A new fourth-order numerical algorithm for a class of three-dimensional nonlinear evolution equations

2012 ◽  
Vol 29 (1) ◽  
pp. 102-130 ◽  
Author(s):  
Dingwen Deng ◽  
Chengjian Zhang
Keyword(s):  
Numerical Algorithm ◽  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations

scholarly journals Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water

2002 ◽  
Vol 43 (4) ◽  
pp. 513-524 ◽  
Author(s):  
Suma Debsarma ◽  
K.P. Das
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Nonlinear Evolution Equations ◽  
Marginal Stability ◽  
Stability And Instability ◽  
The Stability

AbstractFor a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

Construction of the numerical and analytical wave solutions of the Joseph–Egri dynamical equation for the long waves in nonlinear dispersive systems

2020 ◽  
Vol 34 (30) ◽  
pp. 2050289
Author(s):  
Abdulghani R. Alharbi ◽  
M. B. Almatrafi ◽  
Aly R. Seadawy
Keyword(s):  
Solitary Wave ◽  
Numerical Solution ◽  
Evolution Equations ◽  
Dynamical Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability

The Kudryashov technique is employed to extract several classes of solitary wave solutions for the Joseph–Egri equation. The stability of the achieved solutions is tested. The numerical solution of this equation is also investigated. We also present the accuracy and the stability of the numerical schemes. Some two- and three-dimensional figures are shown to present the solutions on some specific domains. The used methods are found useful to be applied on other nonlinear evolution equations.

scholarly journals Multiple Lump Novel and Accurate Analytical and Numerical Solutions of the Three-Dimensional Potential Yu–Toda–Sasa–Fukuyama Equation

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2081
Author(s):  
Mostafa M. A. Khater ◽  
Dumitru Baleanu ◽  
Mohamed S. Mohamed
Keyword(s):  
Evolution Equations ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Dispersive Media ◽  
Analytical And Numerical Solutions ◽  
Potential Form ◽  
Analytical Strategies ◽  
Ytsf Equation

The accuracy of novel lump solutions of the potential form of the three–dimensional potential Yu–Toda–Sasa–Fukuyama (3-Dp-YTSF) equation is investigated. These solutions are obtained by employing the extended simplest equation (ESE) and modified Kudryashov (MKud) schemes to explore its lump and breather wave solutions that characterizes the dynamics of solitons and nonlinear waves in weakly dispersive media, plasma physics, and fluid dynamics. The accuracy of the obtained analytical solutions is investigated through the perspective of numerical and semi-analytical strategies (septic B-spline (SBS) and variational iteration (VI) techniques). Additionally, matching the analytical and numerical solutions is represented along with some distinct types of sketches. The superiority of the MKud is showed as the fourth research paper in our series that has been beginning by Mostafa M. A. Khater and Carlo Cattani with the title “Accuracy of computational schemes”. The functioning of employed schemes appears their effectual and ability to apply to different nonlinear evolution equations.

scholarly journals Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

1997 ◽  
Vol 39 (2) ◽  
pp. 214-229 ◽  
Author(s):  
Sudebi Bhattacharyya ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Maximum Growth ◽  
Single Equation ◽  
Maximum Growth Rate ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Coupled Equations

AbstractTwo coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

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