Beong In Yun

Intuitive approach to the approximate analytical solution for the Blasius problem

For the Blasius problem, we propose an approximate analytical solution in the form of a logarithm of the hyperbolic cosine function which satisfies the given boundary conditions and some known properties of the exact solution. Furthermore, adding some hyperbolic tangent functions to this solution, we obtain much more accurate approximate solution with the relative error less than 0.16% over the whole region. The superiority of the proposed solutions is shown by comparison with the existing approximate analytical solution.

A New Sigmoidal Transformation for Weakly Singular Integrals in the Boundary Element Method

The power of the sigmoidal transformation in weakly singular integrals has been demonstrated by the recent works [A. Sidi, in Numerical Integration IV, H. Brass and G. Hammerlin, eds., Birkhauser--Verlag, Berlin, 1993, pp. 359--373; P. R. Johnston, Internat. J. Numer. Methods Engrg., 45 (1999), pp. 1333--1348; P. R. Johnston, Internat. J. Numer. Methods Engrg., 47 (2000), pp. 1709--1730; D. Elliott, Math. Methods Appl. Sci., 20 (1997), pp. 121--132; D. Elliott, J. Austral. Math. Soc. Ser. B, 40 (1998), pp. E77--E137]. Especially, application of this transformation is useful for efficient numerical evaluation of the singular integrals appearing in the usual boundary element method. In this paper, a new sigmoidal transformation containing a parameter b is presented. It is shown that the present transformation, with the Gauss--Legendre quadrature rule, can improve the asymptotic truncation error of the traditional sigmoidal transformations by controlling the parameter. For some examples, we compare the numerical results of the present method with those of the well-known Sidi- and Elliott-transformations to show the superiority of the former.

An Extended Sigmoidal Transformation Technique for Evaluating Weakly Singular Integrals without Splitting the Integration Interval

In this paper, we present an efficient transformation technique for accurate numerical evaluation of weakly singular integrals with interior singularities. The present transformation technique does not require any division of the integration interval. It is composed of two parts of a sigmoidal transformation whose tails coincide with a singular point smoothly up to the order of the sigmoidal transformation employed. Therefore, in using the standard Gauss quadrature rule, the present transformation has a feature in which the integration points are clustered into the interior singular point very closely, as in the case of endpoint singular integrals. The results of some numerical examples show the superiority of the present method over the existing methods.

Iterative methods based on the signum function approach for solving nonlinear equations

For finding a root of an equation f(x) = 0 on an interval (a, b), we develop an iterative method using the signum function and the trapezoidal rule for numerical integrations based on the recent work (Yun, Appl Math Comput 198:691–699, 2008). This method, so-called signum iteration method, depends only on the signum function

A DERIVATIVE FREE ITERATIVE METHOD FOR FINDING MULTIPLE ROOTS OF NONLINEAR EQUATIONS

{\rm{sgn}}\left(f(x)\right)

Iterative methods for solving nonlinear equations with finitely many roots in an interval

independently of the behavior of f(x), and the error bound of the kth approximation is (b − a)/(2Nk), where N is the number of integration points for the trapezoidal rule in each iteration. In addition we suggest hybrid methods which combine the signum iteration method with usual methods such as Newton, Ostrowski and secant methods. In particular the hybrid method combined with the signum iteration and the secant method is a predictor-corrector type method (Noor and Ahmad, Appl Math Comput 180:167–172, 2006). The proposed methods result in the rapidly convergent approximations, without worry about choosing a proper initial guess. By some numerical examples we show the superiority of the presented methods over the existing iterative methods.

Solving nonlinear equations by a new derivative free iterative method

Abstract For an equation f ( x ) = 0 having a multiple root of multiplicity m > 1 unknown, we propose a transformation which converts the multiple root to a simple root of H ϵ ( x ) = 0 . The transformed function H ϵ ( x ) of f ( x ) with a small ϵ > 0 has appropriate properties in applying a derivative free iterative method to find the root. Moreover, there is no need to choose a proper initial approximation. We show that the proposed method is superior to the existing methods by several numerical examples.

On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals

In this paper we consider a nonlinear equation f(x)=0 having finitely many roots in a bounded interval. Based on the so-called numerical integration method [B.I. Yun, A non-iterative method for solving non-linear equations, Appl. Math. Comput. 198 (2008) 691-699] without any initial guess, we propose iterative methods to obtain all the roots of the nonlinear equation. In the result, an algorithm to find all of the simple roots and multiple ones as well as the extrema of f(x) is developed. Moreover, criteria for distinguishing zeros and extrema are included in the algorithm. Availability of the proposed method is demonstrated by some numerical examples.

Sigmoid-like functions and root finding methods☆

We develop a new simple iteration formula, which does not require any derivatives of f(x), for solving a nonlinear equation f(x) = 0. It is proved that the convergence order of the new method is quadratic. Furthermore, the new method can approximate complex roots. By several numerical examples we show that the presented method will give desirable approximation to the root without a particularly good initial approximation and be efficient for all cases, regardless of the behavior of f(x).

An averaging method for the Fourier approximation to discontinuous functions

The aim of this work is to analyse the stability and the convergence for the quadrature rule of interpolatory-type, based on the trigonometric approximation, for the discretization of the Cauchy principal value integrals ∫-11 f(τ)/(τ - t)dτ. We prove that the quadrature rule has almost optimal stability property behaving in the form O((logN + 1)/sin2x), x=cos t. Using this result, we show that the rule has an exponential convergence rate when the function f is differentiable enough. When f possesses continuous derivatives up to order p ≥ 0 and the derivative f(p)(t) satisfies Holder continuity of order ρ, we can also prove that the rule has the convergence rate of the form O((A + B log N + N2ν)/Np+p), where ν is as small as we like, A and B are constants depending only on x.