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FAMILIES OF NONLINEAR TRANSFORMATIONS FOR ACCURATE EVALUATION OF WEAKLY SINGULAR INTEGRALS
윤병인 한국산업응용수학회 2023 Journal of the Korean Society for Industrial and A Vol.27 No.3
We present families of nonlinear transformations useful for numerical evaluation of weakly singular integrals. First, for end-point singular integrals, we define a prototype func- tion with some appropriate features and then suggest a family of transformations. In addition, for interior-point singular integrals, we develop a family of nonlinear transformations based on the aforementioned prototype function. We take some examples to explore the efficiency of the proposed nonlinear transformations in using the Gauss-Legendre quadrature rule. From the nu- merical results, we can find the superiority of the proposed transformations compared to some existing transformations, especially for the integrals with high singularity strength.
A QUADRATICALLY CONVERGENT ITERATIVE METHOD FOR NONLINEAR EQUATIONS
윤병인,Miodrag S. Petkovic 대한수학회 2011 대한수학회지 Vol.48 No.3
In this paper we propose a simple iterative method for nding a root of a nonlinear equation. It is shown that the new method, which does not require any derivatives, has a quadratic convergence order. In addition, one can nd that a hybrid method combined with the non-iterative method can further improve the convergence rate. To show the efficiency of the presented method we give some numerical examples.
EVALUATION OF SINGULAR INTEGRALS BY HYPERBOLIC TANGENT BASED TRANSFORMATIONS
윤병인 대한수학회 2011 대한수학회지 Vol.48 No.1
We employ a hyperbolic tangent function to construct nonlinear transformations which are useful in numerical evaluation of weakly singular integrals and Cauchy principal value integrals. Results of numerical implementation based on the standard Gauss quadrature rule show that the present transformations are available for the singular integrals and, in some cases, give much better approximations compared with those of existing non-linear transformation methods.
Approximation to the cumulative normal distribution using hyperbolic tangent based functions
윤병인 대한수학회 2009 대한수학회지 Vol.46 No.6
This paper presents a method for approximation of the standard normal distribution by using hyperbolic tangent based functions. The presented approximate formula for the cumulative distribution depends on one numerical coefficient only, and its accuracy is admissible. Furthermore, in some particular cases, closed forms of inverse formulas are derived. Numerical results of the present method are compared with those of an existing method. This paper presents a method for approximation of the standard normal distribution by using hyperbolic tangent based functions. The presented approximate formula for the cumulative distribution depends on one numerical coefficient only, and its accuracy is admissible. Furthermore, in some particular cases, closed forms of inverse formulas are derived. Numerical results of the present method are compared with those of an existing method.
The trapezoidal rule with a nonlinear coordinate transformation for weakly singular integrals
윤병인 대한수학회 2004 대한수학회지 Vol.41 No.6
It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation Ωm(b; x), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation Ωm(b; x), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation Ωm(b; x).