• 대한수학회지
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• 제49권2호
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• pp.315-324
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• 2012

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a long periodic domain. We show by a simple argument that a strong solution exists globally in time when the initial velocity in $H^1$ and the forcing function in $L^p$([0; T);$L^2$), T > 0, $2{\leq}p{\leq}+\infty$ satisfy a certain condition. This condition common appears for the global existence in thin non-periodic domains. Larger and larger initial data and forcing functions satisfy this condition as the thickness of the domain $\epsilon$ tends to zero.

• 충청수학회지
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• 제34권3호
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• pp.221-234
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• 2021

We introduce high-order Hopfield neural networks with Leakage delays. Furthermore, we study the uniqueness and existence of Hopfield artificial neural networks having the weighted pseudo almost periodic forcing terms on finite delay. Our analysis is based on the differential inequality techniques and the Banach contraction mapping principle.

• Ocean and Polar Research
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• 제34권3호
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• pp.287-296
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• 2012

In the theory of source-driven abyssal circulation, the forcing is usually assumed to be steady source (deep-water formation). In many cases, however, the deep-water formation occurs instantaneously and it is not clear whether the theory can be applied well in this case. An attempt is made to resolve this problem by using a simple reduced gravity model. The model basin has large depth change compared for its size, like the East Sea, such that isobaths nearly coincide with geostrophic contours. Deep-water is formed every year impulsively and flows into the model basin through the boundary. It is found that the circulation driven by the impulsive source is generally the same as that driven by a steady source except that the former has a seasonal fluctuation associated with unsteadiness of forcing. The magnitudes of both the annual average and seasonal fluctuations increase with the rate of deep-water formation. The problem can be approximated to that of linear diffusion of momentum with boundary flux, which well demonstrates the essential feature of abyssal circulation spun-up by periodic impulsive source. Although the model greatly idealizes the real situation, it suggests that abyssal circulation can be driven by a periodic impulsive source in the East Sea.

• Journal of applied mathematics & informatics
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• 제8권2호
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• pp.613-625
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• 2001

In this paper, we investigate a simplified model of a particle suspended elastically from two towers by two nonlinear elastic springs, with a restoring force similar to Hooke’s law under extension and with no resistance to compression. Numerical results are presented, showing the solutions can be either of the same period oscillation the forcing term, can be a subharmonic response of multiple period, or can be noisy periodic which is apparently chaotic. Multiplicity of periodic solutions for certain physical parameters are demonstrated.

• 충청수학회지
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• 제35권1호
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• pp.39-52
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• 2022

We introduce Clifford-valued neural networks with leakage delays. Furthermore, we study the uniqueness and existence of Clifford-valued Hopfield artificial neural networks having the Stepanov weighted pseudo almost periodic forcing terms on leakage delay terms. However the noncommutativity of the Clifford numbers' multiplication made our investigation diffcult, so our results are obtained by decomposing Clifford-valued neural networks into real-valued neural networks. Our analysis is based on the differential inequality techniques and the Banach contraction mapping principle.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제15권2호
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• pp.143-150
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• 2011

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a thin periodic domain. We present a simple proof that a strong solution exists globally in time when the initial velocity in $H^1$ and the forcing function in $L^p$(0,${\infty}$;$L^2$), $2{\leq}p{\leq}{\infty}$ satisfy certain condition. This condition is basically similar to that by Iftimie and Raugel[7], which covers larger and larger initial data and forcing functions as the thickness of the domain ${\epsilon}$ tends to zero.

• Wind and Structures
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• 제7권4호
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• pp.265-280
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• 2004

This paper presents some fundamental results on the dynamics of the periodic Karman wake behind a circular cylinder. The wake is treated like a dynamical system. External forcing is then introduced and its effect investigated. The main result obtained is the following. Perturbation of the wake, by controlled cylinder oscillations in the flow direction at a frequency equal to the Karman vortex shedding frequency, leads to instability of the Karman vortex structure. The resulting wake structure oscillates at half the original Karman vortex shedding frequency. For higher frequency excitation the primary pattern involves symmetry breaking of the initially shed symmetric vortex pairs. The Karman shedding phenomenon can be modeled by a nonlinear oscillator. The symmetrical flow perturbations resulting from the periodic cylinder excitation can also be similarly represented by a nonlinear oscillator. The oscillators represent two flow modes. By considering these two nonlinear oscillators, one having inline shedding symmetry and the other having the Karman wake spatio-temporal symmetry, the possible symmetries of subsequent flow perturbations resulting from the modal interaction are determined. A theoretical analysis based on symmetry (group) theory is presented. The analysis confirms the occurrence of a period-doubling instability, which is responsible for the frequency halving phenomenon observed in the experiments. Finally it is remarked that the present findings have important implications for vortex shedding control. Perturbations in the inflow direction introduce 'control' of the Karman wake by inducing a bifurcation which forces the transfer of energy to a lower frequency which is far from the original Karman frequency.

• Korean Journal of Mathematics
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• 제28권2호
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• pp.223-240
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• 2020

The paper is concerned with periodic linear evolution equations of the form x"(t) = A(t)x(t)+f(t), where A(t) is a family of (unbounded) linear operators in a Banach space X, strongly and periodically depending on t, f is an almost (or asymptotic) almost periodic function. We study conditions for this equation to have almost periodic solutions on ℝ as well as to have asymptotic almost periodic solutions on ℝ⁺. We convert the second order equation under consideration into a first order equation to use the spectral theory of functions as well as recent methods of study. We obtain new conditions that are stated in terms of the spectrum of the monodromy operator associated with the first order equation and the frequencies of the forcing term f.

• 한국수자원학회논문집
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• 제47권1호
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• pp.37-47
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• 2014

본 연구에서는 하폭의 주기적인 변화에 의하여 교호사주와 복렬사주가 발생하는 수리학적 조건에서 주기적인 하폭변화에 의한 강제사주의 형성과정과 거동 특성을 2차원 수치모형을 이용하여 파악하였다. 하폭변화의 진폭이 크면, 사주의 파장은 짧아지고, 사주의 이동성은 크게 감소하였다. 교호사주가 발생할 수 있는 영역에서는 하폭의 변화에 대하여 강제효과가 크게 작용하며, 복렬사주가 발생할 영역에서는 하폭의 변화에 의한 강제효과가 상대적으로 작은 특성을 보여주었다. 복렬사주가 발달한 조건에서는 하도변화의 진폭 대 평균하폭의 비인 무차원 진폭이 0.25로 증가할 때 사주의 이동속도는 증가하지만, 이보다 크면 사주의 이동속도가 급격하게 감소하였다. 교호사주가 발생하는 조건에서는 사주의 파장 대하폭 변화의 파장 비인 무차원 사주의 파장이 증가할수록 사주의 이동속도가 증가하지만, 무차원 사주의 파장이 1에 가까운 경우에 사주의 이동속도가 급격하게 감소하였다. 즉, 사주의 파장과 하폭의 파장이 일치할 때, 하폭변화에 의한 강제효과가 강하게 작용하여 사주가 압박되기 때문이다.

• International Journal of Fuzzy Logic and Intelligent Systems
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• 제10권3호
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• pp.184-193
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• 2010

In this paper, we investigate the existence and calculation of the expression of periodic solutions for fuzzy differential equations with three types of forcing terms, by using Hukuhara derivative. In particular, Theorems 3.2, 4.2 and 5.2 are the results of existences of periodic solutions for fuzzy differential equations I, II and III, respectively. These results will help us to study phenomena with periodic peculiarity such as wave or sound.