scholarly journals An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

2020 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
Irfan Nurhidayat ◽  
Zijun Hao ◽  
Chu-chin Hu ◽  
Jein-Shan Chen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Time Integration ◽  
Nonlinear Dynamical System ◽  
Special Technique ◽  
Algebraic Equations ◽  
Approximation Solution ◽  
Nonlinear Dynamical ◽  
Group Preserving Scheme ◽  
Original Time

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

Beating Effects in a Single Nonlinear Dynamical System in a Neighborhood of the Resonance

2016 ◽  
Vol 849 ◽  
pp. 76-83
Author(s):  
Jiří Náprstek ◽  
Cyril Fischer
Keyword(s):  
Harmonic Balance ◽  
Excitation Frequency ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
System Response ◽  
Algebraic Equations ◽  
Single Degree Of Freedom ◽  
Nonlinear Dynamical ◽  
The Difference ◽  
Eigen Frequency

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) nonlinear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs from the system eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. Theabove mentioned phenomena have been identified in many papers including authors’ contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference of excitation and eigen-frequency is still missing. Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system in mono- and multi-harmonic versions. They lead to nonlinear differential and algebraic equations serving as a basis for qualitativeanalytic estimation or numerical description of characteristics of the quasi-periodic system response. Zero, first and second level perturbation techniques are used. Appearance, stability and neighborhood of limit cycles is evaluated. Numerical phases are based on simulation processes and numerical continuation tools. Parametric evaluation and illustrating examples are presented.

scholarly journals A REPRESENTATIVE SOLUTION TO M-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION WITH NONLOCAL CONDITIONS BY GREEN'S FUNCTIONAL CONCEPT

2012 ◽  
Vol 17 (4) ◽  
pp. 571-588 ◽  
Author(s):  
Kemal Ozen ◽  
Kamil Orucoglu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlocal Conditions ◽  
Adjoint System ◽  
Linear Ordinary Differential Equation ◽  
The Other ◽  
Algebraic Equations ◽  
Order Ordinary Differential Equation ◽  
Multipoint Problem ◽  
Nonsmooth Coefficients

In this work, we investigate a linear completely nonhomogeneous nonlocal multipoint problem for an m-order ordinary differential equation with generally variable nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness. A system of m + 1 integro-algebraic equations called the special adjoint system is constructed for this problem. Green's functional is a solution of this special adjoint system. Its first component corresponds to Green's function for the problem. The other components correspond to the unit effects of the conditions. A solution to the problem is an integral representation which is based on using this new Green's functional. Some illustrative implementations and comparisons are provided with some known results in order to demonstrate the advantages of the proposed approach.

Urn models and differential algebraic equations

2003 ◽  
Vol 40 (02) ◽  
pp. 401-412 ◽  
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Random Environment ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Urn Model ◽  
Recurrent Equation ◽  
Strong Laws ◽  
The Stability ◽  
Generalized Pólya Urn

The aim of this paper is to study the distribution of colours, { X n }, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process { X n } is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process { X n }.

Unusual extension–torsion–inflation couplings in pressurized thin circular tubes with helical anisotropy

2018 ◽  
Vol 24 (9) ◽  
pp. 2694-2712 ◽  
Author(s):  
Raushan Singh ◽  
Pranjal Singh ◽  
Ajeet Kumar
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Radial Displacement ◽  
Applied Pressure ◽  
Nonlinear Ordinary Differential Equation ◽  
Algebraic Equations ◽  
Circular Tubes ◽  
Thin Tube ◽  
Analytical Expressions ◽  
Helical Anisotropy

We present a thin tube formulation for coupled extension–torsion–inflation deformation in helically reinforced pressurized circular tubes. Both compressible and incompressible tubes are considered. On applying the thin tube limit, the nonlinear ordinary differential equation to obtain the in-plane radial displacement is converted into a set of two simple algebraic equations for the compressible case and one equation for the incompressible case. This allows us to obtain analytical expressions, in terms of the tube’s intrinsic twist, material constants, and the applied pressure, which can predict whether such tubes would overwind/unwind on being infinitesimally stretched or exhibit positive/negative Poisson’s effect. We further show numerically that such tubes can be tuned to generate initial overwinding followed by rapid unwinding as observed during finite stretching of a torsionally relaxed DNA. Finally, we demonstrate that such tubes can also exhibit usual deflation initially followed by unusual inflation as the tube is finitely stretched.

Improved Adjoint Operator Method and Normal Form of Nonlinear Dynamical System

2013 ◽  
Vol 437 ◽  
pp. 70-75
Author(s):  
Jun Jun Li ◽  
Xiao Qing Liu ◽  
Shi Zhu Yang
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Normal Form ◽  
Polynomial Solution ◽  
Adjoint Operator ◽  
Nonlinear Dynamical System ◽  
Solution Space ◽  
Partial Differential ◽  
Nonlinear Dynamical

An improved adjoint operator based on the adjoint operator concept of linear operator and S-N decomposition is proposed to calculate the normal forms of k order general nonlinear dynamic systems.Firstly, the whole polynomial solution space of homogeneous nilpotent partial differential equation are obtained.Secondly, the polynomial solution mentioned above is introduced into homogeneous semi-simple partial differential equation to find the whole polynomial solution space of a homogeneous linear partial differential equation Therefore, more polynomial first integrals need not be found and the simplest normal form of nonlinear dynamical system can be obtained easily. The example shows that the method is very effective.

Urn models and differential algebraic equations

2003 ◽  
Vol 40 (2) ◽  
pp. 401-412 ◽  
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Random Environment ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Urn Model ◽  
Recurrent Equation ◽  
Strong Laws ◽  
The Stability ◽  
Generalized Pólya Urn

The aim of this paper is to study the distribution of colours, {Xn}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {Xn} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {Xn}.

scholarly journals B-spline Wavelet Method for Solving Fredholm Hammerstein Integral Equation Arising from Chemical Reactor Theory

2018 ◽  
Vol 7 (3) ◽  
pp. 163-169 ◽  
Author(s):  
P. K. Sahu ◽  
A. K. Ranjan ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equation ◽  
Chemical Reactor ◽  
Steady State Solution ◽  
Algebraic Equations ◽  
Hammerstein Integral Equation ◽  
Wavelet Method ◽  
Spline Wavelet ◽  
B Spline

Abstract Mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.

A Simple Model for Prediction of Preheating and Pyrolysis Time of a Thermally Thin Charring Particle

2012 ◽  
Author(s):  
Y. Haseli ◽  
J. A. van Oijen ◽  
L. P. H. de Goey
Keyword(s):  
Differential Equation ◽  
Simple Model ◽  
Solid Particle ◽  
Time Integration ◽  
Quadratic Function ◽  
Conservation Equation ◽  
Virgin Material ◽  
Algebraic Equations ◽  
Energy Conservation Equation ◽  
Pyrolysis Model

The aim of this paper is to present a simple model, based on a time and space integral method, for prediction of preheating and conversion time of a charring solid particle exposed to a non-oxidative hot environment. The main assumptions are 1) thermo-physical properties remain constant throughout the process; 2) temperature profile within the particle is assumed to obey a quadratic function with respect to the space coordinate; 3) pyrolysis initiates when the surface temperature reaches a characteristic pyrolysis temperature; 4) decomposition of virgin material occurs at an infinitesimal thin layer dividing the particle into char and virgin material regions; 5) the volume of the particle remains unaltered; 6) volatiles escape through the pores immediately after formation. Employing assumption (2) allows one to convert the energy conservation equation of the particle, which is basically described in the form of a partial differential equation (PDE), into an ordinary differential equation (ODE) by performing space integration. Next, by applying approximate time integration the ODE is transformed into an algebraic equation. Applying this approach to the preheating and pyrolysis stages of a thermally thin charring solid particle leads to a set of algebraic equations which provides reactor designers with a convenient means for computation of the heating up time, mass loss history and total conversion of particle. The accuracy of the simple model is assessed by comparing its prediction with that of a one-dimensional detailed pyrolysis model. Overall, good agreement is achieved indicating that this new model can be used for engineering and design purposes.

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