• Journal of the Korean Statistical Society
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• 제28권1호
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• pp.93-106
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• 1999

In this paper we consider estimation of a real valued parameter in the drift coefficient of a Hilbert space valued Ito stochastic differential equation. First we consider observation of the corresponding diffusion in a fixed time interval [0, T] and prove the Bernstein - von Mises theorem concerning the convergence of posterior distribution of the parameter given the observation, suitably normalised and centered at the MLE, to the normal distribution as Tlongrightarrow$\infty$. As a consequence, the Bayes estimator of the drift parameter becomes asymptotically efficient and asymptotically equivalent to the MLE as Tlongrightarrow$\infty$. Next, we consider observation in a random time interval where the random time is determined by a predetermined level of precision. We show that the sequential MLE is better than the ordinary MLE in the sense that the former is unbiased, uniformly normally distributed and efficient but is latter is not so.

• Geomechanics and Engineering
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• 제32권2호
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• pp.125-135
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• 2023

Nonlinear vibration analysis of composite beam reinforced by carbon nanotubes resting on the nonlinear viscoelastic foundation is investigated in this study. The material properties of the composite beam is considered as a polymeric matrix by reinforced carbon nanotubes according to different distributions. With using Hamilton's principle, the governing nonlinear partial differential equations are derived based on the Euler-Bernoulli beam theory. In the nonlinear kinematic assumption, the Von Kármán nonlinearity is used. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The nonlinear natural frequency and the nonlinear free response of the system is obtained. In addition, the effects of different patterns of reinforcement, linear and nonlinear damping coefficients of the viscoelastic foundation on the nonlinear vibration responses and phase trajectory of the carbon nanotube reinforced composite beam are investigated.

• Structural Engineering and Mechanics
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• 제91권2호
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• pp.197-209
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• 2024

The vibration of elastically supported bidirectional functionally graded (BDFG) sandwich beams on an elastic foundation is investigated. The sandwich structure is composed of upper and lower layers of BDFG material and the core layer of isotropic material. Material properties of upper and lower layers are assumed to vary continuously along the length and thickness of the beam with a power-law function. Hamilton's principle is used to deduce the vibration equations of motion of the sandwich Timoshenko beam. Then, the partial differential equation of motion is spatially discretized into a time-varying ordinary differential equation in terms of Chebyshev differential matrices. The eigenvalue equation associated with the free vibration is formulated to study the influence of various slenderness ratios, material gradient indexes, thickness ratios, foundation and support spring constants on the vibration frequency of BDFG sandwich beams. The present method can provide researchers with deep insight into the impact of various geometric, material, foundation and support parameters on the vibration behavior of BDFG sandwich beam structures.

• Smart Structures and Systems
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• 제9권3호
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• pp.231-251
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• 2012

The stochastic optimal control for a piezoelectric spherically symmetric shell subjected to stochastic boundary perturbations is constructed, analyzed and evaluated. The stochastic optimal control problem on the boundary stress output reduction of the piezoelectric shell subjected to stochastic boundary displacement perturbations is presented. The electric potential integral as a function of displacement is obtained to convert the differential equations for the piezoelectric shell with electrical and mechanical coupling into the equation only for displacement. The displacement transformation is constructed to convert the stochastic boundary conditions into homogeneous ones, and the transformed displacement is expanded in space to convert further the partial differential equation for displacement into ordinary differential equations by using the Galerkin method. Then the stochastic optimal control problem of the piezoelectric shell in partial differential equations is transformed into that of the multi-degree-of-freedom system. The optimal control law for electric potential is determined according to the stochastic dynamical programming principle. The frequency-response function matrix, power spectral density matrix and correlation function matrix of the controlled system response are derived based on the theory of random vibration. The expressions of mean-square stress, displacement and electric potential of the controlled piezoelectric shell are finally obtained to evaluate the control effectiveness. Numerical results are given to illustrate the high relative reduction in the root-mean-square boundary stress of the piezoelectric shell subjected to stochastic boundary displacement perturbations by the optimal electric potential control.

• Kyungpook Mathematical Journal
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• 제57권4호
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• pp.651-665
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• 2017

Some computational fluid dynamics problems concerning the thin films flow of viscous fluid with a free surface and draining or coating fluid-flow problems can be delineated by third-order ordinary differential equations. In this paper, the aim is to introduce the numerical solutions of the boundary value problems of such equations by variational iteration method. In this paper, it is shown that the third-order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration method. These solutions are gleaned in terms of convergent series. Numerical examples are given to depict the method and their convergence.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제11권1호
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• pp.57-70
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• 2007

The effects of variable viscosity, blowing or suction on mixed convection flow of a viscous incompressible fluid past a semi-infinite horizontal flat plate aligned parallel to a uniform free stream in the presence of the wall temperature distribution inversely proportional to the square root of the distance from the leading edge have been investigated. The equations governing the flow are transformed into a system of coupled non-linear ordinary differential equations by using similarity variables. The similarity equations have been solved numerically. The effect of the viscosity temperature parameter, the buoyancy parameter and the blowing or suction parameter on the velocity and temperature profiles as well as on the skin-friction coefficient and the Nusselt number are discussed.

• 대한수학회지
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• 제37권4호
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• pp.625-643
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• 2000

In this paper we formulate an optimal control problem governed by time-delay Volterra integral equations; the problem includes control constraints as well as terminal equality and inequality constraints on the terminal state variables. First, using a special type of state and control variations, we represent a relatively simple and self-contained method for deriving new necessary conditions in the form of Pontryagin minimum principle. We show that these results immediately yield classical Pontryagin necessary conditions for control processes governed by ordinary differential equations (with or without delay). Next, imposing suitable convexity conditions on the functions involved, we derive Mangasarian-type and Arrow-type sufficient optimality conditions.

• 설비공학논문집
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• 제2권4호
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• pp.268-278
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• 1990

Hydrodynamic stability equations are formulated for natural convection flows adjacent to a heated or cooled, inclined, isothermal surface in pure water at $4^{\circ}C$, where the density variation with temperature becomes nonlinear. The resulting stability equations, when reduced to ordinary differential equations by a similarity transformation, constitute a two-point boundary-value problem, which was solved numerically. It is found from the obtained stability results that the neutral stability curves are systematically shifted to have lower critical Grashof numbers, as the inclination angle of upward-facing plate increases. Also, the nose of the neutral stability curve becomes blunter as the angle increases. It implies that the greater the inclination of the upward-facing plate, the more susceptible of the flow to instability for the wide range of disturbance wave number and frequency.

• Interaction and multiscale mechanics
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• 제4권1호
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• pp.1-16
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• 2011

The inverse problem about the theoretical analysis of a drill string bending in a channel of an inclined bore-hole with localized geometrical imperfections is studied. The system of ordinary differential equations is first derived based on the theory of curvilinear flexible elastic rods. One can then use these equations to investigate the quasi-static effects of the drill string bending that may occur in the process of raising, lowering and rotation of the string inside the bore-hole. The method for numerical solution of the constructed equations is described. With the proposed method, the phenomenon of the drill column movement, its contact interaction with the bore-hole surface, and the frictional seizure can be simulated for different combinations of velocities, directions of rotation and axial motion of the string. Geometrical imperfections in the shape of localized smoothed breaks of the bore-hole axis line are considered. Some numerical examples are presented to illustrate the applicability of the method proposed.

• Korean Chemical Engineering Research
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• 제57권5호
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• pp.723-729
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• 2019

A theoretical analysis was conducted of convective instability driven by buoyancy forces under transient temperature fields in an annular porous medium bounded by coaxial vertical cylinders. Darcy's law and Boussinesq approximation are used to explain the characteristics of fluid motion and linear stability theory is employed to predict the onset of buoyancy-driven motion. The linear stability equations are derived in a global domain, and then cast into in a self-similar domain. Using a spectral expansion method, the stability equations are reformed as a system of ordinary differential equations and solved analytically and numerically. The critical Darcy-Rayleigh number is founded as a function of the radius ratio. Also, the onset time and corresponding wavelength are obtained for the various cases. The critical time becomes smaller with increasing the Darcy-Rayleigh number and follows the asymptotic relation derived in the infinite horizontal porous layer.