Study on convergence and error of a numerical method for solving systems of nonlinear Fredholm–Volterra integral equations of Hammerstein type

The aim of this paper is to present an efficient analytical and numerical procedure for solving systems of nonlinear Fredholm–Volterra integral equations of the Hammerstein type with the aid of fixed point techniques and the usual Schauder basis in an adequate Banach space.     

A Remark on Adaptive Decomp osition for Nonlinear Time-frequency Analysis

In recent study the bank of real square integrable functions that have nonlinear phases and admit a well-behaved Hilbert transform has been constructed for adaptive representation of nonlinear signals. We first show in this paper that the available basic functions are adequate for establishing an ideal adaptive decomposi-tion algorithm. However, we also point out that the best approximation algorithm, which is a common strategy in decomposing a function into a sum of functions in a prescribed class of basis functions, should not be considered as a candidate for the ideal algorithm.     

Chaotic dynamic and control for micro-electro-mechanical systems of massive storage with harmonic base excitation

This paper explores chaotic behaviour and control of micro-electro-mechanical systems (MEMS), which consist of thousands of small read/write probe tips that access gigabytes of data stored in a non-volatile magnetic surface. The model of the system is formed by two masses connected by a nonlinear spring and a viscous damping. The paper shows that, by means of an adequate feedback law, the masses can behave as two coupled Duffing’s oscillators, which may reach chaotic behaviour when harmonic forces are applied. The chaotic motion is destroyed by applying the following control strategies: (i) static output feedback control law with constant forces and (ii) geometric nonlinear control. The aim is to drive the masses to a set point even with harmonic base excitation, by using chaotic dynamics and nonlinear control. The paper shows that it is possible to obtain a positioning time around a few ms with sub-nanometre accuracy, velocities, accelerations and forces, as it appears in the design of present MEMS devices. Numerical simulations are used to verify the mathematical discussions.     

Higher Order Dissipative-Dispersive System and Application

In this paper, a generalized nonlinear dissipative and dispersive equation with time and space-dependent coefficients is considered. We show that the control of the higher order term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method. As an application to this general Theorem, we prove the well-posedness of the Camassa-Holm type equation.     

解非线性规划问题的不精确线性搜索SQP滤子方法

本文用序列二次规划方法(SQP)结合Wolfe-Powell不精确线性搜索准则求解非线性规划问题.Wolfe-Powell准则是一种能够使目标函数获得充分下降而运行时间较省的确定步长方法.不精确线性搜索滤子方法比较其它结合精确线性搜索和信赖域方法求解问题的滤子方法更灵活更易实现.如果目标函数的预测下降量为负,我们的工作将主要利用可行恢复项改善可行性.一般条件下,本文提出的算法较易实现,且具有全局收敛性.数值试验显示了算法的有效性.     

Local well-posedness of a nonlinear KdV-type equation

In this paper, a generalized nonlinear KdV equation with time- and space-dependent coefficients is considered. We show that the control of the dispersive and “diffusion” terms is possible if we use an adequate weight function determined with respect to the dispersive and “diffusion” coefficients to define the energy. We use the dispersive properties of the equation to prove the existence and uniqueness of solutions.     

TRESNEI, a Matlab trust-region solver for systems of?nonlinear equalities and inequalities

The Matlab implementation of a trust-region Gauss-Newton method for bound-constrained nonlinear least-squares problems is presented. The solver, called TRESNEI, is adequate for zero and small-residual problems and handles the solution of nonlinear systems of equalities and inequalities. The structure and the usage of the solver are described and an extensive numerical comparison with functions from the Matlab Optimization Toolbox is carried out.     

Space-Time Transfer Function-Noise Modelling of Rainfall-Runoff Process

An explanatory model class belonging to the family of Space-Time Transfer Function-Noise (STTFN) process is presented. The paper develops a three-stage iterative procedure for building STTFN models of the rainfall-runoff process. Four precipitation and runoff stations located in a watershed in southern Ontario, Canada, sampled at 15-day intervals are used for the numerical analysis. Three STTF models are identified. The model parameters are estimated by the polytope technique, a nonlinear optimization algorithm. Two of the developed space-time models proved adequate in describing the spatio-temporal characteristics of precipitation and runoff time series.     

Nonparametric prediction of stock returns based on yearly data: The long-term view

One of the most studied questions in economics and finance is whether empirical models can be used to predict equity returns or premiums. In this paper, we take the actuarial long-term view and base our prediction on yearly data from 1872 through 2014. While many authors favor the historical mean or other parametric methods, this article focuses on nonlinear relationships between a set of covariates. A bootstrap test on the true functional form of the conditional expected returns confirms that yearly returns on the S&P500 are predictable. The inclusion of prior knowledge in our nonlinear model shows notable improvement in the prediction of excess stock returns compared to a fully nonparametric model. Statistically, a bias and dimension reduction method is proposed to import more structure in the estimation process as an adequate way to circumvent the curse of dimensionality.     

High-order nonlinear initial-value problems countably determined

We give a numerical approximation of the solution of a high-order nonlinear initial-value problem by making use of certain properties of an adequate Schauder basis.     

On the numerical performance of a sharp a posteriori error estimator for some nonlinear elliptic problems

Karátson and Korotov developed a sharp upper global a posteriori error estimator for a large class of nonlinear problems of elliptic type, see J. Karátson, S. Korotov (2009). The goal of this paper is to check its numerical performance, and to demonstrate the efficiency and accuracy of this estimator on the base of quasilinear elliptic equations of the second order. The focus will be on the technical and numerical aspects and on the components of the error estimation, especially on the adequate solution of the involved auxiliary problem.     

Homoclinic solutions for nonlinear general fourth‐order differential equations

This work provides sufficient conditions for the existence of homoclinic solutions of fourth‐order nonlinear ordinary differential equations. Using Green's functions, we formulate a new modified integral equation that is equivalent to the original nonlinear equation. In an adequate function space, the corresponding nonlinear integral operator is compact, and it is proved an existence result by Schauder's fixed point theorem. Copyright © 2017 John Wiley & Sons, Ltd.     

Local and global dynamics in a duopoly with price competition and market share delegation

This paper aims at studying a nonlinear dynamic duopoly model with price competition and horizontal product differentiation augmented with managerial firms, where managers behave according to market share delegation contracts. Ownership and management are then separate and managers are paid through adequate incentives in order to achieve a competitive advantage in the market. In this context, we show that complexity arises, related both to the structure of the attractors of the system and the structure of their basins, as multistability occurs. The study is conducted by combining analytical and numerical techniques, and aims at showing that slight different initial conditions may cause very different long-term outcomes.     

The design of branch and bound algorithms for a class of nonlinear integer programs

This paper is concerned with computational experimentation leading to the design of effective branch and bound algorithms for an important class of nonlinear integer programming problems, namely linearly constrained problems, which are used to model several real-world situations. The main contribution here is a study of the effect of node and branching variable selection and storage reduction strategies on overall computational effort for this class of problems, as well as the generation of a set of adequate test problems. Several node and branching variable strategies are compared in the context of a pure breadth-first enumeration, as well as in a special breadth and depth enumeration combination approach presented herein. Also, the effect of using updated pseudocosts is briefly addressed. Computational experience is presented on a set of eighteen suitably-sized nonlinear test problems, as well as on some random linear integer programs. Some of the new rules proposed are demonstrated to be significantly superior to previously suggested strategies; interestingly, even for linear integer programming problems.     

New Theoretical Results on Recursive Quadratic Programming Algorithms

Recursive quadratic programming is a family of techniques developed by Bartholomew-Biggs and other authors for solving nonlinear programming problems. The first-order optimality conditions for a local minimizer of the augmented Lagrangian are transformed into a nonlinear system where both primal and dual variables appear explicitly. The inner iteration of the algorithm is a Newton-like procedure that updates simultaneously primal variables and Lagrange multipliers. In this way, as observed by Gould, the implementation of the Newton method becomes stable, in spite of the possibility of having large penalization parameters. In this paper, the inner iteration is analyzed from a different point of view. Namely, the size of the convergence region and the speed of convergence of the inner process are considered and it is shown that, in some sense, both are independent of the penalization parameter when an adequate version of the Newton method is used. In other words, classical Newton-like iterations are improved, not only in relation to stability of the linear algebra involved, but also with regard to the ovearll convergence of the nonlinear process. Some numerical experiments suggset that, in fact, practical efficiency of the methods is related to these theoretical results.     

EXISTENCE OF SOLUTIONS OF nTH-ORDER NONLINEAR DIFFERENCE EQUATIONS WITH GENERAL BOUNDARY CONDITIONS

The aim of this paper is to prove the existence of one or multiple solutions of nonlinear difference equations coupled to a general set of boundary conditions. Before to do this, we construct a discrete operator whose fixed points coincide with the solutions of the problem we are looking for. Moreover, we introduce a strong positiveness condition on the related Green's function that allows us to construct suitable cones where to apply adequate fixed point theorems. Once we have the general existence result, we deduce, as a particular case, the existence of solutions of a second order difference equation with nonlocal perturbed Dirichlet conditions.     

Biorthogonal systems for solving Volterra integral equation systems of the second kind

With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.     

External Estimates for Reachable Sets of a Control System with Uncertainty and Combined Nonlinearity

The problem of estimating the trajectory tubes of a nonlinear control dynamic system with uncertainty in the initial data is studied. It is assumed that the dynamic system has a special structure in which the nonlinear terms are defined by quadratic forms in the state coordinates and the values of uncertain initial states and admissible controls are subject to ellipsoidal constraints. The matrix of the linear terms in the velocities of the system is not known exactly; it belongs to a given compact set in the corresponding space. Thus, the dynamics of the system is complicated by the presence of bilinear components in the righthand sides of the differential equations of the system. We consider a complicated case and generalize the author’s earlier results. More exactly, we assume the simultaneous presence in the dynamics of the system of bilinear functions and quadratic forms (without the assumption of their positive definiteness) and we also take into account the uncertainty in the initial data and the impact of the control actions, which may also be treated here as undefined additive disturbances. The presence of all these factors greatly complicates the study of the problem and requires an adequate analysis, which constitutes the main purpose of this study. The paper presents algorithms for estimating the reachable sets of a nonlinear control system of this type. The results are illustrated by examples.     

Stable approximation of solutions to irregular nonlinear operator equations in a Hilbert space under large noise

The class of regularized Gauss-Newton methods for solving inexactly specified irregular nonlinear equations is examined under the condition that additive perturbations of the operator in the problem are close to zero only in the weak topology. By analogy with the well-understood conventional situation where the perturbed and exact operators are close in norm, a stopping criterion is constructed ensuring that the approximate solution is adequate to the errors in the operator.     

Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses

In this paper, the chaotic behavior of a simplest autonomous memristor-based circuit of fractional order is suppressed by periodic impulses applied to one or several state variables. The circuit consists of two passive linear elements, a capacitor and an inductor, as well as a nonlinear memristive element. It is shown that by applying a sequence of adequate (identical or different) periodic impulses to one or several variables, the chaotic behavior can be suppressed. Impulse values and control timing are determined numerically, based on the bifurcation diagram with impulses as bifurcation parameters. Empirically, the probability to have a reasonably wide range of impulses to suppress chaos is quite large, ensuring that chaos suppression can be implemented, as demonstrated by several examples presented.