• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.1
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• pp.55-69
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• 1999

In this paper we analyze the error estimates for a single-phase nonlinear Stefan problem with Neumann boundary conditions. We apply the modified Crank-Nicolson method to get the optimal order of error estimates in the temporal direction.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1445-1465
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• 2015

In this paper, the long time behavior of the convective Cahn-Hilliard equation in two dimensions is considered, semidiscrete and completely discrete spectral approximations are constructed, error estimates of optimal order that hold uniformly on the unbounded time interval $0{\leq}t<{\infty}$ are obtained.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.185-199
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• 2003

In this paper, we apply finite element Galerkin method to a single-phase quasi-linear Stefan problem with a forcing term. We consider the existence and uniqueness of a semidiscrete approximation and optimal error estimates in $L_2$, $L_{\infty}$, $H_1$ and $H_2$ norms for semidiscrete Galerkin approximations we derived.

• Proceedings of the KIEE Conference
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• 2007.11c
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• pp.160-164
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• 2007

MRAS observer, which is based on adaptive control theory, estimates speed and position by using optimal observer gains on the basis of Lyapunov stability theory. However, in case of MRAS theory, position estimation error is in existence because of non-linearity for inductance variation and limit cycles for position estimation. The adaptive sliding observer based on the variable structure control theory estimates the speed and position for zero of estimation error by using the sliding surface equal to the error between speed and position estimation. The binary observer estimates the rotor speed and rotor flux with alleviation of the high-frequency chattering, and retains the benefits achieved in the conventional sliding observer, such as robustness to parameter and disturbance variations. The speed and position sensorless control of SRM under the load and inductance variation is verified by the experimental results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.4
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• pp.337-350
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• 2014

In this paper, we consider discontinuous Galerkin finite element methods with interior penalty term to approximate the solution of nonlinear parabolic problems with mixed boundary conditions. We construct the finite element spaces of the piecewise polynomials on which we define fully discrete discontinuous Galerkin approximations using the Crank-Nicolson method. To analyze the error estimates, we construct an appropriate projection which allows us to obtain the optimal order of a priori ${\ell}^{\infty}(L^2)$ error estimates of discontinuous Galerkin approximations in both spatial and temporal directions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.25-37
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• 2001

We consider the finite element method applied to nonlinear Sobolev equation with smooth data and demonstrate for arbitrary order ($k{\geq}2$) finite element spaces the optimal rate of convergence in $L_{\infty}\;W^{1,{\infty}}({\Omega})$ and $L_{\infty}(L_{\infty}({\Omega}))$ (quasi-optimal for k = 1). In other words, the nonlinear Sobolev equation can be approximated equally well as its linear counterpart. Furthermore, we also obtain superconvergence results in $L_{\infty}(W^{1,{\infty}}({\Omega}))$ for the difference between the approximate solution and the generalized elliptic projection of the exact solution.

• Journal of the Korean Mathematical Society
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• v.60 no.2
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• pp.273-302
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• 2023

In this paper, a fully discrete numerical scheme for the viscoelastic Oldroyd flow is considered with an introduced auxiliary variable. Our scheme is based on the finite element approximation for the spatial discretization and the backward Euler scheme for the time discretization. The integral term is discretized by the right trapezoidal rule. Firstly, we present the corresponding equivalent form of the considered model, and show the relationship between the origin problem and its equivalent system in finite element discretization. Secondly, unconditional stability and optimal error estimates of fully discrete numerical solutions in various norms are established. Finally, some numerical results are provided to confirm the established theoretical analysis and show the performances of the considered numerical scheme.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.223-235
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• 2006

By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations in $L_2,\;H^1$ and $H^2$ normed spaces.

• Asia-Pacific Journal of Business
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• v.13 no.4
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• pp.163-175
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• 2022

Purpose - This study introduces a methodology for finding the optimal tracking error of active stock funds. Tracking error is commonly used in risk budgeting techniques as a concept of cost for alpha creation. Design/methodology/approach - This study uses a post-optimal smart beta portfolio that maximizes alpha under the given tracking error constraint. Findings - As a result of the analysis, the smart beta strategy that maximized alpha under the constraint of 0.15% daily tracking error shows the highest IR. This means the maximum theoretically achievable efficiency. In this regard, a fixed-effect panel regression analysis is conducted to evaluate the active efficiency of domestic stock funds. In addition to control variables based on previous studies, the effect of tracking error on alpha is analyzed. The alpha used in this model is calculated using the smart beta portfolio according to the size of the constraint of the tracking error as a benchmark. Contrary to theoretical estimates, in Korea, the alpha performance is maximized under a daily tracking error of 0.1%. This indicates that the active efficiency of domestic equity funds is lower than the theoretical maximum. Research implications or Originality - Based on this study, it is expected that it can be used for active risk management of pension funds and performance evaluation of active strategies.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.173-192
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• 1999

We study a boundary optimal control problem of the fluid flow governed by the Navier-Stokes equations. the control problem is formulated with the flow through a channel with steps. The first-order optimality condition of the optimal control is derived. Finite element approximations of the solutions of the optimality system are defined and optimal error estimates are derived. finally, we present some numerical results.