An evaluation of a model equation for water waves

1981 ◽  
Vol 302 (1471) ◽  
pp. 457-510 ◽  
Keyword(s):  
Theoretical Model ◽  
Initial Value Problem ◽  
Water Waves ◽  
Laboratory Experiments ◽  
Initial Conditions ◽  
Initial Value ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Spatial Periodicity ◽  
Long Channel

This study assesses a particular model for the unidirectional propagation of water waves, comparing its predictions with the results of a set of laboratory experiments. The equation to be tested is a one-dimensional representation of weakly nonlinear, dispersive waves in shallow water. A model for such flows was proposed by Korteweg & de Vries (1895) and this has provided the theoretical basis for a number of laboratory experiments. Some recent studies that have been made in the area are those of Zabusky & Galvin (1971), Hammack (1973) and Hammack & Segur (1974). In each case the theoretical model gave a good qualitative account of the experiments, but the quantitative comparisons were not very extensive. One of the purposes of this paper is to provide a more detailed quantitative assessment of a particular model than has been given to date. An important aspect of the formulation of the theoretical model is the specification of the initial conditions and the boundary conditions for the equation. Zabusky & Galvin (1971) considered an initial-value problem having spatial periodicity, whereas Hammack (1973) and Hammack & Segur (1974) considered an initial-value problem posed on the real line. In contrast, we shall consider an initial-value problem posed on the half line with boundary data specified at the origin. This problem was chosen to correspond with an experiment in which waves were generated at one end of a long channel, and obviates certain difficulties inherent in the other formulations.

scholarly journals Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

2021 ◽  
Vol 24 (4) ◽  
pp. 32-39
Author(s):  
Hussein M. Sagban ◽  
◽  
Fadhel S. Fadhel ◽  
Keyword(s):  
Approximate Solution ◽  
Rate Of Convergence ◽  
Initial Value Problem ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

STABILITY OF FUZZY DYNAMIC SYSTEMS

2009 ◽  
Vol 17 (01) ◽  
pp. 69-83 ◽  
Author(s):  
MARINA TUYAKO MIZUKOSHI ◽  
LAÉCIO CARVALHO BARROS ◽  
RODNEY CARLOS BASSANEZI
Keyword(s):  
Fuzzy Sets ◽  
Dynamic Systems ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Flow Equation ◽  
Equilibrium Stability ◽  
Initial Value ◽  
Flow Through

In this work we are study the Fuzzy Initial Value Problem (FIVP) with parameters and/or initial conditions given by fuzzy sets. Starting from the flow equation of the deterministic Initial Value Problem (IVP) associates to FIVP, we obtain the FIVP flow, through the principle of Zadeh. Follow, we introduce the concept of fuzzy equilibrium stability of FIVP and some examples are given.

On the Understanding of Chaos in Duffings Equation Including a Comparison With Experiment

1986 ◽  
Vol 53 (1) ◽  
pp. 5-9 ◽  
Author(s):  
E. H. Dowell ◽  
C. Pezeshki
Keyword(s):  
Theoretical Model ◽  
Initial Value Problem ◽  
Physical Experiment ◽  
Chaotic Oscillations ◽  
External Excitation ◽  
Initial Value ◽  
Comparison Of Results ◽  
Buckled Beam ◽  
Comparison With Experiment ◽  
Forced Excitation

The dynamics of a buckled beam are studied for both the initial value problem and forced external excitation. The principal focus is on chaotic oscillations due to forced excitation. In particular, a discussion of their relationship to the initial value problem and a comparison of results from a theoretical model with those from a physical experiment are presented.

scholarly journals Solving a well-posed fractional initial value problem by a complex approach

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Arran Fernandez ◽  
Sümeyra Uçar ◽  
Necati Özdemir
Keyword(s):  
Fixed Point ◽  
Initial Value Problem ◽  
Complex Analysis ◽  
Fixed Point Theory ◽  
Initial Conditions ◽  
Point Theory ◽  
Initial Value ◽  
Complex Approach ◽  
Set Up ◽  
Well Posed

AbstractNonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces. Here we combine fixed point theory with complex analysis, considering spaces of analytic functions and the behaviour of complex powers. It is necessary to study carefully the initial value properties of Riemann–Liouville fractional derivatives in order to set up an appropriate initial value problem, since some such problems considered in the literature are not well-posed due to their initial conditions. The problem that emerges turns out to be dimensionally consistent in an unexpected way, and therefore suitable for applications too.

Some aspects of the initial-value problem for the inviscid motion of a contained, rotating, weakly-stratified fluid

1971 ◽  
Vol 46 (1) ◽  
pp. 1-23 ◽  
Author(s):  
J. S. Allen
Keyword(s):  
Initial Value Problem ◽  
Initial Conditions ◽  
Stratified Fluid ◽  
Rotational Frequency ◽  
Rotating Fluid ◽  
Initial Value ◽  
Homogeneous Fluid ◽  
Relationship Of ◽  
The Relationship ◽  
Rotating Stratified Fluid

The initial-value problem for the linear, inviscid motion of a contained, rotating stratified fluid is considered in the limit of weak stratification, that is, for small values of the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The limiting flow is of interest because, although the initial-value problem has been studied, both for the case of a homogeneous, rotating fluid and for the case of a stratified, rotating fluid, the exact relationship of the two flows, in the limit of vanishing stratification, is not straightforward. For example, the method of determining, from the initial conditions, the steady geostrophic component of the flow of a rotating, stratified fluid does not in general give a motion that reduces, in the limit S → 0, to the steady component of the flow of a homogeneous fluid. By including a consideration of slow unsteady motions that vary on a time scale dependent on the stratification parameter, the relationship of the limiting flow to the flow of a homogeneous fluid is established.

scholarly journals Integral inequalities of Gronwall type for piecewise continuous functions

1997 ◽  
Vol 10 (1) ◽  
pp. 89-94 ◽  
Author(s):  
Drumi D. Bainov ◽  
Snezhana G. Hristova
Keyword(s):  
Differential Equations ◽  
Integral Inequality ◽  
Initial Value Problem ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Continuous Functions ◽  
Integral Inequalities ◽  
Initial Value ◽  
Volterra Type ◽  
Piecewise Continuous

In this paper we generalize the integral inequality of Gronwall and study the continuous dependence of the solution of the initial value problem for nonlinear impulsive integro-differential equations of Volterra type on the initial conditions.

scholarly journals Spatially Quasi-Periodic Water Waves of Infinite Depth

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Jon Wilkening ◽  
Xinyu Zhao
Keyword(s):  
Initial Value Problem ◽  
Water Waves ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Small Scale ◽  
Normal Velocity ◽  
Natural Setting ◽  
Two Dimensional ◽  
Periodic Functions ◽  
Initial Value

AbstractWe formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

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