• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.615-622
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• 2005

In this paper, we discuss the average case errors of some numerical quadratures, namely Trapezoidal and Simpson's, in the numerical integration problem. Our integrands are r-fold Wiener functions from the interval [0,1] and only at finite number of points the function values are evaluated. We study average case errors of these quadratures theoretically and then compare it with our practical (a posteriori) researches. Monte-Carlo simulation is used to perform these empirical researches. Finally we empirically compute the error bounds of studied quadratures for the higher degrees of Wiener functions.

• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.107-117
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• 2003

We study the integration problem in which one wants to compute the approximation to the definite integral in the average case setting. We choose the composite Newton- Cotes quadratures as our algorithm and the function values at equally spaced sample points on the given interval［0, 1］as information. We compute the average case error of composite Newton-Cotes quadratures and show that it is minimal (modulo a multi-plicative constant).

• The KIPS Transactions:PartA
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• v.11A no.5
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• pp.401-406
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• 2004

Among many algorithms for the integration problems in which one wants to compute the approximation to the definite integral in the average case setting, we study the average case errors of numerical integration rules using interpolation. In particular, we choose the composite Newton-Cotes quadratures and the function values at equally spaced sample points on the given interval as information. We compute the average case error of composite Newton-Cotes quadratures and show that it is minimal(modulo a multiplicative constant).

• Kyungpook Mathematical Journal
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• v.25 no.1
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• pp.49-60
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• 1985

• Bulletin of the Society of Naval Architects of Korea
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• v.16 no.4
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• pp.13-21
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• 1979

A method of evaluating Kelvin wave pattern is presented in this paper. The mathematical manipulation of x-derivative of the Green function of the Havelock source by the use of contour integration on the complex plane has resulted in the expression that can be readily incorporated with computer program. The efficiency and accuracy that can be secured by the use of the present mathematical expressions seem to be excellent when suitable numerical quadratures are employed. The wave patterns for particular submergences of the singularity are presented.

• Journal of applied mathematics & informatics
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• v.2 no.2
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• pp.13-20
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• 1995

The goal of this paper is to show that the linear approxi-mation in evaluation of functions may be effectively replaced by the ex-ponential approximation formulas obtained by numerical quadratures and in the instance the relative errors can be estimated without know-ing the true values.

• Communications for Statistical Applications and Methods
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• v.26 no.3
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• pp.315-323
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• 2019

Panel data sets have recently been developed in various areas, and many recent studies have analyzed panel, or longitudinal data sets. Often a dichotomous dependent variable occur in survival analysis, biomedical and epidemiological studies that is analyzed by a generalized linear mixed effects model (GLMM). The most common estimation method for the binary panel data may be the maximum likelihood (ML). Many statistical packages provide ML estimates; however, the estimates are computed from numerically approximated likelihood function. For instance, R packages, pglm (Croissant, 2017) approximate the likelihood function by the Gauss-Hermite quadratures, while Rchoice (Sarrias, Journal of Statistical Software, 74, 1-31, 2016) use a Monte Carlo integration method for the approximation. As a result, it can be observed that different packages give different results because of different numerical computation methods. In this note, we discuss the pros and cons of numerical methods compared with the exact computation method.

• Structural Engineering and Mechanics
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• v.85 no.5
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• pp.621-633
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• 2023

The major objective of this paper is to study the receding contact problem between two functional graded layers under a flat indenter. The gravity is assumed negligible, and the shear moduli of both layers are assumed to vary exponentially along the thickness direction. In the absence of body forces, the problem is reduced to a system of Fredholm singular integral equations with the contact pressure and contact size as unknowns via Fourier integral transform, which is transformed into an algebraic one by the Gauss-Chebyshev quadratures and polynomials of both the first and second kinds. Then, an iterative speediest descending algorithm is proposed to numerically solve the system of algebraic equations. Both semi-analytical and finite element method, FEM solutions for the presented problem validate each other. To improve the accuracy of the numerical result of FEM, a graded FEM solution is performed to simulate the FGM mechanical characteristics. The results reveal the potential links between the contact stress/size and the indenter size, the thickness, as well as some other material properties of FGM.

• The Journal of the Acoustical Society of Korea
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• v.27 no.8
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• pp.411-417
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• 2008

An alternative formulation of the Helmholtz integral equation derived to express the pressure field explicitly in terms of the velocity vector of a radiating surface is used to solve acoustic radiation and fluid/structure interaction problems. This formulation, derived for arbitrary sources, is similar in form to the Rayleigh's formula for planar sources. Because the surface pressure field is expressed explicitly as a surface integral of the surface velocity, which can be implemented numerically using standard Gaussian quadratures, there is no need to use BEM to solve a set of simultaneous equations for the surface pressure at the discretized nodes. Furthermore the non-uniqueness problem inherent in methods based on Helmholtz integral equation is avoided. Validation of this formulation is demonstrated for some simple geometries.