• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.215-228
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• 2007

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange's characteristics method (a new approach) for solving non linear fractional partial differential equations. The key of this results is the fractional Taylor's series $f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha})f(x)$ where $E_{\alpha}(.)$ is the Mittag-Leffler function.

• 소음진동
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• 제7권3호
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• pp.489-498
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• 1997

In this paper, chaotic motions of a curved pipe conveying oscillatory flow are theoretically investigated. The nonliear partial differential equation of motion is derived by Newton's method. The transformed nonlinear ordinary differential equation is a type of Hill's equation, which has the external and parametric excitation with a same frequency. Bifurcation curves of chaotic motion of the piping systems are obtained by applying Melnikov's method. Numerical simulations are performed to demonstrate theoretical results and show the strange attractor of the chaotic motion.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 1996년도 추계학술대회논문집; 한국과학기술회관, 8 Nov. 1996
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• pp.288-294
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• 1996

In this paper, Chaotic motions of a curved pipe conveying oscillatory flow are theoretically investigated. The nonlinear partial differential equation of motion is derived by Newton's method. The transformed nonlinear ordinary differential equation is a type of Hill's equation, which have the parametric and external excitation. Bifurcation curves of chaotic motion of the piping systems are obtained by applying Melnikov's method. Poincare maps numerically demonstrate theoretical results and show transverse homoclinic orbit of the chaotic motion.

• Advances in nano research
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• 제13권5호
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• pp.465-474
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• 2022

Based on the nonlocal strain gradient (NSG) theory and considering the influence of moment of inertia, the governing equations of motion of porous functionally graded (FG) nanoplates with four edges clamped are established; The Galerkin method is applied to eliminate the spatial variables of the partial differential equation, and the partial differential governing equation is transformed into an ordinary differential equation with time variables. By satisfying the boundary conditions and solving the characteristic equation, the dispersion relations of the porous FG strain gradient nanoplates with four edges fixed are obtained. It is found that when the wave number is very small, the influences of nonlocal parameters and strain gradient parameters on the dispersion relation is very small. However, when the wave number is large, it has a great influence on the group velocity and phase velocity. The nonlocal parameter represents the effect of stiffness softening, and the strain gradient parameter represents the effect of stiffness strengthening. In addition, we also study the influence of power law index parameter and porosity on guided wave propagation.

• Nonlinear Functional Analysis and Applications
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• 제27권3호
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• pp.679-689
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• 2022

In this paper, we establish sufficient conditions for the existence and uniqueness of solution for a class of initial boundary value problems with Dirichlet condition in regard to a category of fractional-order partial differential equations. The results are established by a method based on the theorem of Lax Milligram.

• 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.

• 대한교통학회지
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• 제20권5호
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• pp.67-79
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• 2002

교통량, 교통밀도, 교통류 속도 등, 교통류 변수에 대한 현재까지의 불확실한 정의와 연속적 파동방정식의 거시적 교통류 해석상의 문제점을 지적하고 이를 개선하기 위해 교통류 변수들에 대한 새로운 확률적 정의를 제시하고 이들의 성격을 규명하였다. 이러한 새로운 교통류 변수들에 대한 새로운 정의를 바탕으로 미시적 운전자 행동을 세밀하게 수용할 수 있고 많은 교통환경에서 연속적 파동 방정식을 대체하여 교통류 변수들과 통행시간을 예측할 수 있는 미분방정식 체계를 확률 미분방적식을 이용하여 도출하였다. 도출된 미분 방정식을 단일 차량의 시공 괘적에 적용해 보았다.

• 한국컴퓨터산업학회논문지
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• 제3권5호
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• pp.569-574
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• 2002

유계된 정의역에서 sine-Gordon 방정식인 현수교의 과격한 운동의 모델을 만든다. 유한 차분법을 이용하여 비선형 미분방정식을 수치 해석학적으로 풀다. 이 미분방정식은 다중 주기근을 가지고 있다. 다리가 큰 진폭이나 작은 진폭으로 진동하는 것은 초기의 변위와 속도에만 달려있다. 게다가, 많은 현상들이 Tacoma Narrows가 붕괴된 날에 관찰되는 것과 일치하고 있다.

• International Journal of Precision Engineering and Manufacturing
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• 제3권4호
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• pp.87-93
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• 2002

In this study, basic concepts of magnet were introduced, and dynamic characteristics of magnet coupling were explored. Based on these characteristics, it was proposed how to analyze transverse and torsional vibrations of a spindle system with magnet coupling. Proposed theoretical approaches were applied to a precision power transmission system machined for this study, and the transverse and torsional vibrations were simulated. The force on magnet coupling was shown as a form of nonlinear function of the gap and the eccentricity. Also, the form of torque transmitted by magnet coupling was considered as a sinusoidal function. Main spindle connected to a coupling of a follower part was assumed to be a rigid body. Nonlinear partial differential equation was derived to be as a function of angular displacement. By using the equation, torsional vibration analysis of a spindle system with magnet coupling was performed. Free and forced vibration analyses of a spindle system with magnetic coupling were explored by using FEM.

• 대한수학회지
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• 제34권2호
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• pp.395-405
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• 1997

A globality of the Hopf bifurcation in a free boundary problem for a parabolic partial differential equation is investigated in this paper. We shall examine the global behavior of the Hopf critical eigenvalues and and apply the center-index theory to show the globality.