• Bulletin of the Korean Chemical Society
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• v.7 no.4
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• pp.283-288
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• 1986

A nonlinear analysis is presented for the treatment of fluctuations near the critical point in the presence of diffusion in the Schlogl models. The two time scaling method is used to obtain an evolution equation for the amplitude of fluctuations. It is shown that the fluctuations decay to zero in the stable region and they are enhanced to a finite value as time goes to infinity in the unstable region.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.1031-1042
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• 2000

The equations prescribed in Ω⊂R(sup)n are called loaded, if they contain some operations of the traces of desired solution on manifolds (of dimension which is strongly less than n) from closure Ω. These equations result from approximations of nonlinear equations by linear ones, in the problems of optimal control when the control when the control actions depends on a part of independent variables, in investigations of the inverse problems and so on. In present work we study the nonlocal boundary value problems for first-order loaded differential operator equations. Criterion of unique solvability is established. We illustrate the obtained results by examples.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1725-1739
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• 2016

Abstract. Numerical solutions for the evolutionary space fractional order differential equations are considered. A predictor corrector method is applied in order to obtain numerical solutions for the equation without solving nonlinear systems iteratively at every time step. Theoretical error estimates are performed and computational results are given to show the theoretical results.

• Journal of the Society of Naval Architects of Korea
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• v.28 no.1
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• pp.53-59
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• 1991

By employing multiple-scale expansion techniques, the diffraction of sinusoidally-modulated nonlinear Stokes waves by a stationary thin wedge has been studied within the framework of potential theory. It is found that the evolution of diffracted waves can be described by the Zakharov equation to the leading order and it can be replaced by the cubic $Schr\ddot{o}dinger$ equation with an additional linear term for stable modulations. Computations are made for the cubic $Schr\ddot{o}dinger$ equation with different values of nonlinear and dispersion parameters. Numerical results well reflect the experimental findings in the amplitude and width of generated stem waves. It is numerically confirmed that the nonlinearity dominates the wave field, while the dispersion hardly affects the wave evolution, and stem waves are likely to be formed for steep incident waves in the case of stable sinusoidal modulations.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.669-688
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• 2021

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1998.04a
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• pp.101-109
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• 1998

Genetic Algorithm(GA), which is based on the theory of natural evolution, has been evaluated highly for their robust performances. The optimization problems on truss structures under the prescribed displacement are solved by using GA. In this paper, the homologous deformation of structures was proposed as the prescribed displacement. The shape analysis of structures is a kind of inverse problems different from stress analysis, and the governing equation becomes nonlinear. In this regard, GA was used to solve the nonlinear equation. In this study, the shape analysis method in which not only the positions of the objective nodes but also the layout and sectional area of the member are encoded to strings in the GA as design parameters of the structures is proposed.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.23 no.5
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• pp.335-344
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• 2011

A curvilinear non-hydrostatic free surface model is developed to investigate nonlinear wave interactions in a circular channel. The proposed model solves the unsteady Navier-Stokes equations in a three-dimensional domain with a pressure correction method, which is one of fractional step methods. A hybrid staggered-grid layout in the vertical direction is implemented, which renders relatively simple resulting pressure equation as well as free surface closure. Numerical accuracy with respect to wave nonlinearity is tested against the fifth-order Stokes solution in a two-dimensional numerical wave tank. Numerical applications center on the evolution of nonlinear waves including diffraction and reflection affected by the curvature of side wall in a circular channel comparing with linear waves. Except for a highly nonlinear bichrmatic wave, the model's results are in good agreement with superimposed analytical solution that neglects nonlinear effects. Through the numerical simulation of the highly nonlinear bichramatic wave, the model shows its capability to investigate the evolution of nonlinear wave groups in a circular channel.

• 제어로봇시스템학회:학술대회논문집
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• 2004.08a
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• pp.1769-1775
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• 2004

In presence of collision between two rigid bodies, they exhibit impulsive behavior to generate physically feasible state. When the frictional impulse is involved, collision resolution can not be easily made based on a simple Newton's law or Poisson's law, mainly due to possible change of collision mode during collision, For example, sliding may change to sticking, and then sliding resumes. We first examine two conventional methods: the method of mode evolution by differential equation, and the other by linear complementarity programming. Then, we propose a new method for mode evolution by solving only algebraic equations defining mode changes. Further, our method attains the original nonlinear impulse cone constraint. The numerical simulation will elucidate the advantage of the proposed method as an alternative to conventional ones.

• Kyungpook Mathematical Journal
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• v.62 no.4
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• pp.729-736
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• 2022

To help solving intractable nonlinear evolution equations (NLEEs) of waves in the field of fluid dynamics we develop an algorithm to find new high order solutions of the class of Abel, Bernoulli, Chini and Riccati equations of the form y' = ayⁿ + by + c, n > 1, with constant coefficients a, b, c. The role of this class of equations in NLEEs is explained in the introduction below. The basic algorithm to compute the coefficients of the power series solutions of the class, emerged long ago and is further developed in this paper. Practical application for hitherto unknown solutions is exemplified.

• International Journal of Aeronautical and Space Sciences
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• v.7 no.2
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• pp.26-32
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• 2006

This paper targets a direct and quantitative prediction of characteristics of unstable waves in a combustion chamber, which employs the governing equations derived in terms of amplification factors of flow variables. A freshly formulated nonlinear acoustic equation is obtained and the analysis of unsteady waves in a rocket engine is attempted. In the present formalism, perturbation method decomposes the variables into time-averaged part that can be obtained easily and accurately and time-varying part which is assumed to be harmonic. Excluding the use of conventional spatially sinusoidal eigenfunctions, a direct numerical solution of wave equation replaces the initial spatial distribution of standing waves and forms the nonlinear space-averaged terms. Amplification factor is also calculated independently by the time rate of changes of fluctuating variables, and is no longer an explicit function for compulsory representation. Employing only the numerical computation, major assumptions inevitably inherent, and in erroneous manner, in up to date analytical methods could be avoided. With two definitions of amplification factor, 1-D stable wave and 3-D unstable wave are examined, and clearly demonstrated the potentiality of a suggested theoretical-numerical method of combustion instability.