• 대한수학회논문집
• /
• 제23권1호
• /
• pp.61-65
• /
• 2008

In this paper we prove that every positive solution of the third order rational difference equation $$x_{n+1}\;=\;\frac{{\beta}x_n\;+\;x_{n-2}}{A\;+\;Bx_n\;+\;x_{n-2}}$ converges to the positive equilibrium point $$\bar{x}\;=\;\frac{{\beta}\;+\;1\;-\;A}{B\;+\;1}$, where $0\;<\;{\beta}\;{\leq}\;B$, $1\;<\;A\;<\;{\beta}\;+\;1$

• 호남수학학술지
• /
• 제44권3호
• /
• pp.391-401
• /
• 2022

In this study, we investigate the explicit characterization of spherical curves using the Flc (Frenet like curve) frame in Euclidean 3-space. Firstly, the axis of curvature and the osculating sphere of a polynomial space curve are calculated using Flc frame invariants. It is then shown that the axis of curvature is on a straight line. The position vector of a spherical curve is expressed with curvatures connected to the Flc frame. Finally, a differential equation is obtained from the third order, which characterizes a spherical curve.

• 충청수학회지
• /
• 제28권4호
• /
• pp.513-523
• /
• 2015

We present a local convergence analysis of some Newton-like methods of R-order at least three in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second $Fr{\acute{e}}chet$-derivative of the operator involved. These conditions are weaker that the corresponding ones given by Hernandez, Romero [10] and others [1], [4]-[9] requiring hypotheses up to the third $Fr{\acute{e}}chet$ derivative. Numerical examples are also provided in this study.

• 한국수학교육학회지시리즈A:수학교육
• /
• 제57권3호
• /
• pp.231-246
• /
• 2018

In this paper, we applied Sfard's reification theory to analyze the secondary students' level of conceptualization with regard to the formula of quadratic equation. Through the generation and development of mathematical concepts from a historical perspective, Sfard classified the formulation process into three stages of interiorization, condensation, and reification, and proposed levels of formulation. Based on this theory, we constructed a test tool reflecting the reversibility of the nature of manipulation of Piaget's theory as a criterion of content judgement in order to grasp students' conceptualization level of the formula of quadratic equation. By applying this tool, we analyzed the conceptualization level of the formula of quadratic equation of the $9^{th}$ and $10^{th}$ graders. The main results are as follows. First, approximately 45% of $9^{th}$ graders can not memorize the formula of quadratic equation, or even if they memorize, they do not have the ability of accurate calculation to apply for it. Second, high school curriculum requires for students to use the formula of the quadratic equation, but about 60% of $10^{th}$ graders have not reached at the level of reification that they can use the formula of quadratic equation. Third, as a result of imaginarily correcting the error of the previous concept, there was a change in the levels of $9^{th}$ graders, and there was no change in $10^{th}$ graders.

• Coupled systems mechanics
• /
• 제1권2호
• /
• pp.155-163
• /
• 2012

This paper presents analytical approximate solutions for the initial post-buckling deformation of the pipes in oil and gas wells. The governing differential equation with sinusoidal nonlinearity can be reduced to form a third-order-polynomial nonlinear equation, by coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. The linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations, unlike the classical method of harmonic balance. We are hence able to establish analytical approximate solutions. The approximate formulae for load along axis, and periodic solution are established for derivative of the helix angle at the end of the pipe. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

• Journal of Mechanical Science and Technology
• /
• 제16권10호
• /
• pp.1320-1326
• /
• 2002

In a one-dimensional heat conduction domain with heated and insulated walls, an integral approach is proposed to estimate time-dependent boundary heat flux without internal measurements. It is assumed that the expression of the heat flux is not known a priori. Hence, the present inverse heat conduction problem is classified as a function estimation problem. The spatial temperature distribution is approximated as a third-order polynomial of position, whose four coefficients are determined from the heat fluxes and the temperatures at both ends at each measurement. After integrating the heat conduction equation over spatial and time domain, respectively, a simple and non-iterative recursive equation to estimate the time-dependent boundary heat flux is derived. Several examples are introduced to show the effectiveness of the present approach.

• Kyungpook Mathematical Journal
• /
• 제57권2호
• /
• pp.251-263
• /
• 2017

The aim of this paper is to investigate the dynamical behavior of the difference equation $$x_{n+1}={\frac{{\alpha}x_n}{{\beta}+{\gamma}x^p_{n-1}x^q_{n-2}}},\;n=0,1,{\ldots}$$, where the parameters ${\alpha}$, ${\beta}$, ${\gamma}$, p, q are non-negative numbers and the initial values $x_{-2}$, $x_{-1}$, $x_0$ are positive numbers. Also, some numerical examples are given to verify our theoretical results.

• 한국전산유체공학회지
• /
• 제12권4호
• /
• pp.20-27
• /
• 2007

Two-dimensional stagnation flow toward a plane wall coated with magnetic fluid of uniform thickness is investigated. The flow field is represented as a similarity solution of the Navier-Stokes equation for this incompressible laminar flow. The resulting third order ordinary differential equation is solved numerically by using the shooting method and by determining two shooting parameters so as to satisfy the boundary and interface conditions. Features of the flow including streamline patterns are investigated for the varying values of density ratio, viscosity ratio, and Reynolds number. An adverse flow with double eddy pair in magnetic fluid region is found to emerge as the Reynolds number becomes higher than a threshold value. The results for the interface velocity, interface and wall shear stress, and boundary layer and displacement thickness are also presented.

• Bulletin of the Korean Chemical Society
• /
• 제35권1호
• /
• pp.51-56
• /
• 2014

The solvolysis rate constants of 2,4-dimethoxybenzenesulfonyl chloride (1) in 30 different solvents are well correlated with the extended Grunwald-Winstein equation, using the $N_T$ solvent nucleophilicity scale and $Y_{Cl}$ solvent ionizing scale, with sensitivity values of $0.93{\pm}0.14$ and $0.65{\pm}0.06$ for l and m, respectively. These l and m values can be considered to support a $S_N2$ reaction pathway. The activation enthalpies (${\Delta}H^{\neq}$) were 12.4 to $14.6kcal{\cdot}mol^{-1}$ and the activation entropies (${\Delta}S^{\neq}$) were -15.5 to -$32.3kcal{\cdot}mol^{-1}{\cdot}K^{-1}$, which is consistent with the proposed bimolecular reaction mechanism. The solvent kinetic isotope effects (SKIE) were 1.74 to 1.86, which is also in accord with the $S_N2$ mechanism and was possibly assisted using a general-base catalysis. The values of product selectivity (S) for solvolyses of 1 in alcohol/water mixtures was 0.57 to 6.5, which is also consistent with the proposed bimolecular reaction mechanism. Third-order rate constants, $k_{ww}$ and $k_{aa}$, were calculated from the rate constants ($k_{obs}$), together with $k_{aw}$ and $k_{wa}$ calculated from the intercept and slope of the plot of 1/S vs. [water]/[alcohol]. The calculated rate constants, $k_{calc}$ ($k_{ww}$, $k_{aw}$, $k_{wa}$ and $k_{aa}$), are in satisfactory agreement with the experimental values, supporting the stoichiometric solvation effect analysis.

• Smart Structures and Systems
• /
• 제21권1호
• /
• pp.113-122
• /
• 2018

In this article, an exact analytical solution for mechanical buckling analysis of magnetoelectroelastic plate resting on pasternak foundation is investigated based on the third-order shear deformation plate theory. The in-plane electric and magnetic fields can be ignored for plates. According to Maxwell equation and magnetoelectric boundary condition, the variation of electric and magnetic potentials along the thickness direction of the plate is determined. The von Karman model is exploited to capture the effect of nonlinearity. Navier's approach has been used to solve the governing equations for all edges simply supported boundary conditions. Numerical results reveal the effects of (i) lateral load, (ii) electric load, (iii) magnetic load and (iv) higher order shear deformation theory on the critical buckling load have been investigated. These results must be the analysis of intelligent structures constructed from magnetoelectroelastic materials.