A tau method for nonlinear dynamical systems

In this work we present a new Tau method for the solution of nonlinear systems of differential equations which are linear in the derivative of highest order and polynomial in the remaining. We avoid the linearization of the problem by associating to it a nonlinear algebraic system and combine a forward substitution with the Tau method. We develop an adaptive step by step version of this alternative nonlinear tau method and we apply it to several nonlinear dynamical systems.     

Computation of the canonical polynomials and applications to some optimal control problems

The Tau method is a numerical technique that consists in constructing polynomial approximate solutions for ordinary differential equations. This method has two approaches: operational and recursive. The former converts the differential problem to a matrix problem and produces approximations in terms of a prescribed orthogonal polynomials basis. In the recursive approach, we construct approximate solutions in terms of a special set of polynomials {Q _k (t); k?=?0, 1, 2...} called canonical polynomials basis. In some cases, the Q _k ??s can be obtained explicitly through a recursive formula. But no analogous formulae are reported in the literature for the general cases. In this paper, utilizing the operational Tau method, we develop an algorithm that allows to generate those canonical polynomials iteratively and explicitly. In addition, we demonstrate the capability of the operational Tau method in treating quadratic optimal control problems governed by ordinary differential equations.     

Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method

A general form of numerical piecewise approximate solution of linear integro-differential equations of Fredholm type is discussed. It is formulated for using the operational Tau method to convert the differential part of a given integro-differential equation, or IDE for short, to its matrix representation. This formulation of the Tau method can be useful for such problems over long intervals and also can be used as a good and simple alternative algorithm for other piecewise approximations such as splines or collocation. A Tau error estimator is also adapted for piecewise application of the Tau method. Some numerical examples are considered to demonstrate the implementation and general effect of application of this (segmented) piecewise Chebyshev Tau method.     

An error analysis of the tau method for a class of singularly perturbed problems for differential equations

By using classical results of Poincaré and Birkhoff we discuss the existence and uniqueness of solution for a class of singularly perturbed problems for differential equations. The Tau method formulation of Ortiz [6] is applied to the construction of approximate solutions of these problems. Sharp error bounds are deduced. These error bounds are applied to the discussions of a model problem, a simple one-dimensional analogue of Navier-Stokes equation, which has been considered recently by several authors (see [2], [3], [8], [10]). Numerical results for this problem [8] show that the Tau method leads to more accurate approximations than specially designed finite difference or finite element schemes.     

Incremental self-consistent approach for the estimation of nonlinear material behavior of metal matrix composites

The application of homogenization methods to compute the macroscopic material response of metal matrix composites is a possibility to save memory and computation time in comparison to full field simulations. This paper deals with a method to extend the self-consistent scheme from linear elasticity theory to nonlinear problems. The idea is to approximate the nonlinear problem by an incrementally linear one. Since time discretization of the deformation process implies a certain linearization, we use the algorithmic consistent tangent operator of the composite for defining the linear comparison material in each time step. This is in contrast to classical incremental self-consistent approaches which use continuum tangent or secant operators. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The L<Superscript>2</Superscript>-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations

The operational Tau method, a well known method for solving functional equations is employed to approximate the solution of nonlinear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. The unique solvability of the linear Tau algebraic system is discussed. In addition, we provide a rigorous convergence analysis for the Legendre Tau method which indicate that the proposed method converges exponentially provided that the data in the given FIDE are smooth. To do so, Sobolev inequality with some properties of Banach algebras are considered. Some numerical results are given to clarify the efficiency of the method.     

Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations

In this work, we propose an extension of the algebraic formulation of the Tau method for the numerical solution of the nonlinear Volterra-Hammerstein integral equations. This extension is based on the operational Tau method with arbitrary polynomial basis functions for constructing the algebraic equivalent representation of the problem. This representation is an special semi lower triangular system whose solution gives the components of the vector solution. We will show that the operational Tau matrix representation for the integration of the nonlinear function can be represented by an upper triangular Toeplitz matrix. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results. Our numerical experiments show that the accuracy of the Tau approximate solution is independent of the selection of the basis functions.     

An alternative linearization approach applicable to hysteretic systems

In this paper a method is proposed for equivalent linearization of nonlinear restoring forces being governed by differential equations in weakly nonlinear systems. These types of restoring forces cannot be linearized by employing conventional approximate approaches. Two analytical examples are used to show the accuracy of the proposed method. The application of the method to hysteretic systems is examined by constructing equivalent linear representation for Bouc–Wen model in its general formulation. Numerical investigations reveal that the proposed method is efficient in dynamic behavior analysis of weakly nonlinear hysteretic systems.     

Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems

Many global optimization approaches for solving signomial geometric programming problems are based on transformation techniques and piecewise linear approximations of the inverse transformations. Since using numerous break points in the linearization process leads to a significant increase in the computational burden for solving the reformulated problem, this study integrates the range reduction techniques in a global optimization algorithm for signomial geometric programming to improve computational efficiency. In the proposed algorithm, the non-convex geometric programming problem is first converted into a convex mixed-integer nonlinear programming problem by convexification and piecewise linearization techniques. Then, an optimization-based approach is used to reduce the range of each variable. Tightening variable bounds iteratively allows the proposed method to reach an approximate solution within an acceptable error by using fewer break points in the linearization process, therefore decreasing the required CPU time. Several numerical experiments are presented to demonstrate the advantages of the proposed method in terms of both computational efficiency and solution quality.     

Matrix displacement mappings in the numerical solution of functional and nonlinear differential equations with the tau method

The use of matrix displacement mappings reduces most matrix operations required in the construction of an approximate solution of a functional or differential equation by means of Ortiz' formulation of the Tau method to index shifts. The coefficient vector of the approximate solution is defined implicitly by a very sparse system of linear algebraic equations. The contributions of the differential or functional operator, and of the supplementary conditions of the problem (initial, boundary, or multipoint conditions) are treated with a single and versatile procedure of remarkable simplicity, which can be easily implemented in a computer. We give two nontrivial examples on the application of this approach: the first is a nonlinear boundary value problem with a continuous locus of singular points and multiple solutions, where stiffness is present, the second is a functional differential equation arising in analytic number theory. In both cases we obtain results of nigh accuracy.     

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.     

非线性回归方法的应用与比较

比较了非线性回归3种方法的数学原理:曲线直线化方法、非线性最小二乘方法、近似非线性法.说明了用方差分析确定回归模型的统计学意义、用决定系数R2描述曲线的拟合效果的理论依据.通过对同一问题用3种方法分析得出结论:非线性回归与近似非线性拟合方法决定系数相近(0.9966与0.9965),而曲线直线化决定系数为0.9738.因为近似非线性拟合方法无需选初值.建议应用近似非线性拟合方法.     

Direct shooting method,linearization, and nonlinear algebraic equations

The direct shooting method for the solution of boundary-value problems for ordinary, nonlinear differential equations is analyzed from the point of view of linearization. Some relations between this method and perturbation methods are established. The relations between the direct shooting algorithm and Whittaker's algorithm are also established, considering the problem from the point of view of solving nonlinear algebraic equations.     

A power method for nonlinear operators

In this article we consider the problem of computing the dominant eigenvalue of the linearization of a nonlinear operator. We define a power method that converges under natural conditions on the nonlinear operator. This nonlinear power method does not require the linearization itself, but only the action of the nonlinear operator on arbitrary functions. We apply this method to investigate the stability of equilibrium solutions of differential equations.     

The alternative Legendre Tau method for solving nonlinear multi-order fractional differential equations

In this paper, the alternative Legendre polynomials (ALPs) are used to approximate the solution of a class of nonlinear multi-order fractional differential equations (FDEs). First, the operational matrix of fractional integration of an arbitrary order and the product operational matrix are derived for ALPs. These matrices together with the spectral Tau method are then utilized to reduce the solution of the mentioned equations into the one of solving a system of nonlinear algebraic equations with unknown ALP coefficients of the exact solution. The fractional derivatives are considered in the Caputo sense and the fractional integration is described in the Riemann-Liouville sense. Numerical examples illustrate that the present method is very effective for linear and nonlinear multi-order FDEs and high accuracy solutions can be obtained only using a small number of ALPs.     

Higher order B-spline differential quadrature rule to approximate generalized Rosenau-RLW equation

In this article, B-spline-based collocation method is employed to approximate the usual and modified Rosenau-RLW nonlinear equations. The weighted extended B-spline (WEB-spline) is used as the modified form of B-spline as the usual B-splines fail to obey the Dirichlet boundary conditions. The WEB method is more general method that allows to discretize the domain into finite number of elements not necessarily start from the boundary points of the domain. Our method omits the linearization process of the nonlinear partial differential equation (PDE). Different cases are discussed by setting the parameter p=2,3,4, and 6 that appears in Rosenau-RLW equations. The error estimation is calculated, which gives good agreement of the exact solution.     

A note on convergence rate of a linearization method for the discretization of stochastic differential equations

This note discusses convergence rate of a linearization method for the discretization of stochastic differential equations with multiplicative noise. The method is to approximate the drift coefficient by the local linearization method and the diffusion coefficient by the Euler method. The mixed method guarantees the approximated process converges to the original one with the rate of convergence Δt, where Δt is the time interval of discretization.     

Functional differential equations

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. In this paper we apply this technique to functional differential problems. It is shown that linear iterations converge to the unique solution and this convergence is superlinear.     

Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

Perturbation theory for nonlinear feedback control systems and spencer-goldschmidt integrability of linear partial differential equations

This paper aims to develop the differential-geometric and Lie-theoretic foundations of perturbation theory for control systems, extending the classical methods of Poincaré from the differential equation-dynamical system level where they are traditionally considered, to the situation where the element of control is added. It will be guided by general geometric principles of the theory of differential systems, seeking approximate solutions of the feedback linearization equations for nonlinear affine control systems. In this study, certain algebraic problems of compatibility of prolonged differential systems are encountered. The methods developed by D. C. Spencer and H. Goldschmidt for studying over-determined systems of partial differential equations are needed. Work in the direction of applying theio theory is presented.Supported by grants from the Ames Research Center of NASA and the Applied Mathematics and Systems Research Programs of the National Science Foundation