Iterative numerical methods for nonlinear systems

2012 ◽  
Vol 19 (2) ◽  
pp. 64-66
Author(s):  
Marinka Zitnik
Keyword(s):  
Numerical Methods ◽  
Nonlinear Systems ◽  
Iterative Numerical Methods

scholarly journals Validation of Simulation Models without Knowledge of Parameters Using Differential Algebra

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Björn Haffke ◽  
Riccardo Möller ◽  
Tobias Melz ◽  
Jens Strackeljan
Keyword(s):  
System Identification ◽  
Numerical Methods ◽  
Nonlinear Systems ◽  
Differential Algebra ◽  
External Validation ◽  
Simulation Models ◽  
Input And Output ◽  
Iterative Numerical Methods

This study deals with the external validation of simulation models using methods from differential algebra. Without any system identification or iterative numerical methods, this approach provides evidence that the equations of a model can represent measured and simulated sets of data. This is very useful to check if a model is, in general, suitable. In addition, the application of this approach to verification of the similarity between the identifiable parameters of two models with different sets of input and output measurements is demonstrated. We present a discussion on how the method can be used to find parameter deviations between any two models. The advantage of this method is its applicability to nonlinear systems as well as its algorithmic nature, which makes it easy to automate.

scholarly journals Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

2021 ◽  
pp. 66-70
Author(s):  
S. Homeniuk ◽  
S. Grebenyuk ◽  
D. Gristchak
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Methods ◽  
Nonlinear Systems ◽  
Analytical Solution ◽  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Numerical Solutions ◽  
Hybrid Approach ◽  
Time Dependent ◽  
Forced Oscillations

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters. The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters. The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions. Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions. Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.

scholarly journals On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1200
Author(s):  
Sanda Micula
Keyword(s):  
Numerical Methods ◽  
Integral Equations ◽  
Trapezoidal Rule ◽  
Picard Iteration ◽  
Fredholm Integral Equations ◽  
Convergence Conditions ◽  
One Dimensional ◽  
Cubature Formulas ◽  
Uniqueness Of The Solution ◽  
Iterative Numerical Methods

In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra–Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also provide successive iterations of the solution) and suitable cubature formulas for the numerical approximations. We discuss in detail a method using Picard iteration and the two-dimensional composite trapezoidal rule, giving convergence conditions and error estimates. The paper concludes with numerical experiments and a discussion of the methods proposed.

scholarly journals Numerical Simulation of a Class of Hyperchaotic System Using Barycentric Lagrange Interpolation Collocation Method

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Xiaofei Zhou ◽  
Junmei Li ◽  
Yulan Wang ◽  
Wei Zhang
Keyword(s):  
Numerical Simulation ◽  
Numerical Methods ◽  
Nonlinear Systems ◽  
Collocation Method ◽  
Present Method ◽  
Lagrange Interpolation ◽  
Hyperchaotic System ◽  
Research Subject ◽  
The Past ◽  
Active Research

Hyperchaotic system, as an important topic, has become an active research subject in nonlinear science. Over the past two decades, hyperchaotic system between nonlinear systems has been extensively studied. Although many kinds of numerical methods of the system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, this paper introduces another novel numerical method to solve a class of hyperchaotic system. Barycentric Lagrange interpolation collocation method is given and illustrated with hyperchaotic system (x˙=ax+dz-yz,y˙=xz-by,  0≤t≤T,z˙=cx-z+xy,w˙=cy-w+xz,) as examples. Numerical simulations are used to verify the effectiveness of the present method.

scholarly journals Comparative Study of Various Iterative Numerical Methods for Computation of Approximate Root of the Polynomials

2021 ◽  
Vol 11 (2) ◽  
pp. 6-11
Author(s):  
Dr. Roopa K M ◽  
◽  
Venkatesha P ◽  
Keyword(s):  
Numerical Methods ◽  
Comparative Study ◽  
Iterative Methods ◽  
Numerical Comparison ◽  
Polynomial Equations ◽  
Chebyshev Method ◽  
Newton Raphson ◽  
Raphson Method ◽  
Iterative Numerical Methods

The aim of this article is to present a brief review and a numerical comparison of iterative methods applied to solve the polynomial equations with real coefficients. In this paper, four numerical methods are compared, namely: Horner’s method, Synthetic division with Chebyshev method (Proposed Method), Synthetic division with Modified Newton Raphson method and Birge-Vieta method which will helpful to the readers to understand the importance and usefulness of these methods.

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