Numerical continuation of solution at a singular point of high codimension for systems of nonlinear algebraic or transcendental equations

2016 ◽  
Vol 56 (9) ◽  
pp. 1551-1564 ◽  
Author(s):  
S. D. Krasnikov ◽  
E. B. Kuznetsov
Keyword(s):  
Singular Point ◽  
Numerical Continuation ◽  
High Codimension ◽  
Transcendental Equations

scholarly journals An approach for constructing one-point iterative methods for solving nonlinear equations of one variable

2015 ◽  
pp. 298-306
Author(s):  
А.Н. Громов
Keyword(s):  
Singular Point ◽  
Convergence Rate ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Iterative Process ◽  
Initial Approximation ◽  
Higher Order ◽  
Convergence Criterion ◽  
Real Roots ◽  
Transcendental Equations

Предложен подход к построению одноточечных итерационных методов для решения нелинейных уравнений одного переменного. Подход основан на использовании понятия полюса в качестве особой точки и на применении критерия сходимости Коши. Показано, что такой подход приводит к новым итерационным процессам высшего порядка, которые имеют более широкую область сходимости по сравнению с известными методами. Доказаны теоремы сходимости и получены оценки скорости сходимости. Для многочленов, имеющих только действительные корни, итерационный процесс сходится для любого начального приближения. В общем случае для действительных корней трансцендентных уравнений сходимость имеет место при выборе начального приближения в окрестности корня. An approach for constructing one-point iterative methods for solving nonlinear equations of one variable is proposed. This approach is based on the concept of a pole as a singular point and on using Cauchy's convergence criterion. It is shown that such an approach leads to new iterative processes of higher order with larger convergence domains compared to the known iterative methods. Convergence theorems are proved and convergence rate estimates are obtained. For polynomials having only real roots, the iterative process converges for any initial approximation to the sought root. Generally, in the case of real roots of transcendental equations, the convergence takes place when an initial approximation is chosen near the sought root.

scholarly journals Kinematic Singularity and Bifurcation Analysis of Sidestay Landing Gear Locking Mechanisms

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Kui Xu ◽  
Yin Yin ◽  
Yixin Yang ◽  
Hong Nie ◽  
Xiaohui Wei
Keyword(s):  
Singular Point ◽  
Structural Parameters ◽  
Continuation Method ◽  
High Sensitivity ◽  
Optimization Method ◽  
Landing Gear ◽  
Singular Points ◽  
Numerical Continuation ◽  
Kinematic Optimization ◽  
Kinematic Mechanism

A dual-sidestay landing gear is prone to locking failure in the deployed state due to the restriction of movement between two sidestays. However, the principle of its locking movement still remains unclear. The present study is aimed at investigating the synchronous locking performance of the dual-sidestay landing gear based on the singularity and bifurcation theory. From the perspective of the kinematic mechanism, the reason for high sensitivity to structural dimensions in the locking process is explained, and the locked position is investigated by employing the numerical continuation method in the case of a single-sidestay landing gear. Afterwards, the reason for the locking failure of the dual-sidestay landing gear is analyzed, and a kinematic optimization method for the synchronous locking is proposed. The results reveal that the lock links must reach the lower overcenter singular point to fully lock the landing gear, and the singular point is sensitively affected by structural parameters. Owing to the different positions of singular points, the movements of fore and aft sidestays seriously restrict each other, causing locking failure of the dual-sidestay landing gear. The singular points of two sidestays can be optimized to be approximately identical, making their movements more coordinated.

First order over determined system with singular point

2007 ◽  
Vol 18 (1) ◽  
pp. 65-80
Author(s):  
Adel Nasim Adib ◽  
Nusrat Rajabov
Keyword(s):  
Singular Point ◽  
First Order

scholarly journals A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 920
Author(s):  
Chukwuma Ogbonnaya ◽  
Chamil Abeykoon ◽  
Adel Nasser ◽  
Ali Turan
Keyword(s):  
Numerical Models ◽  
Problem Formulation ◽  
Science And Engineering ◽  
Physical Systems ◽  
Engineering Problems ◽  
Parametric Studies ◽  
Independent Variable ◽  
Transcendental Equations ◽  
The Waves ◽  
Modelling Approach

A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to x1 = 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems.

Boundary Value Problems of Electroelasticity with Concentrated Singularities

1994 ◽  
Vol 1 (5) ◽  
pp. 459-467
Author(s):  
T. Buchukuri ◽  
D. Yanakidi
Keyword(s):  
Singular Point ◽  
Boundary Value Problems ◽  
Power Function ◽  
Boundary Value ◽  
Linearly Independent

Abstract We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighbourhood of infinity or in the neighbourhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.

Swept volumes via spacetime numerical continuation

2021 ◽  
Vol 40 (4) ◽  
pp. 1-11
Author(s):  
Silvia Sellán ◽  
Noam Aigerman ◽  
Alec Jacobson
Keyword(s):  
Numerical Continuation ◽  
Swept Volumes

scholarly journals Two dimensionless parameters and a mechanical analogue for the HKB model of motor coordination

2021 ◽  
Author(s):  
J. F. Cass ◽  
S. J. Hogan
Keyword(s):  
Motor Coordination ◽  
Nonlinear Damping ◽  
Numerical Continuation ◽  
Arbitrary Parameter ◽  
Relevant Parameter ◽  
Coupling Coefficients ◽  
Dimensionless Parameters ◽  
Biologically Relevant ◽  
Mechanical Analogue ◽  
Partner Interaction

AbstractThe widely cited Haken–Kelso–Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself.

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