Showing papers on "Nonlinear system published in 2002"

Finite Volume Methods for Hyperbolic Problems

Abstract: Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

Nonlinear Dynamics and Chaos

Abstract: Preface. Preface to the First Edition. Acknowledgements from the First Edition. Introduction PART I: BASIC CONCEPTS OF NONLINEAR DYNAMICS An overview of nonlinear phenomena Point attractors in autonomous systems Limit cycles in autonomous systems Periodic attractors in driven oscillators Chaotic attractors in forced oscillators Stability and bifurcations of equilibria and cycles PART II ITERATED MAPS AS DYNAMICAL SYSTEMS Stability and bifurcation of maps Chaotic behaviour of one--and two--dimensional maps PART III FLOWS, OUTSTRUCTURES AND CHAOS The Geometry of Recurrence The Lorenz system Rosslers band Geometry of bifurcations PART IV APPLICATIONS IN THE PHYSICAL SCIENCES Subharmonic resonances of an offshore structure Chaotic motions of an impacting system Escape from a potential well Appendix. Illustrated Glossary. Bibliography. Online Resource. Index.

Stable clustering, the halo model and nonlinear cosmological power spectra

Abstract: We present the results of a large library of cosmological N-body simulations, using power-law initial spectra. The nonlinear evolution of the matter power spectra is compared with the predictions of existing analytic scaling formulae based on the work of Hamilton et al. The scaling approach has assumed that highly nonlinear structures obey `stable clustering' and are frozen in proper coordinates. Our results show that, when transformed under the self-similarity scaling, the scale-free spectra define a nonlinear locus that is clearly shallower than would be required under stable clustering. Furthermore, the small-scale nonlinear power increases as both the power-spectrum index n and the density parameter Omega decrease, and this evolution is not well accounted for by the previous scaling formulae. This breakdown of stable clustering can be understood as resulting from the modification of dark-matter haloes by continuing mergers. These effects are naturally included in the analytic `halo model' for nonlinear structure; using this approach we are able to fit both our scale-free results and also our previous CDM data. This approach is more accurate than the commonly-used Peacock--Dodds formula and should be applicable to more general power spectra. Code to evaluate nonlinear power spectra using this method is available from this http URL Following publication, we will make the power-law simulation data available through the Virgo website this http URL

The world of the complex Ginzburg-Landau equation

Abstract: The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory. The authors give an overview of various phenomena described by the complex Ginzburg-Landau equation in one, two, and three dimensions from the point of view of condensed-matter physicists. Their aim is to study the relevant solutions in order to gain insight into nonequilibrium phenomena in spatially extended systems.

Handbook of Exact Solutions for Ordinary Differential Equations

Abstract: Annotation Foreward Some Remarks and Notation First Order Differential Equations Simplest Equations with Arbitrary Functions Integrable in a Closed Form Riccati Equations: g(y)y'x = f2(x)y2 + f1(x)y + f0(x) Abel Equations of the Second Kind Equations Containing Polynomial Functions of y Nonlinear Equations of the Form f(x,y)y'x = g(x,y) Containing Arbitrary Parameters Equations Not Solved for Derivative Equations of the Form F(x,y)y'x = G(x,y) Containing Arbitrary Functions Equations of the Form F(x,y,y'x) = 0 Not Solved for the Derivative and Containing Arbitrary Functions Second Order Differential Equations Linear Equations Autonomous Equations y"xx = F(y,y'x) Emden-Fowler Equation y"xx = Axnym Equations of the Form y"xx = A1xn1ym1 + A2xn2ym2 Generalized Emden-Fowler Equation y"xx = Axnym(y'x)l Equations of the Form y"xx = A1xn1ym1(y'x)l1 + A2xn2ym2(y'x)l2 Equations of the Form y"xx = f(x)g(y)h(y'x) Some Nonlinear Equations with Arbitrary Parameters Equations Containing Arbitrary Functions Third Order Differential Equations Linear Equations Equations of the Form y'"xxx = Axayss(y'x)g(y"xx)d Equations of the Form y'"xxx = f(y)g(y'x)h(y"xx) Some Nonlinear Equations with Arbitrary Parameters Nonlinear Equations Containing Arbitrary Functions Fourth Order Differential Equations Linear Equations Nonlinear Equations Higher Order Differential Equations Linear Equations Nonlinear Equations Supplement 1. Some Elementary Functions and Their Properties Trigonometric Functions Hyperbolic Functions Inverse Trigonometric Functions Inverse Hyperbolic Functions Some Conventional Symbols Supplement 2. Some Special Functions Gamma-Function Bessel Functions Jn and Yn Modified Bessel Functions In and Kn Degenerate Hypergeometric Functions Legendre Functions The Weierstrass Function References Index

Sliding mode control on electro-mechanical systems

Abstract: The first sliding mode control application may be found in the papers back in the 1930s in Russia. With its versatile yet simple design procedure the methodology is proven to be one of the most powerful solutions for many practical control designs. For the sake of demonstration this paper is oriented towards application aspects of sliding mode control methodology. First the design approach based on the regularization is generalized for mechanical systems. It is shown that stability of zero dynamics should be taken into account when the regular form consists of blocks of second-order equations. Majority of applications in the paper are related to control and estimation methods of automotive industry. New theoretical methods are developed in the context of these studies: sliding made nonlinear observers, observers with binary measurements, parameter estimation in systems with sliding mode control.

Balanced Model Reduction via the Proper Orthogonal Decomposition

Abstract: A new method for performing a balanced reduction of a high-order linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshotsisused to obtainlow-rank,reduced-rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain.Theapproximationsarethenusedtoobtainabalancedreducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized high-order system that models unsteady motion of a two-dimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.

Dynamics of Evolutionary Equations

Abstract: Preface * 1 The Evolution of Evolutionary Systems * 2 Dynamical Systems: Basic Theory * 3 Linear Semigroups * 4 Basic Theory of Evolutionary Equations * 5 Nonlinear Partial Differential Equations * 6 Navier Stokes Dynamics * 7 Basic Principles of Dynamics * 8 Inertial Manifolds and the Reduction Principle * Appendices: Basics of Functional Analysis * Bibliography * Notation Index * Subject Index

Time-Splitting Methods for Elastic Models Using Forward Time Schemes

Abstract: Two time-splitting methods for integrating the elastic equations are presented. The methods are based on a third-order Runge‐Kutta time scheme and the Crowley advection schemes. The schemes are combined with a forward‐backward scheme for integrating high-frequency acoustic and gravity modes to create stable splitexplicit schemes for integrating the compressible Navier‐Stokes equations. The time-split methods facilitate the use of both centered and upwind-biased discretizations for the advection terms, allow for larger time steps, and produce more accurate solutions than existing approaches. The time-split Crowley scheme illustrates a methodology for combining any pure forward-in-time advection schemes with an explicit time-splitting method. Based on both linear and nonlinear tests, the third-order Runge‐Kutta-based time-splitting scheme appears to offer the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models.

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

Abstract: This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

Linear and nonlinear structural mechanics

Abstract: 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.

Geophysical Inverse Theory and Regularization Problems

Abstract: Preface. I. Introduction to Inversion Theory. 1. Forward and inverse problems in geophysics. 1.1 Formulation of forward and inverse problems for different geophysical fields. 1.2 Existence and uniqueness of the inverse problem solutions. 1.3 Instability of the inverse problem solution. 2. Ill-posed problems and the methods of their solution. 2.1 Sensitivity and resolution of geophysical methods. 2.2 Formulation of well-posed and ill-posed problems. 2.3 Foundations of regularization methods of inverse problem solution. 2.4 Family of stabilizing functionals. 2.5 Definition of the regularization parameter. II. Methods of the Solution of Inverse Problems. 3. Linear discrete inverse problems. 3.1 Linear least-squares inversion. 3.2 Solution of the purely under determined problem. 3.3 Weighted least-squares method. 3.4 Applying the principles of probability theory to a linear inverse problem. 3.5 Regularization methods. 3.6 The Backus-Gilbert method. 4. Iterative solutions of the linear inverse problem. 4.1 Linear operator equations and their solution by iterative methods. 4.2 A generalized minimal residual method. 4.3 The regularization method in a linear inverse problem solution. 5. Nonlinear inversion technique. 5.1 Gradient-type methods. 5.2 Regularized gradient-type methods in the solution of nonlinear inverse problems. 5.3 Regularized solution of a nonlinear discrete inverse problem. 5.4 Conjugate gradient re-weighted optimization. III. Geopotential Field Inversion. 6. Integral representations in forward modeling of gravity and magnetic fields. 6.1 Basic equations for gravity and magnetic fields. 6.2 Integral representations of potential fields based on the theory of functions of a complex variable. 7. Integral representations in inversion of gravity and magnetic data. 7.1 Gradient methods of gravity inversion. 7.2 Gravity field migration. 7.3 Gradient methods of magnetic anomaly inversion. 7.4 Numerical methods in forward and inverse modeling. IV. Electromagnetic Inversion. 8. Foundations of electromagnetic theory. 8.1 Electromagnetic field equations. 8.2 Electromagnetic energy flow. 8.3 Uniqueness of the solution of electromagnetic field equations. 8.4 Electromagnetic Green's tensors. 9. Integral representations in electromagnetic forward modeling. 9.1 Integral equation method. 9.2 Family of linear and nonlinear integral approximations of the electromagnetic field. 9.3 Linear and non-linear approximations of higher orders. 9.4 Integral representations in numerical dressing. 10. Integral representations in electromagnetic inversion. 10.1 Linear inversion methods. 10.2 Nonlinear inversion. 10.3 Quasi-linear inversion. 10.4 Quasi-analytical inversion. 10.5 Magnetotelluric (MT) data inversion. 11. Electromagnetic migration imaging. 11.1 Electromagnetic migration in the frequency domain. 11.2 Electromagnetic migration in the time domain. 12. Differential methods in electromagnetic modeling and inversion. 12.1 Electromagnetic modeling as a boundary-value problem. 12.2 Finite difference approximation of the boundary-value problem. 12.3 Finite element solution of boundary-value problems. 12.4 Inversion based on differential methods. V. Seismic Inversion. 13. Wavefield equations. 13.1 Basic equations of elastic waves. 13.2 Green's functions for wavefield equations. 13.3 Kirchhoff integral formula and its analogs. 13.4 Uniqueness of the solution of the wavefield equations. 14. Integral representations in wavefield theory. 14.1 Integral equation method in acoustic wavefield analysis. 14.2 Integral approximations of the acoustic wavefield. 14.3 Method of integral equations in vector wavefield analysis. 14.4 Integral approximations of the vector wavefield. 15. Integral representations in wavefield inversion. 15.1 Linear inversion methods. 15.2 Quasi-linear inversion. 15.3 Nonlinear inversion. 15.4 Principles of wavefield migration. 15.5 Elastic field inversion. A. Functional spaces of geophysical models and data. A.1 Euclidean space. A.2 Metric space. A.3 Linear vector spaces. A.4 Hilbert spaces. A.5 Complex Euclidean and Hilbert spaces. A.6 Examples of linear vector spaces. B. Operators in the spaces of models and data. B.1 Operators in functional spaces. B.2 Linear operators. B.3 Inverse operators. B.4 Some approximation problems in the Hilbert spaces of geophysical data. B.5 Gram - Schmidt orthogonalization process. C. Functionals in the spaces of geophysical models. C.1 Functionals and their norms. C.2 Riesz representation theorem. C.3 Functional representation of geophysical data and an inverse problem. D. Linear operators and functionals revisited. D.1 Adjoint operators. D.2 Differentiation of operators and functionals. D.3 Concepts for variational calculus. E. Some formulae and rules from matrix algebra. E.1 Some formulae and rules of operation on matrices. E.2 Eigenvalues and eigenvectors. E.3 Spectral decomposition of a symmetric matrix. E.4 Singular value decomposition (SVD). E.5 The spectral Lanczos decomposition method. F. Some formulae and rules from tensor calculus. F.1 Some formulae and rules of operation on tensor functions. F.2 Tensor statements of the Gauss and Green's formulae. F.3 Green's tensor and vector formulae for Lame and Laplace operators. Bibliography. Index.

Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique

Abstract: A harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented. The analysis exploits the fact that many unsteady e ows of interest in turbomachinery are periodic in time. Thus, the unsteady e ow conservation variables may be represented by a Fourier series in time with spatially varying coefe cients. This assumption leads to a harmonic balance form of the Euler or Navier ‐Stokes equations, which, in turn, can be solved efe ciently as a steady problem using conventional computational e uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efe cient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous e ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear e ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the e ow. Furthermore, in some cases, e uid nonlinearities are found to be important for surprisingly small blade vibrations.

Adaptive dynamic programming

Abstract: Unlike the many soft computing applications where it suffices to achieve a "good approximation most of the time," a control system must be stable all of the time. As such, if one desires to learn a control law in real-time, a fusion of soft computing techniques to learn the appropriate control law with hard computing techniques to maintain the stability constraint and guarantee convergence is required. The objective of the paper is to describe an adaptive dynamic programming algorithm (ADPA) which fuses soft computing techniques to learn the optimal cost (or return) functional for a stabilizable nonlinear system with unknown dynamics and hard computing techniques to verify the stability and convergence of the algorithm. Specifically, the algorithm is initialized with a (stabilizing) cost functional and the system is run with the corresponding control law (defined by the Hamilton-Jacobi-Bellman equation), with the resultant state trajectories used to update the cost functional in a soft computing mode. Hard computing techniques are then used to show that this process is globally convergent with stepwise stability to the optimal cost functional/control law pair for an (unknown) input affine system with an input quadratic performance measure (modulo the appropriate technical conditions). Three specific implementations of the ADPA are developed for 1) the linear case, 2) for the nonlinear case using a locally quadratic approximation to the cost functional, and 3) the nonlinear case using a radial basis function approximation of the cost functional; illustrated by applications to flight control.

A subspace approach to balanced truncation for model reduction of nonlinear control systems

Abstract: In this paper, we introduce a new method of model reduction for nonlinear control systems. Our approach is to construct an approximately balanced realization. The method requires only standard matrix computations, and we show that when it is applied to linear systems it results in the usual balanced truncation. For nonlinear systems, the method makes use of data from either simulation or experiment to identify the dynamics relevant to the input}output map of the system. An important feature of this approach is that the resulting reduced-order model is nonlinear, and has inputs and outputs suitable for control. We perform an example reduction for a nonlinear mechanical system.

Adaptive Nonlinear System Identification with Echo State Networks

Abstract: Echo state networks (ESN) are a novel approach to recurrent neural network training. An ESN consists of a large, fixed, recurrent "reservoir" network, from which the desired output is obtained by training suitable output connection weights. Determination of optimal output weights becomes a linear, uniquely solvable task of MSE minimization. This article reviews the basic ideas and describes an online adaptation scheme based on the RLS algorithm known from adaptive linear systems. As an example, a 10-th order NARMA system is adaptively identified. The known benefits of the RLS algorithms carry over from linear systems to nonlinear ones; specifically, the convergence rate and misadjustment can be determined at design time.

Fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation

Abstract: In this paper, the relationship between bilateral filtering and anisotropic diffusion is examined. The bilateral filtering approach represents a large class of nonlinear digital image filters. We first explore the connection between anisotropic diffusion and adaptive smoothing, and then the connection between adaptive smoothing and bilateral filtering. Previously, adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation. We extend adaptive smoothing to make it consistent, thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation.

Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates

Abstract: Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.

Handbook of linear partial differential equations for engineers and scientists

Abstract: INTRODUCTION: SOME DEFINITIONS, FORMULAS, METHODS, AND SOLUTIONS Classification of Second Order Partial Differential Equations Basic Problems of Mathematical Physics Properties and Particular Solutions of Linear Equations Separation of Variables Method Integral Transforms Method Representation of the Solution of the Cauchy Problem via the Fundamental Solution Nonhomogeneous Boundary Value Problems with One Space Variable Nonhomogeneous Boundary Value Problems with Many Space Variables Construction of the Green's Functions: General Formulas and Relations Duhamel's Principles in Nonstationary Problems Transformation Simplifying Initial and boundary Conditions EQUATIONS OF PARABOLIC TYPE WITH ONE SPACE VARIABLE Constant Coefficient Equations Heat Equation with Axial or Central Symmetry and Related Equations Equations Containing Power Functions and Arbitrary Parameters Equations Containing Exponential Functions and Arbitrary Parameters Equations Containing Hyperbolic Functions and Arbitrary Parameters Equations Containing Logarithmic Functions and Arbitrary Parameters Equations Containing Trigonometric Functions and Arbitrary Parameters Equations Containing Arbitrary Functions Equations of Special Form PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES Heat Equation Heat Equation with a Source Other Equations PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Heat Equation Heat Equation with a Source Other Equations with Three Space Variables Equations with n Space Variables HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE Constant Coefficient Equations Wave Equation with Axial or Central Symmetry Equations Containing Power Functions and Arbitrary Parameters Equations Containing the First Time Derivative Equations Containing Arbitrary Functions HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES Wave Equation Nonhomogeneous Wave Equation Telegraph Equation Other Equation with Two Space Variables HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Wave Equation Nonhomogeneous Wave Equation Telegraph Equation Other Equations with Three Space Variables Equations with n Space Variables ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES Laplace Equation Poisson Equation Helmholtz Equation Other Equations ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Laplace Equation Poisson Equation Helmholtz Equation Other Equations Equations with n Space Variables HIGHER ORDER PARTIAL DIFFERENTIAL EQUATIONS Third Order Partial Differential Equations Fourth Order One-Dimensional Nonstationary Equations Two-Dimensional Nonstationary Fourth Order Equations Fourth Order Stationary Equations Higher Order Linear Equations with Constant Coefficients Higher Order Linear Equations with Variable Coefficients SUPPLEMENT A: Special Functions and Their Properties SUPPLEMENT B: Methods of Generalized and Functional Separation of Variables in Nonlinear Equations of Mathematical Physics REFERENCES INDEX

New State Transition Matrix for Relative Motion on an Arbitrary Elliptical Orbit

Abstract: A new state transition matrix is described for the nonlinear problem of relative motion on an arbitrary elliptical orbit. A linearization is performed, leading to a set of linear differential equations with time-dependentcoefficients. A new and simpler solution to those equations is represented in a convenient state transition matrix form. This new state transition matrix is valid for arbitrary elliptical orbits of 0≤e<1. The state propagation using the new state transition matrix shows good agreement with numerical results.

On the construction of Lyapunov functions using the sum of squares decomposition

Abstract: A relaxation of Lyapunov's direct method has been proposed elsewhere that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In the paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain non-polynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain non-polynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.

Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

Abstract: Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations Such equations were first derived by Boussinesq to describe the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal These systems arise also when modeling the propagation of long-crested waves on large lakes or the ocean and in other contexts Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations In the present script, we derive a four-parameter family of Boussinesq systems from the two-dimensional Euler equations for free-surface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation The analysis of the systems according to these criteria is initiated

Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm

Abstract: Considers the problem of global stabilization by output feedback, for a family of nonlinear systems that are dominated by a triangular system satisfying a linear growth condition. The problem has remained unsolved due to the violation of the commonly assumed conditions in the literature. Using a feedback domination design method which is not based on the separation principle, we explicitly construct a linear output compensator making the closed-loop system globally exponentially stable.

Computation of Multiphase Systems with Phase Field Models

Abstract: Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn-Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn-Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier-Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.

The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory

Abstract: We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-α) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, l ∈ , as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the long-time behavior of the solutions to these equations is bounded from above by (L/l ∈ )3, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau–Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NS-α equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier–Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NS-α model to the NSE by proving a convergence theorem, that as the length scale α 1 tends to zero a subsequence of solutions of the NS-α equations converges to a weak solution of the three dimensional NSE.