• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1109-1127
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• 2021

We study the long-term dynamics for a family of incompressible three-dimensional Leray-α-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-α model when varying two nonnegative parameters 𝜃₁ and 𝜃₂. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-α-like models into a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.275-289
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• 2009

In this paper, we investigate the global behavior of the difference equation $x_{n+1}\;=\;\frac{a+bx_{n-1}}{A+Bx^k_n}$, n=0,1,..., where a,b,$B\;{\in}\;[0,\infty)$ and A, $k\;{\in}\;(0,\infty)$ with non-negative initial conditions.

• Proceedings of the KSME Conference
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• 2003.11a
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• pp.634-639
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• 2003

It has been studied the flow near a rotating disk with surface topography. The system Ekman number is assumed very small, i.e., $E[{\equiv}\frac{\nu}{{\Omega}^{\ast}L^{\ast2}}]<<1$ in which $L^{\ast}$ denotes a disk radius, ${\nu}$ kinematic viscosity of the fluid and ${\Omega}^{\ast}$ angular velocity of the basic state. Disk surface has a sinusoidal topographic variation along radial coordinate, i.e., $z={\delta}cos(2{\pi}{\omega}r)$, where ${\delta}$ and ${\omega}$ are, respectively, nondimensional amplitude and wave number of the disk surface. Analytic solutions, being useful over the parametric ranges of ${\delta}{\sim}O$( $E^{1/2}$ ) and ${\omega}{\leq}O$ ( $E^{1/2}$ ), are secured in a series-function form of Fourier-Bessel type. An asymptotic behavior, when $E{\rightarrow}0$, is clarified as : for a disk with surface roughness, in contrast to the case of a flat disk, the azimuthal velocity increases in magnitude, together with the thickening boundary layer. The radial velocity, however, decreases in magnitude as the amplitude of surface waviness increases. Consequently, the overall Ekman pumping at the edge of the boundary layer remains unchanged, maintaining the constant value equal to that of the flat disk.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.289-304
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• 2012

In this paper, we study asymptotic behaviour of solutions of the following higher order nonlinear dynamic equations $$S_n^{\Delta}(t,x)+{\delta}p(t)f(x(g(t)))=0$$ and $$S_n^{\Delta}(t,x)+{\delta}p(t)f(x(h(t)))=0$$ on an arbitrary time scale $\mathbb{T}$ with sup $\mathbb{T}={\infty}$, where n is a positive integer, ${\delta}=1$ or -1 and $$S_k(t,x)=\{\array x(t),\;if\;k=0,\\a_k(t)S_{{\kappa}-1}^{\Delta}(t),\;if\;1{\leq}k{\leq}n-1,\\a_n(t)[S_{{\kappa}-1}^{\Delta}(t)]^{\alpha},\;if\;k=n,$$ with ${\alpha}$ being a quotient of two odd positive integers and every $a_k$ ($1{\leq}k{\leq}n$) being positive rd-continuous function. We obtain some sufficient conditions for the equivalence of the oscillation of the above equations.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.6 no.2
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• pp.139-149
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• 1994

In this paper, a finite element technique is applied to the prediction of the wave resonance phenomena in harbors. The mild-slope equation is used with a partial reflection boundary condition introduced to model the energy dissipating effects on the solid boundary. For an efficient modeling of the radiation condition at infinity, a new infinite element is developed. The shape function of the infinite element is derived from the asymptotic behavior of the first kind of the Hankel's function in the analytical boundary series solutions. For the computational efficiency, the system matrices of the element are constructed by performing the relevant integrations in the infinite direction analytically. Comparisons with the results from experiments and other solution methods show that the present model gives fairly good results. Numerical experiments are also carried out to determine the proper distance to the infinite elements from the mouth of the halter, which directly affect the accuracy and efficiency of the solution.