AN IMPROVED EXPONENTIAL REGULA FALSI METHODS WITH CUBIC CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS

Hoda Ibrahim 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.5

The aim of this paper is to propose a cubic convergent regula falsi iterative method for solving the nonlinear equation f(x) = 0, where f : [a,b] ⊂ □ → □ is a continuously di®erentiable. In [3,6] a quadratically convergent regula falsi iterative methods for solving this nonlinear equations is proposed. It is shown there that both the sequences of diameters and iterative points sequence converge to zero simultaneously. So The aim of this paper is to accelerate further the convergence of these methods from quadratic to cubic. This is done by replacing the parameter p in the iteration of [3,5,6] by a function p(x) defined suitably. The convergence analysis is carried out for the method. The method is tested on number of numerical examples and results obtained shows that our methods are better and more effective and comparable to well-known methods.

AN IMPROVED EXPONENTIAL REGULA FALSI METHODS WITH CUBIC CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS

Ibrahim, S.A. Hoda The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.5

The aim of this paper is to propose a cubic convergent regula falsi iterative method for solving the nonlinear equation f(x) = 0, where f : [a,b] $\subset$ R $\rightarrow$ R is a continuously differentiable. In [3,6] a quadratically convergent regula falsi iterative methods for solving this nonlinear equations is proposed. It is shown there that both the sequences of diameters and iterative points sequence converge to zero simultaneously. So The aim of this paper is to accelerate further the convergence of these methods from quadratic to cubic. This is done by replacing the parameter p in the iteration of [3,5,6] by a function p(x) defined suitably. The convergence analysis is carried out for the method. The method is tested on number of numerical examples and results obtained shows that our methods are better and more effective and comparable to well-known methods.

The Role of the Asymmetry of a Dispersion Map in a Dispersion Managed Optical Communication System Possessing Quintic Nonlinearity

Manoj Mishra,홍우표 한국물리학회 2011 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.58 No.6

The role of the asymmetry of a dispersion map in the dynamics of a propagating dispersionmanaged (DM) optical soliton is investigated in a medium possessing cubic-quintic nonlinearities. The propagation of the optical pulse in the medium is numerically studied based on the modified nonlinear Schr쮗dinger equation. A variational method is employed to find the initial pulse profile of the nonlinear Schr쮗dinger equation, and its dynamical behavior in the nonlinear medium is numerically simulated using the spilt-step Fourier method. The numerical simulations predict that when the ratio of the anomalous fiber length to the map length is equal to 0.7, a significant improvement in the intra-pulse collision distance is achieved.

제3 고조파를 이용한 비선형 파라미터의 측정 연구 – 1. 이론

정현조(Hyunjo Jeong),최성호(Sungho Choi) 한국비파괴검사학회 2021 한국비파괴검사학회지 Vol.41 No.1

제2 고조파의 발생 및 측정을 통하여 재료의 비선형 물성 또는 손상 정도를 평가할 수 있는 2차 비선형 파라미터의 측정과 관련된 이론 및 실험은 비교적 잘 정립되어 있다. 이에 비하여 제3 고조파를 이용한 3차 비선형 파라미터 측정은 아직까지 상대적 측정이 주를 이루고 있고, 절대적 측정을 위한 이론과 실험은 잘 개발되어 있지 않다. 본 논문에서는 비선형 등방성 재료에서 제3 고조파를 이용하여 비선형 파라미터를 정확하게 측정하기 위한 이론적인 내용을 다룬다. 먼저 변위 기반 평면파의 파동방정식에 섭동해를 적용하여 제3 고조파의 진폭을 구하고 이로부터 3차 비선형 파라미터 공식을 정의한다. 실제로 측정되는 초음파의 진폭은 회절 및 감쇠로 인하여 평면파에서 많이 벗어나므로 이에 대한 보정이 필요하다. 감쇠 및 회절보정은 3차원 Westervelt 방정식의 적분해에 기초한 다중 가우시안 빔 모델로부터 구한다. 감쇠와 회절이 보정된 3차 비선형 파라미터 공식을 제시하며, 이 식은 추후 논문에서 다룰 3차 비선형 파라미터 측정에 적용될 것이다. Theories and experiments related to the measurement of quadratic nonlinear parameters capable of evaluating the degree of damage or nonlinear material properties through the generation and measurement of second harmonics are relatively well established. In contrast, measurement of cubic nonlinear parameters using third harmonics is limited to relative measurement, and theories and experiments for absolute measurement have not been well developed. This paper deals with the theoretical aspects for accurately measuring cubic nonlinear parameters using third harmonics in isotropic materials. First, the amplitude of the third harmonic is obtained by applying the perturbation solution to the displacement-based wave equation, and the nonlinear parameter formula is defined. Since the amplitude of the ultrasonic wave actually measured deviates from the plane wave due to diffraction and attenuation, corrections of these effects are required. The attenuation and diffraction corrections are obtained from a multi-Gaussian beam model based on the integral solutions for the three-dimensional Westervelt equation. We present a formula for the cubic nonlinear parameter corrected for diffraction and attenuation. The results shown in this paper will be applied to the measurement of cubic nonlinear parameter in a separate paper.

Variants of Newton's method using fifth-order quadrature formulas: Revisited

Muhammad Aslam Noor,Muhammad Waseem 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5

In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results. In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results.

A LINEARIZED FINITE-DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION OF THE NONLINEAR CUBIC SCHRODINGER EQUATION

Bratsos, A.G. 한국전산응용수학회 2001 The Korean journal of computational & applied math Vol.8 No.3

A linearized finite-difference scheme is used to transform the initial/boundary-value problem associated with the nonlinear Schrodinger equation into a linear algebraic system. This method is developed by replacing the time and the nonlinear term by an appropriate parametric linearized scheme based on Taylor’s expansion. The resulting finite-difference method is analysed for stability and convergence. The results of a number of numerical experiments for the single-soliton wave are given. AMS Mathematics Subject Classification : 65J15, 47H17, 49D15

Effect of sweep angle on bifurcation analysis of a wing containing cubic nonlinearity

Irani, Saied,Amoozgar, Mohammadreza,Sarrafzadeh, Hamid Techno-Press 2016 Advances in aircraft and spacecraft science Vol.3 No.4

Limit cycle oscillations (LCO) as well as nonlinear aeroelastic analysis of a swept aircraft wing with cubic restoring moments in the pitch degree of freedom is investigated. The unsteady aerodynamic loading applied on the wing is modeled by using the strip theory. The harmonic balance method is used to calculate the LCO frequency and amplitude for the swept wing. Finally the super and subcritical Hopf bifurcation diagrams are plotted. It is concluded that the type of bifurcation and turning point location is sensitive to the system parameters such as wing geometry and sweep angle.

3차 비용함수에 의한 이익곡선 도출과 산업 라이프사이클의 수리적 검증: 우리나라 전 산업을 중심으로

배후석,임채관 한국품질경영학회 2023 품질경영학회지 Vol.51 No.4

Purpose: The main theme of this study is to derive a profit curve by a cubic cost function for nonlinear CVP analysis. According to the analytical approach to derive a nonlinear profit function in this study, it is possible with only the existing cost structure to calculate the profit maximization and downtime point sales unlike the classical CVP analysis. Furthermore, the profit curve by the mathematical model of this study could serve as a tool to quantify the qualitative evaluation of each stage of the industry life cycle. Methods: This study followed the mathematical approach from the cubic cost function model of microeconomics, and using real data of the Bank of Korea Results: The nonlinear profit function suggested by this study is as follows; Conclusion: The process and results of this study would be able to contribute not only in practice of nonlinear CVP analysis required in the management accounting or financial management, but also in cost theory of microeconomics. Also, since the life cycle of all industries in Korea was verified to the growth or mature stage, decision makers should pay careful attention to determining life cycle stages and consider the profit curve by the average variable cost ratio over multi periods.

Time series prediction of nonlinear ship structural responses in irregular seaways using a third-order Volterra model

Kim, Y.,Kim, J.H.,Kim, Y. Academic Press 2014 Journal of fluids and structures Vol.49 No.-

To predict the nonlinear structural responses of a ship traveling through irregular waves, a third-order Volterra model was applied based on the given irregular data. A nonlinear wave-body interaction system was identified using the nonlinear autoregressive with exogenous input (NARX) technique, which is one of the most commonly used nonlinear system identification schemes. The harmonic probing method was applied to extract the first-, second- and third-order frequency response functions of the system. To achieve this, a given set of time history data of both the irregular wave excitation and the corresponding midship vertical bending moment for a certain sea state was fed into the three-layer perceptron neural network. The network parameters are determined based on the supervised training. Next, the harmonic probing method was applied to the identified system to extract the frequency response function of each order. While applying the harmonic probing method, the nonlinear activation function (i.e., the hyperbolic tangent function) was expanded into a Taylor series for harmonic component matching. After the frequency response functions were obtained, the structural responses of the ship under an arbitrary random wave excitation were easily calculated with rapidity using a third-order Volterra series. Additionally, the methodology was validated through the in-depth analysis of a nonlinear oscillator model for a weak quadratic and cubic stiffness term, whose analytic solutions are known. It was confirmed that the current method effectively predicts the nonlinear structural response of a large container carrier under arbitrary random wave excitation.

The Cauchy problem for an integrable generalized Camassa-Holm equation with cubic nonlinearity

Bin Liu,Lei Zhang 대한수학회 2018 대한수학회보 Vol.55 No.1

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B_{p,r}^s$ with $s>\max\{\frac{1}{p},\frac{1}{2},1-\frac{1}{p}\}$, $p,r\in [0,\infty]$. Second, we prove that the equation is locally well-posed in $B_{2,r}^s$ with the critical index $s=\frac{1}{2}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is H\"{o}lder continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of $m$ along the characteristics.