Fully discrete approximation of a second order linear evolution equation related to the water wave problem

1993 ◽  
pp. 361-380
Author(s):  
Ushijima Teruo ◽  
Matsuki Mihoko
Keyword(s):  
Evolution Equation ◽  
Water Wave ◽  
Discrete Approximation ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Order Linear ◽  
Fully Discrete ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Fully Discrete Approximation

New Form of the Hamiltonian Equations for the Nonlinear Water-Wave Problem, Based on a New Representation of the DTN Operator, and Some Applications

2015 ◽  
Author(s):  
Gerassimos A. Athanassoulis ◽  
Christos E. Papoutsellis
Keyword(s):  
Water Wave ◽  
Evolution Equations ◽  
Hamiltonian Formulation ◽  
Series Representation ◽  
Free Surface Elevation ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Linear Evolution ◽  
Dirichlet To Neumann Operator ◽  
Local Depth

We present a new Hamiltonian formulation for the non-linear evolution of surface gravity waves over a variable impermeable bottom. The derivation is based on Luke’s variational principle and the use of an exact (convergent up to the boundaries) infinite-series representation of the unknown wave potential, in terms of a system of prescribed vertical functions (explicitly dependent on the local depth and the local free-surface elevation) and unknown horizontal modal amplitudes. The key idea of this approach is the introduction of two unconventional modes ensuring a rapid convergence of the modal series. The fully nonlinear water-wave problem is reformulated as two evolution equations, essentially equivalent with the Zakharov-Craig-Sulem formulation. The Dirichlet-to-Neumann operator (DtN) over arbitrary bathymetry is determined by means of a few first modes, the two unconventional ones being most important. While this formulation is exact, its numerical implementation, even for general domains, is not much more involved than that of the various simplified models (Boussinesq, Green-Nagdhi) widely used in engineering applications. The efficiency of this formulation is demonstrated by the excellent agreement of the numerical and experimental results for the case of the classical Beji-Battjes experiment. A more complicated bathymetry is also studied.

scholarly journals Fractional approximation of solutions of evolution equations

Analysis ◽  
2016 ◽  
Vol 36 (2) ◽  
Author(s):  
Anatoly N. Kochubei ◽  
Yuri G. Kondratiev
Keyword(s):  
Evolution Equation ◽  
Analytic Continuation ◽  
Evolution Equations ◽  
Order Linear ◽  
First Order ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Fractional Approximation ◽  
Fractional Equation

AbstractWe show how to approximate a solution of the first order linear evolution equation, together with its possible analytic continuation, using a solution of the time-fractional equation of order

scholarly journals An alternative approach to study irrotational periodic gravity water waves

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Biswajit Basu ◽  
Calin I. Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Stagnation Points ◽  
Alternative Approach ◽  
New Formulation ◽  
Bounded Below ◽  
Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

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