Applications of Fractional Lower Order Time-frequency Representation to Machine Bearing Fault Diagnosis

The machinery fault signal is a typical non-Gaussian and non-stationary process. The fault signal can be described by SαS distribution model because of the presence of impulses. Time-frequency distribution is a useful tool to extract helpful information of the machinery fault signal. Various fractional lower order (FLO) time-frequency distribution methods have been proposed based on fractional lower order statistics, which include fractional lower order short time Fourier transform (FLO-STFT), fractional lower order Wigner-Ville distributions (FLO-WVDs), fractional lower order Cohen class time-frequency distributions (FLO-CDs), fractional lower order adaptive kernel time-frequency distributions (FLO-AKDs) and adaptive fractional lower order time-frequency auto-regressive moving average (FLO-TFARMA) model time-frequency representation method. The methods and the exiting methods based on second order statistics in SαS distribution environments are compared, simulation results show that the new methods have better performances than the existing methods. The advantages and disadvantages of the improved time-frequency methods have been summarized. Last, the new methods are applied to analyze the outer race fault signals, the results illustrate their good performances.      

Note on derivation of order conditions for ARKN methods for perturbed oscillators

J.M. Franco derived the sufficient order conditions as well as the necessary and sufficient order conditions for his Adapted Runge-Kutta-Nyström methods (in short notation ARKN methods) based on the B-series theory [J.M. Franco, Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770-787]. Unfortunately, some fundamental mistakes have been made in the derivation of order conditions in that paper. In view of importance of the algebraic order theory for ARKN methods, a new and correct derivation of the order conditions for the ARKN methods is presented in this short note.     

A geometric construction of iterative functions of order three to solve nonlinear equations

In this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations. This construction can be applied to any iteration function of order two to develop an iteration function of order three. Some examples are given of deriving several third-order iteration methods, and several numerical results follow to illustrate the performance of the derived methods.     

Initial value problems: numerical methods and mathematics

A number of questions and results concerning Runge-Kutta and general linear methods are surveyed. These include order conditions and order bounds for Runge-Kutta methods, the A-stability of implicit Runge-Kutta methods based on Gaussian quadrature and transformation methods of implementation which lead to singly-implicit methods. The sections dealing with general linear methods include a discussion of the order conditions and an algebraic structure for carrying out order analyses as well as an introduction to a special function associated with parallel methods for stiff problems.     

On implicit Runge–Kutta methods obtained as a result of the inversion of explicit methods

We consider methods that are the inverse of the explicit Runge–Kutta methods. Such methods have some advantages, while their disadvantage is the low (first) stage order. This reduces the accuracy and the real order in solving stiff and differential-algebraic equations. New methods possessing properties of methods of a higher stage order are proposed. The results of the numerical experiments show that the proposed methods allow us to avoid reducing the order.     

A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.     

Higher order operator splitting methods via Zassenhaus product formula: Theory and applications

In this paper, we contribute higher order operator splitting methods improved by Zassenhaus product. We apply the contribution to classical and iterative splitting methods. The underlying analysis to obtain higher order operator splitting methods is presented. While applying the methods to partial differential equations, the benefits of balancing time and spatial scales are discussed to accelerate the methods.The verification of the improved splitting methods are done with numerical examples. An individual handling of each operator with adapted standard higher order time-integrators is discussed. Finally, we conclude the higher order operator splitting methods.     

Finite difference methods of order two and four for 2-d non-linear biharmonic problems of first kind

For the numerical integration of the 2-D nonlinear biharmonic problems of first kind, we report two difference methods of order two and four over a rectangular domain. These methods use only the nine grid points and do not require fictitious points in order to approximate the boundary conditions. Derivatives of the solution are obtained as a by-product of the methods. In numerical experiments, the new second and fourth order formulas are compared with the exact solutions.     

A family of Numerov-type exponentially fitted methods for the numerical integration of the Schrödinger equation

A family of predictor-corrector exponential Numerov-type methods is developed for the numerical integration of the one-dimensional Schrödinger equation. The Numerov-type methods considered contain free parameters which allow it to be fitted to exponential functions. The new fourth algebraic order methods are very simple and integrate more exponential functions than both the well-known fourth order Numerov-type exponentially fitted methods and the sixth algebraic order Runge-Kutta-type methods. Numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above.     

A family of simultaneous zero finding methods

A class of iterative methods with arbitrary high order of convergence for the simultaneous approximation of multiple complex zeros is considered in this paper. A special attention is paid to the fourth order method and its modifications because of their good computational efficiency. The order of convergence of the presented methods is determined. Numerical examples are given.     

Stability of nonequidistant variable order multistep methods for stiff systems

The order, stability and convergence of the nonequidistant variable order multistep methods (NVOMMs) for stiff systems are discussed. The stability properties of certain class of variable step multistep methods (VSMMs) depending on one free parameter β^* for some order will be discussed, some theorems and lemmas are proved. New effective techniques restricted by larger interval of β^* to get strongly stable methods are determined     

High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.     

One-step explicit methods for the numerical integration of perturbed oscillators

In this paper, we present a class of one-step explicit zero-dissipative nonlinear methods for the numerical integration of perturbed oscillators, which have second algebraic order and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.     

Partitioned and Implicit–Explicit General Linear Methods for Ordinary Differential Equations

Implicit–explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiff components. IMEX Runge–Kutta methods and IMEX linear multistep methods have been studied in the literature. In this paper we study new implicit–explicit methods of general linear type. We develop an order conditions theory for high stage order partitioned general linear methods (GLMs) that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order up to three. Numerical results confirm the theoretical findings.     

Multiderivative extended Runge–Kutta–Nyström methods for multi-frequency oscillatory systems

ABSTRACT

In this paper, a class of multistage and multivalue numerical methods called multiderivative extended Runge–Kutta–Nyström (MERKN) methods are presented for the integration of multi-frequency oscillatory systems. The order conditions of these schemes are derived by using the theory of extended Nyström trees and B-series. To simplify the construction of the new methods, the order conditions are simplified by omitting the redundant trees. Based on the simplified-order conditions, a family of three-stage and one-parameter explicit MERKN methods of order five is constructed. Meanwhile, the order of energy preservation, the stability and the phase properties of the new methods are analysed, and a three-stage method with minimal dispersion error is obtained. The results of numerical experiments demonstrate the efficiency of our new methods in comparison with some other RKN-type methods.     

A generator of dissipative methods for the numerical solution of the Schrödinger equation

In this paper a generator of hybrid methods with minimal phase-lag is developed for the numerical solution of the Schrödinger equation and related problems. The generator's methods are dissipative and are of eighth algebraic order. In order to have minimal phase-lag with the new methods, their coefficients are determined automatically. Numerical results obtained by their application to some well known problems with periodic or oscillating solutions and to the coupled differential equations of the Schrödinger type indicate the efficiency of these new methods.     

Splitting methods for fourth order parabolic partial differential equations

In this paper, several splitting methods are discussed which can be used to solve fourth order parabolic partial differential equations that are given in some suitable first order system form. The methods are generalisations of splitting methods for (second order) parabolic PDE's. For all methods which are considered, stability or instability is studied for problems in 2 and in 3 or more spatial dimensions.     

On high order strong stability preserving runge-kutta and multi step time discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.     

A comparative evaluation of nonlinear dynamics methods for time series prediction

A key problem in time series prediction using autoregressive models is to fix the model order, namely the number of past samples required to model the time series adequately. The estimation of the model order using cross-validation may be a long process. In this paper, we investigate alternative methods to cross-validation, based on nonlinear dynamics methods, namely Grassberger–Procaccia, Kégl, Levina–Bickel and False Nearest Neighbors algorithms. The experiments have been performed in two different ways. In the first case, the model order has been used to carry out the prediction, performed by a SVM for regression on three real data time series showing that nonlinear dynamics methods have performances very close to the cross-validation ones. In the second case, we have tested the accuracy of nonlinear dynamics methods in predicting the known model order of synthetic time series. In this case, most of the methods have yielded a correct estimate and when the estimate was not correct, the value was very close to the real one.     

Efficient Jarratt-like methods for solving systems of nonlinear equations

We present the iterative methods of fourth and sixth order convergence for solving systems of nonlinear equations. Fourth order method is composed of two Jarratt-like steps and requires the evaluations of one function, two first derivatives and one matrix inversion in each iteration. Sixth order method is the composition of three Jarratt-like steps of which the first two steps are that of the proposed fourth order scheme and requires one extra function evaluation in addition to the evaluations of fourth order method. Computational efficiency in its general form is discussed. A comparison between the efficiencies of proposed techniques with existing methods of similar nature is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in the examples. It is shown that the present methods are more efficient than their existing counterparts, particularly when applied to the large systems of equations.