• 대한기계학회:학술대회논문집
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• 대한기계학회 2001년도 춘계학술대회논문집E
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• pp.760-765
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• 2001

Some multidimentional generalizations of the Fokker-Planck Equation used by Friedrich and Peinke for description of a turbulent cascade was solved by A.A.Donkov, A.D.Donkov, and G.I.Grancharova. The solutions are two types, isotropic and anisotropic diffusion case. We introduce their methods to solve the Equation and solutions. Furthermore we get the more generalized exact solution as combination of two cases and plot to compare those to experimental results for the isotropic case.

• 항공우주시스템공학회지
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• 제10권1호
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• pp.15-20
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• 2016

The optimal estimation of a general continuous-discrete system can be achieved through the solution of the Fokker-Planck equation and the Bayesian update. Due the high nonlinearity of the equation of motion of the system and the measurement model, it is necessary to linearize the both equation. To avoid linearization, the filter based on Fokker-Planck equation is designed. with the unscented transformation update mechanism, in which the associated Fokker-Planck equation was solved efficiently and accurately via discrete quadrature and the measurement update was done through the unscented transformation update mechanism. This filter based on the Direct Quadrature Moment of Method(DQMOM) and the unscented transformation update is applied to the bearing only target tracking problem. The proposed filter can still provide more accurate estimation of the state than those of the extended Kalman filter especially when measurements are sparse. Simulation results indicate that the advantages of the proposed filter based on the DQMOM and the unscented transformation update make it a promising alternative to the extended Kalman filter.

• 한국전산구조공학회논문집
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• 제12권2호
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• pp.251-264
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• 1999

본 연구에서는 추계론적 동적시스템의 응답거동을 예측할 수 있는 반해석적 절차를 개발하였으며, 이를 이용하여 구분적선형시스템의 동적거동특성을 확률적 영역에서 분석하였다. 반 해석적 절차는 시스템의 추계론적 미분방정식에 상응하는 Fokker-Planck 방정식을 path-integral solotion을 이용하여 풂으로써 구할 수 있다. 결합확률밀도함수의 시간에 따른 전개과정을 통하여 시스템의 동적 응답거동 특성의 예측과 분석을 하고 시스템의 거동에 미치는 외부노이즈의 영향 또한 조사하였다. 반 해석적 방법은 위상면 상에서 결합확률밀도 함수를 통하여 응답거동의 예측은 물론 거동특성에 대하여 적절한 정보를 제공하는 것을 밝혔다. 혼돈거동의 특성은 외부노이즈가 존재하는 상황에서도 시스템의 응답 안에 잔재하는 것을 밝혔다.

• Journal of the Optical Society of Korea
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• 제12권4호
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• pp.240-243
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• 2008

Using the well-known Fokker-Planck method, this paper presents a standard way to find the joint-characteristic function for the first- and second-order polarization-mode-dispersion vectors originally derived by Foschini and Shepp. Compared with the Foschini and Shepp's approach, the Fokker-Planck approach gives a more accurate and straightforward way to find the joint-characteristic function.

• 한국전산구조공학회논문집
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• 제13권4호
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• pp.427-436
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• 2000

이 논문은 불규칙 가진력을 받는 회전-고정보의 비선형 응답특성을 나타낸다. 불규칙 가진력은 두 번째 고유모드의 절점과 최대변위점에 가했다. 비선형 편미분 방정식과 경계조건으로 표현되는 이 문제를 Galerkin의 방법을 이용하여 연립 비선형 상미분방정식으로 변환하였다. 이 상미분방정식으로부터 Fokker-Planck방정식과 모멘트 방정식을 얻은 후 Gaussian closure 방법 및 non-Gaussian closure 방법을 이용하여 3 모드 근사시 각각 27개 및 209개의 자율 상미분방정식을 구하였다. Gaussian closure 방법과 non-Gaussian closure 방법으로 2 모드 및 3 모드 근사해석을 수행하였고 해석적 결과들은 Monte Carlo 시뮬레이션 결과와 비교되었다. 해석결과 2 모드 근사해와 3 모드 근사해가 거의 일치하였고 2 모드 내부공진만 고려하여도 해석결과에 별 영향을 주지 않는다는 것을 알 수 있다.

• 항공우주시스템공학회지
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• 제10권1호
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• pp.8-14
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• 2016

The optimal estimation of a bearing only target tracking problem be achieved through the solution of the Fokker-Planck equation and the Bayesian update. Recently, a nonlinear filtering algorithm using a direct quadrature method of moments in which the associated Fokker-Planck equation can be propagated efficiently and accurately was proposed. Although this approach has demonstrated its promising in the field of nonlinear filtering in several examples, the "degeneracy" phenomenon, similar to that which exists in a typical particle filter, occasionally appears because only the weights are updated in the modified Bayesian rule in this algorithm. Therefore, in this paper to enhance the performance, a more stable measurement update process based upon the update equation in the Extended Kalman filters and a more accurate initialization and re-sampling strategy for weight and abscissas are proposed. Simulations are used to show the effectiveness of the proposed filter and the obtained results are promising.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1157-1169
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• 2010

This paper presents the analytical solution of the fractional Fokker-Planck equation by Adomian decomposition method. By using initial conditions, the explicit solution of the equation has been presented in the closed form and then the numerical solution has been represented graphically. Two different approaches have been presented in order to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.

• International Journal of Aeronautical and Space Sciences
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• 제18권4호
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• pp.816-826
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• 2017

In this paper, a generalized discretization scheme is proposed that can derive general-order finite difference equations representing the joint probability density function of dynamic response of stochastic systems. The various order of finite difference equations are applied to solutions of the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference equations derived by the proposed method can greatly increase accuracy even at the tail parts of the probability density function, giving accurate reliability estimations. Compared with exact solutions and finite element solutions, the generalized finite difference method showed increasing accuracy as the order increases. With the proposed method, it is allowed to use different orders and types (i.e. forward, central or backward) of discretization in the finite difference method to solve FPK and other partial differential equations in various engineering fields having requirements of accuracy or specific boundary conditions.

• Bulletin of the Korean Chemical Society
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• 제9권1호
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• pp.36-40
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• 1988

The time correlation functions of concentration fluctuations due to the random forces near the steady state are evaluated for a general two-component nonlinear chemical system by solving the corresponding two dimensional Fokker-Planck equation. The approximate method of solving the Fokker-Planck equation is based on the eigenfunction expansion and the corresponding eigenvalues for both the linear and nonlinear Fokker-Planck operators are obtained near the steady state. The general results are applied to the Lotka model near the oscillatory marginal steady state and the comparison is made between linear and nonlinear cases.

• 항공우주시스템공학회지
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• 제10권1호
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• pp.21-28
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• 2016

Orbit determination problems have been interest of many researchers for long time. Due to the high nonlinearity of the equation of motion and the measurement model, it is necessary to linearize the both equations. To avoid linearization, the filter based on Fokker-Planck equation is designed. with the extended Kalman filter update mechanism, in which the associated Fokker-Planck equation was solved efficiently and accurately via discrete quadrature and the measurement update was done through the extended Kalman filter update mechanism. This filter based on the DQMOM and the EKF update is applied to the orbit determination problem with appropriate modification to mitigate the filter smugness. Unlike the extended Kalman filter, the hybrid filter based on the DQMOM and the EKF update does not require the burdensome evaluation of the Jacobian matrix and Gaussian assumption for the system, and can still provide more accurate estimations of the state than those of the extended Kalman filter especially when measurements are sparse. Simulation results indicate that the advantages of the hybrid filter based on the DQMOM and the EKF update make it a promising alternative to the extended Kalman filter for orbit estimation problems.