NEW BOUNDED TRAVELING WAVES OF CAMASSA–HOLM EQUATION

2004 ◽  
Vol 14 (10) ◽  
pp. 3541-3556 ◽  
Author(s):  
ZHENGRONG LIU ◽  
QIXIU LI ◽  
QINGMEI LIN
Keyword(s):  
Differential Equations ◽  
Traveling Waves ◽  
Simulation Method ◽  
Numerical Results ◽  
Bifurcation Method ◽  
Holm Equation ◽  
Kink Waves ◽  
Planar Systems ◽  
Implicit Expressions

In this paper, the bifurcation method of planar systems and simulation method of differential equations are employed to investigate the bounded traveling waves of the Camassa–Holm equation. Some new bounded traveling waves are found and their implicit expressions are obtained. Both qualitative and numerical results show that they possess the properties of compactons or generalized kink waves.

GENERALIZED KINK WAVES IN A GENERAL COMPRESSIBLE HYPERELASTIC ROD

2005 ◽  
Vol 15 (08) ◽  
pp. 2671-2679 ◽  
Author(s):  
ZHENGRONG LIU ◽  
YAO LONG
Keyword(s):  
Numerical Simulation ◽  
Qualitative Analysis ◽  
Traveling Waves ◽  
Planar Graphs ◽  
Kink Waves ◽  
Implicit Expressions

In this paper, we employ both qualitative analysis and numerical simulation to investigate bounded traveling waves in a general compressible hyperelastic rod. Some new bounded traveling waves are found. Their implicit expressions are obtained. Also, their planar graphs are simulated. Since they possess some properties of kink waves, we call them generalized kink waves.

PERIODIC WAVES AND THEIR LIMITS FOR THE CAMASSA–HOLM EQUATION

2006 ◽  
Vol 16 (08) ◽  
pp. 2261-2274 ◽  
Author(s):  
ZHENGRONG LIU ◽  
ALI MOHAMMED KAYED ◽  
CAN CHEN
Keyword(s):  
Periodic Wave ◽  
Periodic Waves ◽  
Implicit Functions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Holm Equation ◽  
Implicit Expressions ◽  
Periodic Cusp Waves ◽  
Theoretical Results

In this paper, the bifurcation method of dynamical systems is employed to study the Camassa–Holm equation [Formula: see text] We investigate the periodic wave solutions of form u = φ(ξ) which satisfy φ(ξ + T) = φ(ξ), here ξ = x - ct and c, T are constants. Their six implicit expressions and two explicit expressions are obtained. We point out that when the initial values are changed, the periodic waves may become three waves, periodic cusp waves, smooth solitary waves and peakons. Our results give an explanation to the appearance of periodic cusp waves and peakons. Moreover, three sets of graphs of the implicit functions are drawn, and three sets of numerical simulations are displayed. The identity of these graphs and simulations imply the correctness of our theoretical results.

scholarly journals Bounded Traveling Waves of the (2+1)-Dimensional Zoomeron Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Yuqian Zhou ◽  
Shanshan Cai ◽  
Qian Liu
Keyword(s):  
Dynamical System ◽  
Traveling Waves ◽  
Simulation Method ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Parameter Bifurcation

The bifurcation method of dynamical system and numerical simulation method of differential equation are employed to investigate the (2+1)-dimensional Zoomeron equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. According to these phase portraits, all bounded traveling waves are identified and simulated, including solitary wave solutions, shock wave solutions, and periodic wave solutions. Furthermore, all exact expressions of these bounded traveling waves are given. Among them, the elliptic function periodic wave solutions are new solutions.

Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed M. Khader
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Numerical Results ◽  
Convergence Order ◽  
Variable Order ◽  
Numerical Treatment ◽  
Dynamical Models ◽  
Fractional Modeling ◽  
Dynamics Problems

AbstractThis paper is devoted to introduce a numerical treatment using the generalized Adams-Bashforth-Moulton method for some of the variable-order fractional modeling dynamics problems, such as Riccati and Logistic differential equations. The fractional derivative is described in Caputo variable-order fractional sense. The obtained numerical results of the proposed models show the simplicity and efficiency of the proposed method. Moreover, the convergence order of the method is also estimated numerically.

Exact Elasticity Solutions for Stresses and Deflections in Curved Beams and Rings of Exponential and T-Sections

1991 ◽  
Author(s):  
Cemil Bagci
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Plane Stress ◽  
Numerical Results ◽  
Neutral Axis ◽  
Curved Beams ◽  
Equilibrium Stress ◽  
Different Boundary Conditions ◽  
Stress Functions ◽  
Elasticity Solutions

Abstract Exact elasticity solutions for stresses and deflections (displacements) in curved beams and rings of varying thicknesses are developed using polar elasticity and state of plane stress. Basic forms of differential equations of equilibrium, stress functions, and differential equations of compatibility are given. They are solved to develop expressions for radial, tangential, and shearing stresses for moment, force, and combined loadings. Neutral axis location for each type of loading is determined. Expressions for displacements are developed utilizing strain-displacement relationships of polar elasticity satisfying boundary conditions on displacements. In case of full rings stresses are as in curved beams with properly defined moment loading, but displacements differ satisfying different boundary conditions. The developments for constant thicknesses are used to develop solutions for curved beams and rings with T-sections. Comparative numerical results are given.

scholarly journals A comparison of Adomian decomposition method and RK4 algorithm on Volterra integro differential equations of 2nd kind

2015 ◽  
Vol 4 (4) ◽  
pp. 481
Author(s):  
Kekana M.C ◽  
Shatalov M.Y ◽  
Moshokoa S.P
Keyword(s):  
Differential Equations ◽  
Infinite Series ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Numerical Results ◽  
Adomian Decomposition

In this paper, Volterra Integro differential equations are solved using the Adomian decomposition method. The solutions are obtained in form of infinite series and compared to Runge-Kutta4 algorithm. The technique is described and illustrated with examples; numerical results are also presented graphically. The software used in this study is mathematica10.

scholarly journals Nonstationary flow of roiling liquid

1984 ◽  
Vol 6 (4) ◽  
pp. 12-20
Author(s):  
Duong Ngoc Hai
Keyword(s):  
Mass Transfer ◽  
Differential Equations ◽  
Phase Boundary ◽  
Heat And Mass Transfer ◽  
Numerical Results ◽  
Linear Differential Equations ◽  
Nonstationary Flow ◽  
One Dimensional ◽  
Boiling Liquid ◽  
Linear Differential

Steady one-dimensional nonstationary flow of boiling liquid from finite or infinit pipe in a consideration of the effect of the phase-boundary heat and mass transfer. The Received system of quasi-linear differential equations has been decided by the modificati on of Lax - wendroff method in IBM. Numerical results are compared as xperimental data.

scholarly journals A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 07 (01) ◽  
pp. 1350034 ◽  
Author(s):  
M. B. Suleiman ◽  
H. Musa ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Error Constant

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.

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