Water Waves in a Nonhomogeneous Incompressible Fluid

1977 ◽  
Vol 44 (4) ◽  
pp. 523-528 ◽  
Author(s):  
A. E. Green ◽  
P. M. Naghdi
Keyword(s):  
Surface Tension ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Incompressible Fluid ◽  
Water Waves ◽  
Nonlinear Differential Equations ◽  
Long Waves ◽  
Direct Approach ◽  
Hydraulic Jumps ◽  
Two Dimensional

After a brief discussion of some undesirable features of a number of different partial differential equations often employed in the existing literature on water waves, a relatively simple restricted theory is constructed by a direct approach which is particularly suited for applications to problems of fluid sheets. The rest of the paper is concerned with a derivation of a system of nonlinear differential equations (which may include the effects of gravity and surface tension) governing the two-dimensional motion of incompressible in-viscid fluids for propagation of fairly long waves in a nonhomogeneous stream of water of variable initial depth, as well as some new results pertaining to hydraulic jumps. The latter includes an additional class of possible solutions not noted previously.

On the evolution of packets of water waves

1979 ◽  
Vol 92 (4) ◽  
pp. 691-715 ◽  
Author(s):  
Mark J. Ablowitz ◽  
Harvey Segur
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Evolution Equations ◽  
Wave Packets ◽  
Short Wave ◽  
Similarity Solutions ◽  
Two Dimensions ◽  
Long Waves ◽  
Two Dimensional ◽  
One Dimensional

We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separate models are discussed, depending on whether or not the waves are long in comparison with the fluid depth. These models are two-dimensional generalizations of the Korteweg-de Vries equation (for long waves) and the cubic nonlinear Schrödinger equation (for short waves). In either case, we find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth. In particular, for the long waves, one-dimensional (KdV) solitons become unstable with respect to even longer transverse perturbations when the surface-tension parameter becomes large enough, i.e. in very thin sheets of water. Two-dimensional long waves (‘lumps’) that decay algebraically in all horizontal directions and interact like solitons exist only when the one-dimensional solitons are found to be unstable.The most dramatic consequence of surface tension and depth, however, occurs for capillary-type waves in sufficiently deep water. Here a packet of waves that are everywhere small (but not infinitesimal) and modulated in both horizontal dimensions can ‘focus’ in a finite time, producing a region in which the wave amplitudes are finite. This nonlinear instability should be stronger and more apparent than the linear instabilities examined to date; it should be readily observable.Another feature of the evolution of short wave packets in two dimensions is that all one-dimensional solitons are unstable with respect to long transverse perturbations. Finally, we identify some exact similarity solutions to the evolution equations.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

scholarly journals SOLVING SYSTEMS OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA TWO DIMENSIONAL LAPLACE TRANSFORMS

2020 ◽  
Vol 5 (12) ◽  
pp. 406-420
Author(s):  
A. Aghili ◽  
M.R. Masomi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Heat Equations ◽  
Two Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Homogeneous Systems ◽  
Fractional Pde

In this article, the authors used two dimensional Laplace transform to solve non - homogeneous sub - ballistic fractional PDE and homogeneous systems of time fractional heat equations. Constructive examples are also provided.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

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