey Model with Landscape Features

2015 ◽  
Vol 25 (09) ◽  
pp. 1550117 ◽  
Author(s):  
Ana Yun ◽  
Jaemin Shin ◽  
Yibao Li ◽  
Seunggyu Lee ◽  
Junseok Kim
Keyword(s):  
Boundary Condition ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Dirichlet Boundary ◽  
Numerical Technique ◽  
Traveling Wave Solutions ◽  
Predator Prey ◽  
Landscape Features ◽  
Wave Solutions ◽  
Periodic Traveling Wave

We numerically investigate periodic traveling wave solutions for a diffusive predator–prey system with landscape features. The landscape features are modeled through the homogeneous Dirichlet boundary condition which is imposed at the edge of the obstacle domain. To effectively treat the Dirichlet boundary condition, we employ a robust and accurate numerical technique by using a boundary control function. We also propose a robust algorithm for calculating the numerical periodicity of the traveling wave solution. In numerical experiments, we show that periodic traveling waves which move out and away from the obstacle are effectively generated. We explain the formation of the traveling waves by comparing the wavelengths. The spatial asynchrony has been shown in quantitative detail for various obstacles. Furthermore, we apply our numerical technique to the complicated real landscape features.

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 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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