• Journal of the Korean Statistical Society
• /
• v.36 no.4
• /
• pp.479-488
• /
• 2007

The elliptic random field is an extension to the Gaussian random field. We proved a theorem which characterizes the elliptic random field. We proposed a heuristic approach to derive an approximation to the distribution of the size of one connected component of its excursion set above a high threshold. We used this approximation to approximate the distribution of the largest cluster size. We used simulation to compare the approximation with the exact distribution.

• 한국전산유체공학회:학술대회논문집
• /
• 2003.10a
• /
• pp.156-157
• /
• 2003

A simple method is proposed to split the flux vector of the Euler equations by introducing two artificial wave speeds. The direction of wave propagation can be adjusted by these two wave speeds. This idea greatly simplifies the upwinding, and leads to a new family of upwind schemes. Numerical flux function for multi-dimensional Euler equations is formulated for any grid system, structured or unstructured. A remarkable simplicity of the scheme is that it successfully achieves one-sided approximation for all waves without recourse to any matrix operation. Moreover, its accuracy is comparable with the exact Riemann solver. For 1-D Euler equations, the scheme actually surpasses the exact solver in avoiding expansion shocks without any additional entropy fix. The scheme can exactly resolve stationary contact discontinuities, and it is also freed of the carbuncle problem in multi­dimensional computations.

• Coupled systems mechanics
• /
• v.11 no.2
• /
• pp.167-198
• /
• 2022

In this paper we deal with classical instability problems of heterogeneous Euler beam under conservative loading. It is chosen as the model problem to systematically present several possible solution methods from simplest deterministic to more complex stochastic approach, both of which that can handle more complex engineering problems. We first present classical analytic solution along with rigorous definition of the classical Euler buckling problem starting from homogeneous beam with either simplified linearized theory or the most general geometrically exact beam theory. We then present the numerical solution to this problem by using reduced model constructed by discrete approximation based upon the weak form of the instability problem featuring von Karman (virtual) strain combined with the finite element method. We explain how such numerical approach can easily be adapted to solving instability problems much more complex than classical Euler's beam and in particular for heterogeneous beam, where analytic solution is not readily available. We finally present the stochastic approach making use of the Duffing oscillator, as the corresponding reduced model for heterogeneous Euler's beam within the dynamics framework. We show that such an approach allows computing probability density function quantifying all possible solutions to this instability problem. We conclude that increased computational cost of the stochastic framework is more than compensated by its ability to take into account beam material heterogeneities described in terms of fast oscillating stochastic process, which is typical of time evolution of internal variables describing plasticity and damage.

• Journal of applied mathematics & informatics
• /
• v.34 no.5_6
• /
• pp.421-434
• /
• 2016

The main aim of this paper is to discuss the difference between the Euler-Maruyama's approximate solutions and the accurate solution to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and weakened linear growth condition. Furthermore, we give the pth moment continuous of the approximate solution for the delay equation.

• Journal of applied mathematics & informatics
• /
• v.22 no.1_2
• /
• pp.113-124
• /
• 2006

In this paper we consider the numerical solution of the sunflower equation. We prove that if the sunflower equation has a Hopf bifurcation point at a = ao, then the numerical solution with the Euler-method of the equation has a Hopf bifurcation point at ah = ao + O(h).

• Honam Mathematical Journal
• /
• v.35 no.1
• /
• pp.101-107
• /
• 2013

The Gamma function ${\Gamma}$ which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, and has been introduced in most of calculus textbooks. In this note, our major aim is to explain the convergence of the Euler's Gamma function expressed as an improper integral by using some elementary properties and a fundamental axiom holding on the set of real numbers $\mathbb{R}$, in a detailed and instructive manner. A brief history and origin of the Gamma function is also considered.

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

• Communications for Statistical Applications and Methods
• /
• v.17 no.3
• /
• pp.367-376
• /
• 2010

Recently various methods were suggested and reviewed for estimating diffusion processes. Out of suggested estimation method, we mainly concerns on the estimation method using saddlepoint approximation method, and we suggest a term saddlepoint approximation(ASP) method which is the simplest saddlepoint approximation method. We will show that ASP method provides fast estimator as much as Euler approximation method(EAM) in computing, and the estimator also has good statistical properties comparable to the maximum likelihood estimator(MLE). By simulation study we compare the properties of ASP estimator with MLE and EAM, for Ornstein-Uhlenbeck diffusion processes.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.20 no.2
• /
• pp.127-136
• /
• 2007

In this paper, the equations of the scaled boundary finite element method are derived for non-homogeneous half plane and analyzed numerically In the scaled boundary finite element method, partial differential equations are weaken in the circumferential direction by approximation scheme such as the finite element method, and the radial direction of equations remain in analytical form. The scaled boundary equations of non-homogeneous half plane, its elastic modulus varies as power function, are newly derived by the virtual work theory. It is shown that the governing equation of this problem is the Euler-Cauchy equation, therefore, the logarithm mode used in the half plane problem is not valid in this problem. Two numerical examples are analysed for the verification and the feasibility.

• The Korean Journal of Applied Statistics
• /
• v.24 no.6
• /
• pp.1007-1020
• /
• 2011

Since the research of Black and Scholes (1973), modeling methods using diffusion processes have performed principal roles in financial engineering. In modern financial theories, various types of diffusion processes were suggested and applied in real situations. An estimation of the model parameters is an indispensible step to analyze financial data using diffusion process models. Many estimation methods were suggested and their properties were investigated. This paper reviews the statistical properties of the, Euler approximation method, New Local Linearization(NLL) method, and Generalized Methods of Moment(GMM) that are known as the most practical methods. From the simulation study, we found the NLL and Euler methods performed better than GMM. GMM is frequently used to estimate the parameters because of its simplicity; however this paper shows the performance of GMM is poorer than the Euler approximation method or the NLL method that are even simpler than GMM. This paper shows the performance of the GMM is extremely poor especially when the parameters in diffusion coefficient are to be estimated.