• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.141-152
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• 2007

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.4
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• pp.353-361
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• 2022

For an initial value problem, using a weighted average between two adjacent approximates, we propose a simple one-step method based on the Euler method. This method is useful for solving stiff initial value problem, even when the step size is not very small. Moreover, it can be seen that the proposed method with some selected weights results in improved approximation errors.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.3
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• pp.138-155
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• 2022

To solve the initial value problem we present a new single-step implicit method based on the Euler method. We prove that the proposed method has convergence order 2. In practice, numerical results of the proposed method for some selected examples show an error tendency similar to the second-order Taylor method. It can also be found that this method is useful for stiff initial value problems, even when a small number of nodes are used. In addition, we extend the proposed method by using weighted averages with a parameter and show that its convergence order becomes 2 for the parameter near $\frac{1}{2}$. Moreover, it can be seen that the extended method with properly selected values of the parameter improves the approximation error more significantly.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.7 no.2
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• pp.284-292
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• 2003

In this paper, we carried out the numerical examination of the initial value setting-up method to extend the range of optimal step parameter in a adaptive system which is controlled by LMS algorithm. For initial value setting-up methods, the general method which select the initial value randomly and the other method which applies the approximate value obtained from the direct method to initial value, were used. And then, we compared to the ranges of step parameter setting, the convergence speeds of mean-square-error, and the stabilities during the convergence process when the initial values were applied to the optimal directivity synthesis problem. According to the numerical simulation results, the initial value setting-up method by means of the direct method provides wider range for the step parameter, more efficient capability for convergence and stability, and more error correction ability than the general method.

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.131-145
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• 2013

In this paper, we present an initial value technique for solving singularly perturbed differential difference equations with a boundary layer at one end point. Taylor's series is used to tackle the terms containing shift provided the shift is of small order of singular perturbation parameter and obtained a singularly perturbed boundary value problem. This singularly perturbed boundary value problem is replaced by a pair of initial value problems. Classical fourth order Runge-Kutta method is used to solve these initial value problems. The effect of small shift on the boundary layer solution in both the cases, i.e., the boundary layer on the left side as well as the right side is discussed by considering numerical experiments. Several numerical examples are solved to demonstate the applicability of the method.

• Journal of applied mathematics & informatics
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• v.36 no.3_4
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• pp.331-348
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• 2018

In this paper, an initial value method for solving a class of singularly perturbed delay differential equations with layer behavior is proposed. In this approach, first the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay using Taylor series expansion. Then from the modified problem, two explicit Initial Value Problems which are independent of the perturbation parameter, ${\varepsilon}$, are produced: the reduced problem and boundary layer correction problem. Finally, these problems are solved analytically and combined to give an approximate asymptotic solution to the original problem. To demonstrate the efficiency and applicability of the proposed method three linear and one nonlinear test problems are considered. The effect of the delay on the layer behavior of the solution is also examined. It is observed that for very small ${\varepsilon}$ the present method approximates the exact solution very well.

• Communications for Statistical Applications and Methods
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• v.27 no.1
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• pp.97-108
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• 2020

This paper compares several methods for estimating parameters of normal inverse Gaussian distribution. Ordinary maximum likelihood estimation and the method of moment estimation often do not work properly due to restrictions on parameters. We examine the performance of adjusted estimation methods along with the ordinary maximum likelihood estimation and the method of moment estimation by simulation and real data application. We also see the effect of the initial value in estimation methods. The simulation results show that the ordinary maximum likelihood estimator is significantly affected by the initial value; in addition, the adjusted estimators have smaller root mean square error than ordinary estimators as well as less impact on the initial value. With real datasets, we obtain similar results to what we see in simulation studies. Based on the results of simulation and real data application, we suggest using adjusted maximum likelihood estimates with adjusted method of moment estimates as initial values to estimate the parameters of normal inverse Gaussian distribution.

• Journal of the Korean Society of Fisheries and Ocean Technology
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• v.33 no.1
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• pp.68-75
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• 1997

In this paper, we propose the improved initial image estimation method for a fast fractal image decoding. When the correlation between a domain and a range is given as the linear equation, the value of initial image estimation using the conventional method is the intersection between its linear equation and y=x. If the gradient of linear equation is large, that the difference of the value between each adjacent pixels is large, the conventional method has disadvantage which has the impossibility of exact estimation. The method of the proposed initial image estimation performs well by two steps. he first step can improve the disadvantage of the conventional method. The second step upgrades the range value which was found previous step by referring information of its domain. Though the computational complexity for the initial image estimation increses slightly, the total computational complexity decreases by 30% than that of the conventional method because of diminishing in the number of iterations.

• Journal of the Korean Data and Information Science Society
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• v.15 no.2
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• pp.331-337
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• 2004

In the analysis of massive volume data, a neural network model is a useful tool. To implement the Neural network model, it is important to select initial value. Since the initial values are generally used as random value in the neural network, the convergent performance and the prediction rate of model are not stable. To overcome the drawback a possible method use samples randomly selected from the whole data set. That is, coefficients estimated by logistic regression based on the samples are the initial values.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.311-327
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• 2016

In this paper, we propose an error embedded Runge-Kutta method to improve the traditional embedded Runge-Kutta method. The proposed scheme can be applied into most explicit embedded Runge-Kutta methods. At each integration step, the proposed method is comprised of two equations for the solution and the error, respectively. These solution and error are obtained by solving an initial value problem whose solution has the information of the error at each integration step. The constructed algorithm controls both the error and the time step size simultaneously and possesses a good performance in the computational cost compared to the original method. For the assessment of the effectiveness, the van der Pol equation and another one having a difficulty for the global error control are numerically solved. Finally, a two-body Kepler problem is also used to assess the efficiency of the proposed algorithm.