Homotopy perturbation technique

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.

Variational iteration method – a kind of non-linear analytical technique: some examples

This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

Some asymptotic methods for strongly nonlinear equations

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Following your need to always fulfil the inspiration to obtain everybody is now simple. Connecting to the internet is one of the short cuts to do. There are so many sources that offer and connect us to other world condition. As one of the products to see in internet, this website becomes a very available place to look for countless perturbation methods sources. Yeah, sources about the books from countries in the world are provided.

Perturbation Methods

Algebraic Equations. Integrals. The Duffing Equation. The Linear Damped Oscillator. Self-Excited Oscillators. Systems with Quadratic and Cubic Nonlinearities. General Weakly Nonlinear Systems. Forced Oscillations of the Duffing Equation. Multifrequency Excitations. The Mathieu Equation. Boundary-Layer Problems. Linear Equations with Variable Coefficients. Differential Equations with a Large Parameter. Solvability Conditions. Appendices. Bibliography. Index.

/pdf/introduction-to-perturbation-techniques-2wwej84vkj.pdf

Introduction to perturbation techniques

Applied nonlinear dynamics

Perturbation Methods Dynamical Systems and Equilibrium Solutions Dynamic Solutions Tools to Characterize Different Motions Two-to-One Internal Resonances Combination Internal Resonances Three-to-One Internal Resonances Combination Internal Resonances Systems with Quadratic and Cubic Nonlinearities Gyroscopic Systems Systems with More than One Internal Resonance Random Excitations

Applied nonlinear dynamics : analytical, computational, and experimental methods

Preface. 1. Introduction. 1.1 Structural Elements. 1.2 Nonlinearities. 1.3 Composite Materials. 1.4 Damping. 1.5 Dynamic Characteristics of Linear Discrete Systems. 1.6 Dynamic Characteristics of Nonlinear Discrete Systems 1.7 Analyses of Linear Continuous Systems. 1.8 Analyses of Nonlinear Continuous Systems. 2. Elasticity. 2.1 Principles of Dynamics. 2.2 Strain--Displacement Relations. 2.3 Transformation of Strains and Stresses. 2.4 Stress--Strain Relations. 2.5 Governing Equations. 3. Strings and Cables. 3.1 Modeling of Taut Strings. 3.2 Reduction of String Model to Two Equations. 3.3 Nonlinear Response of Strings. 3.4 Modeling of Cables. 3.5 Reduction of Cable Model to Two Equations. 3.6 Natural Frequencies and Modes of Cables. 3.7 Discretization of the Cable Equations. 3.8 Single--Mode Response with Direct Approach. 3.9 Single--Mode Response with Discretization Approach. 3.10 Extensional Bars. 4. Beams. 4.1 Introduction. 4.2 Linear Euler--Bernoulli Beam Theory. 4.3 Linear Shear--Deformable Beam Theories. 4.4 Mathematics for Nonlinear Modeling. 4.5 Nonlinear 2--D Euler--Bernoulli Beam Theory. 4.6 Nonlinear 3--D Euler--Bernoulli Beam Theory. 4.7 Nonlinear 3--D Curved Beam Theory Accounting for Warpings. 5. Dynamics of Beams. 5.1 Parametrically Excited Cantilever Beams. 5.2 Transversely Excited Cantilever Beams. 5.3 Clamped--Clamped Buckled Beams. 5.4 Microbeams. 6. Surface Analysis. 6.1 Initial Curvatures. 6.2 Inplane Strains and Deformed Curvatures. 6.3 Orthogonal Virtual Rotations. 6.4 Variation of Curvatures. 6.5 Local Displacements and Jaumann Strains. 7. Plates. 7.1 Introduction. 7.2 Linear Classical Plate Theory. 7.3 Linear Shear--Deformable Plate Theories. 7.4 Nonlinear Classical Plate Theory. 7.5 Nonlinear Modeling of Rectangular Surfaces. 7.6 General Nonlinear Classical Plate Theory. 7.7 Nonlinear Shear--Deformable Plate Theory. 7.8 Nonlinear Layerwise Shear--Deformable Plate Theory. 8. Dynamics of Plates. 8.1 Linear Vibrations of Rectangular Plates. 8.2 Linear Vibrations of Membranes. 8.3 Linear Vibrations of Circular and Annular Plates. 8.4 Nonlinear Vibrations of Circular and Annular Plates. 8.5 Nonlinear Vibrations of Rotating Disks. 8.6 Nonlinear Vibrations of Near--Square Plates. 8.7 Micropumps. 8.8 Thermally Loaded Plates. 9. Shells. 9.1 Introduction. 9.2 Linear Classical Shell Theory. 9.3 Linear Shear--Deformable Shell Theories. 9.4 Nonlinear Classical Theory for Double--Curved Shells. 9.5 Nonlinear Shear--Deformable Theories for Circular Cylindrical Shells. 9.6 Nonlinear Layerwise Shear--Deformable Shell Theory. 9.7 Nonlinear Dynamics of Infinitely Long Circular Cylindrical Shells. 9.8 Nonlinear Dynamics of Axisymmetric Motion of Closed Spherical Shells. Bibliography. Subject Index.

/pdf/linear-and-nonlinear-structural-mechanics-320pjc1fuk.pdf

Ali H. Nayfeh

Papers

Perturbation Methods

Introduction to perturbation techniques

Applied nonlinear dynamics

Applied nonlinear dynamics : analytical, computational, and experimental methods

Linear and nonlinear structural mechanics