• Korean Journal of Mathematics
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• 제22권1호
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• pp.123-131
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• 2014

The purpose of this paper is to study the perturbation analysis of the matrix equation $X=I-A^*X^{-1}A+B^*X^{-1}B$. Based on the matrix differentiation, we give a precise perturbation bound for the positive definite solution. A numerical example is presented to illustrate the shrpness of the perturbation bound.

• Coupled systems mechanics
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• 제11권6호
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• pp.485-504
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• 2022

This paper presents nonlinear oscillations of a carbon nanotube reinforced composite beam subjected to lateral harmonic load with damping effect based on the modified couple stress theory. As reinforcing phase, three different types of single walled carbon nanotubes distribution are considered through the thickness in polymeric matrix. The non-linear strain-displacement relationship is considered in the von Kármán nonlinearity. The governing nonlinear dynamic equation is derived with using of Hamilton's principle.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 frequency response equation and the forced vibration response of the system are obtained. Effects of patterns of reinforcement, volume fraction, excitation force and the length scale parameter on the nonlinear responses of the carbon nanotube reinforced composite beam are investigated.

• Steel and Composite Structures
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• 제46권5호
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• pp.637-647
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• 2023

Nonlinear free vibration and stability responses of a carbon nanotube reinforced composite beam under temperature rising are investigated in this paper. The material of the beam is considered as a polymeric matrix by reinforced the single-walled carbon nanotubes according to different distributions with temperature-dependent physical properties. With using the Hamilton's principle, the governing nonlinear partial differential equation is 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 critical buckling temperatures, the nonlinear natural frequencies and the nonlinear free response of the system is obtained. The effect of different patterns of reinforcement on the critical buckling temperature, nonlinear natural frequency, nonlinear free response and phase plane trajectory of the carbon nanotube reinforced composite beam investigated with temperature-dependent physical property.

• 대한수학회지
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• 제55권1호
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• pp.73-82
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• 2018

In this article we consider certain types of nonlinear matrix equations including the stochastic rational Riccati equation and show the existence and uniqueness of the positive definite solution by using Bhaskar-Lakshmikantham's coupled fixed point theorem.

• East Asian mathematical journal
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• 제35권3호
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• pp.341-349
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• 2019

In this paper, we are concerned with the minimal positive solution to system of the nonlinear matrix equations $A_1X^2+B_1Y +C_1=0$ and $A_2Y^2+B_2X+C_2=0$, where $A_i$ is a positive matrix or a nonnegative irreducible matrix, $C_i$ is a nonnegative matrix and $-B_i$ is a nonsingular M-matrix for i = 1, 2. We apply Newton's method to system and present a modified Newton's iteration which is validated to be efficient in the numerical experiments. We prove that the sequences generated by the modified Newton's iteration converge to the minimal positive solution to system of nonlinear matrix equations.

• 대한전기학회논문지
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• 제45권3호
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• pp.399-406
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• 1996

This paper presents a method of hierarchical state estimation of the time-varying linear systems via Block-pulse function(BPF). When we estimate the state of the systems where noise is considered, it is very difficult to obtain the solutions because minimum error variance matrix having a form of matrix nonlinear differential equations is included in the filter gain calculation. Therefore, hierarchical approach is adapted to transpose matrix nonlinear differential equations to a sum of low order state space equation from and Block-pulse functions are used for solving each low order state space equation in the form of simple and recursive algebraic equation. We believe that presented methods are very attractive nd proper for state estimation of time-varying linear systems on account of its simplicity and computational convenience. (author). 13 refs., 10 figs.

• 대한수학회지
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• 제54권6호
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• pp.1667-1682
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• 2017

In this paper, the Hermitian positive definite solutions of the matrix equation $X^s+A^*X-^tA=Q$, where Q is an $n{\times}n$ Hermitian positive definite matrix, A is an $n{\times}n$ nonsingular complex matrix and $s,t{\in}[1,{\infty})$ are discussed. We find a matrix interval which contains all the Hermitian positive definite solutions of this equation. Also, a necessary and sufficient condition for the existence of these solutions is presented. Iterative methods for obtaining the maximal and minimal Hermitian positive definite solutions are proposed. The theoretical results are illustrated by numerical examples.

• 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.

• 대한전기학회논문지
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• 제39권8호
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• pp.792-804
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• 1990

This paper describes an efficient optimization algorithm by calculating sensitivity function for power system stabilization. In power system, the dynamic performance of exciter, governor etc. following a disturbance can be presented by a nonlinear differential equation. Since a nonlinear equation can be linearized for small disturbances, the state equation is expressed by a system matrix with system parameters. The objective function for power system operation will be related to the system parameter and the initial state at the optimal control condition for control or stabilization. The object function sensitivity to the system parameter can be considered to be effective in selecting the optimal parameter of the system.

• East Asian mathematical journal
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• 제28권5호
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• pp.587-595
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• 2012

In this paper we consider the quadratic matrix equation which can be defined be $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown real matrix; A,B and C are $n{\times}n$ given matrices with real elements. Newton's method is considered to find the skew-symmetric solvent of the nonlinear matrix equations Q(X). We also show that the method converges the skew-symmetric solvent even if the Fr$\acute{e}$chet derivative is singular. Finally, we give some numerical examples.