• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.157-172
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• 1998

A numerical method is proposed for dynamical systems. We utilize the fact that special matrix exponentials can be exactly evaluated by the intrinsic library functions. Numerical examples are given, which show that the relative error s of the proposed method converge to a small constant and that the method faithfully approximates the dynamics of the nonlinear differential equations.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.31-40
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• 1992

We introduce the generalized Runge-Kutta methods with the exponentially dominant order .omega. in [3], and the convergence theorems of the generalized explicit Euler method are derived in [4]. In this paper we will study the convergence of the generalized implicit Euler method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.238-266
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• 2013

The Interaction Picture (IP) method is a valuable alternative to Split-step methods for solving certain types of partial differential equations such as the nonlinear Schr$\ddot{o}$dinger equation or the Gross-Pitaevskii equation. Although very similar to the Symmetric Split-step (SS) method in its inner computational structure, the IP method results from a change of unknown and therefore do not involve approximation such as the one resulting from the use of a splitting formula. In its standard form the IP method such as the SS method is used in conjunction with the classical 4th order Runge-Kutta (RK) scheme. However it appears to be relevant to look for RK scheme of higher order so as to improve the accuracy of the IP method. In this paper we investigate 5th order Embedded Runge-Kutta schemes suited to be used in conjunction with the IP method and designed to deliver a local error estimation for adaptive step size control.

• The Journal of the Acoustical Society of Korea
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• v.25 no.1E
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• pp.32-35
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• 2006

The screech tone of underexpanded jet is numerically calculated without any specific modeling for the screech tone itself. Fourth-order optimized compact scheme and fourth-order Runge-Kutta method are used to solve the 2D axisymmetric Euler equation. Adaptive nonlinear artificial dissipation model and generalized characteristic boundary condition are also used. The screech tone, generated by a closed loop between instability waves and quasi-periodic shock cells at the near field, is reasonably analyzed with present numerical methods for the underexpanded jet having Mach number 1.13. First of all, the centerline mean pressure distribution is calculated and compared with experimental and other numerical results. The instantaneous density contour plot shows Mach waves due to mixing layer convecting supersonically, which propagate downstream. The pressure signal and its Fourier transform at upstream and downstream shows the directivity pattern of screech tone very clearly. Most of all, we can simulate the axisymmetric mode change of screech tone very precisely with present method. It can be concluded that the basic phenomenon of screech tone including the frequency can be calculated by using high-order and high-resolution schemes without any specific numerical modeling for screech tone feedback loop.

• Journal of the Korean Institute of Intelligent Systems
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• v.13 no.3
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• pp.259-259
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• 2003

Genetic algorithms using a Process of genetic evolution of an organism are appropriate for hard problems that have not been solved by any deterministic method. Up to now, the controller design method has been made with the frequency dependent specification but the design method with the time specification has gotten little progress. In this paper, we study the controller design to satisfy the performance of a plant using the generalized Manabe standard form. When dealing with a controller design in the case of two parameter configurations, there are some situations that neither a known pseudo inverse technique nor the inverse method can be applicable. In this case, we propose two methods of designing a controller by the gradient algorithm and the new pseudo inverse method so that the desired closed polynomials are either equalized to or approximated to the designed polynomial. Design methods of the proposed controller are implemented in Java.