• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.81-108
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• 2003

We consider the Korteweg-de Vries-Burgers (KdVB) equation on the domain [0,11]. We derive a control law which guarantees $L^2$-global exponential stability, $H^3$-global asymptotic stability, and $H^3$-semiglobal exponential stability.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.351-367
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• 2011

In this article, we construct exact traveling wave solutions for nonlinear PDEs in mathematical physics via the (1+1)- dimensional combined Korteweg- de Vries and modified Korteweg- de Vries (KdV-mKdV) equation, the (1+1)- dimensional compouned Korteweg- de Vries Burgers (KdVB) equation, the (2+1)- dimensional cubic Klien- Gordon (cKG) equation, the Generalized Zakharov- Kuznetsov- Bonjanmin- Bona Mahony (GZK-BBM) equation and the modified Korteweg- de Vries - Zakharov- Kuznetsov (mKdV-ZK) equation, by using the (($\frac{G'}{G}$) -expansion method combined with the Riccati equation, where G = $G({\xi})$ satisfies the Riccati equation $G'({\xi})=A+BG^2$ and A, B are arbitrary constants.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.14 no.4
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• pp.308-318
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• 2002

The requirements to the numerical model of wind-generated waves in shallow water are discussed in the framework of the Korteweg-de Vries equation. The weakness of nonlinearity and dispersion required for the Korteweg-de Vries equation applicability is considered for fully developed sea, non-stationary wind waves and swell, including some experimental data. We note for sufficient evaluation of the freak wave statistics it is necessary to consider more than about 10,000 waves in the wave record, and this leads to the limitation of the numerical domain and number of realizations. The numerical modelling of irregular water waves is made to demonstrate the possibility of effective evaluation of the statistical properties of freak waves with heights equal to 2-2.3 significant wave height.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.5 no.4
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• pp.307-317
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• 1993

The problem of sea wave transformation in the coastal zone taking into account effects of nonlinearity and disperison has been studied. Mathematical model for description of regular wave transformation is based on the method of nonlinear ray theory. The equations for rays and wave field have been produced. Nonlinear wave field is described by the modified Korteweg-de Vries equation. Some analytical solutions of this equation are obtained. Caustic transformation and dissipation effects are included in the mathematical model. Numerical algorithm of solution of the Korteweg-de Vries equation and its stability criterion are described. Results of nonlinear transformation of sea waves in the coastal zone are demonstrated.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.565-572
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• 2019

In this paper, we consider a Korteweg-de Vries equation in noncylindrical domain. This work is devoted to prove existence and uniqueness of global solutions employing Faedo-Galerkin's approximation and transformation of the noncylindrical domain with moving boundary into cylindrical one. Moreover, we estimate the exponential decay of solutions in the asymptotically cylindrical domain.

• Honam Mathematical Journal
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• v.43 no.3
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• pp.373-383
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• 2021

In this study, a numerical method deals with the Chebyshev wavelet collocation and Adomian decomposition methods are proposed for solving Korteweg-de Vries equation. Integration of the Chebyshev wavelets operational matrices is derived. This problem is reduced to a system of non-linear algebraic equations by using their operational matrix. Thus, it becomes easier to solve KdV problem. The error estimation for the Chebyshev wavelet collocation method and ADM is investigated. The proposed method's validity and accuracy are demonstrated by numerical results. When the exact and approximate solutions are compared, for non-linear or linear partial differential equations, the Chebyshev wavelet collocation method is shown to be acceptable, efficient and accurate.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.557-573
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• 1996

When we attempt to describe the propagation of nonlinear dispersive waves, it is frequently necessary to take account of diffipative effects.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.5 no.3
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• pp.191-197
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• 1993

Mathematical models of nonlinear waves due to disturbances moving with the near critical velocity in a basin of variable depth are discussed. A two-dimensional model for waves of arbitrary amplitude is developed. In the case of small perturbation it is shown that nonlinear ray method can be applied to obtain the generalized forced Korteweg-de Vries equation.

• Bulletin of the Society of Naval Architects of Korea
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• v.24 no.2
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• pp.55-61
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• 1987

The wave system due to a moving submerged body is investigated both theoretically and numerically. Boussinesq equation, which is derived under the assumption that the effects of nonlinearity and wave dispersion are of the same order, is generalized to take the forcing agency into account. Furthermore, under the more restrive assumption that the disturbance is of higher order, inhomogeneous Korteweg-de Vries equation is derived. These equations are solved numerically to obtain the generated wave system and the wave-making resistance. These results are compared with those given by the linear theory.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.1017-1034
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• 2023

In this paper, we work two different problems. First, we investigate a new class of fractional differential equations involving Caputo sequential derivative with some (e₁, e₂, θ)-periodic conditions. The existence and uniqueness of solutions are proven. The stability of solutions is also discussed. The second part includes studying traveling wave solutions of a conformable fractional Korteweg-de Vries-Burger (KdV Burger) equation through the Tanh method. Graphs of some of the waves are plotted and discussed, and a conclusion follows.