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Showing papers on "Nonlinear system published in 1996"


Book
01 Feb 1996
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Abstract: Preface 1. Introduction. Problems to be considered Characteristics of 'real-world' problems Finite-precision arithmetic and measurement of error Exercises 2. Nonlinear Problems in One Variable. What is not possible Newton's method for solving one equation in one unknown Convergence of sequences of real numbers Convergence of Newton's method Globally convergent methods for solving one equation in one uknown Methods when derivatives are unavailable Minimization of a function of one variable Exercises 3. Numerical Linear Algebra Background. Vector and matrix norms and orthogonality Solving systems of linear equations-matrix factorizations Errors in solving linear systems Updating matrix factorizations Eigenvalues and positive definiteness Linear least squares Exercises 4. Multivariable Calculus Background Derivatives and multivariable models Multivariable finite-difference derivatives Necessary and sufficient conditions for unconstrained minimization Exercises 5. Newton's Method for Nonlinear Equations and Unconstrained Minimization. Newton's method for systems of nonlinear equations Local convergence of Newton's method The Kantorovich and contractive mapping theorems Finite-difference derivative methods for systems of nonlinear equations Newton's method for unconstrained minimization Finite difference derivative methods for unconstrained minimization Exercises 6. Globally Convergent Modifications of Newton's Method. The quasi-Newton framework Descent directions Line searches The model-trust region approach Global methods for systems of nonlinear equations Exercises 7. Stopping, Scaling, and Testing. Scaling Stopping criteria Testing Exercises 8. Secant Methods for Systems of Nonlinear Equations. Broyden's method Local convergence analysis of Broyden's method Implementation of quasi-Newton algorithms using Broyden's update Other secant updates for nonlinear equations Exercises 9. Secant Methods for Unconstrained Minimization. The symmetric secant update of Powell Symmetric positive definite secant updates Local convergence of positive definite secant methods Implementation of quasi-Newton algorithms using the positive definite secant update Another convergence result for the positive definite secant method Other secant updates for unconstrained minimization Exercises 10. Nonlinear Least Squares. The nonlinear least-squares problem Gauss-Newton-type methods Full Newton-type methods Other considerations in solving nonlinear least-squares problems Exercises 11. Methods for Problems with Special Structure. The sparse finite-difference Newton method Sparse secant methods Deriving least-change secant updates Analyzing least-change secant methods Exercises Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel) Appendix B. Test Problems (by Robert Schnabel) References Author Index Subject Index.

6,831 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present a unified approach to impulse response analysis which can be used for both linear and nonlinear multivariate models and demonstrate the use of these measures for a nonlinear bivariate model of US output and the unemployment rate.

3,821 citations


Journal ArticleDOI
TL;DR: The authors represent a nonlinear plant with a Takagi-Sugeno fuzzy model with a model-based fuzzy controller design utilizing the concept of the so-called "parallel distributed compensation" and presents a design methodology for stabilization of a class of nonlinear systems.
Abstract: Presents a design methodology for stabilization of a class of nonlinear systems. First, the authors represent a nonlinear plant with a Takagi-Sugeno fuzzy model. Then a model-based fuzzy controller design utilizing the concept of the so-called "parallel distributed compensation" is employed. The main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system. The design procedure is conceptually simple and natural. Moreover, the stability analysis and control design problems can be reduced to linear matrix inequality (LMI) problems. Therefore, they can be solved efficiently in practice by convex programming techniques for LMIs. The design methodology is illustrated by application to the problem of balancing and swing-up of an inverted pendulum on a cart.

2,534 citations


Book
28 May 1996
TL;DR: Introduction.
Abstract: Introduction. A Simple Model Problem. Abstract Nonlinear Equations. Finite Element Discretizations of Elliptic PDEs. Practical Implementation. Bibliography. Subject Index.

2,253 citations



Book
16 Aug 1996
TL;DR: In this article, a small gain and passivity of input-output maps are discussed. But the authors focus on the Hamiltonian system as passive systems and do not consider the Hamilton-Jacobi Inequalities.
Abstract: 1 Input-Output Stability.- 2 Small-gain and Passivity of Input-Output Maps.- 3 Dissipative Systems Theory.- 4 Hamiltonian Systems as Passive Systems.- 5 Passivity by Feedback.- 6 Factorizations of Nonlinear Systems.- 7 Nonlinear H? Control.- 8 Hamilton-Jacobi Inequalities.

1,909 citations


Book
26 Sep 1996
TL;DR: In this paper, the existence and compactness of solution semiflows of linear systems are investigated. But the authors focus on the nonhomogeneous systems and do not consider the linearized stability of non-homogeneous solutions.
Abstract: 1. Preliminaries.- 1.1 Semigroups and generators.- 1.2 Function spaces, elliptic operators, and maximal principles.- Bibliographical Notes.- 2. Existence and Compactness of Solution Semiflows.- 2.1 Existence and compactness.- 2.2 Local existence and global continuation.- 2.3 Extensions to neutral partial functional differential equations.- Bibliographical Notes.- 3. Generators and Decomposition of State Spaces for Linear Systems.- 3.1 Infinitesimal generators of solution semiflows of linear systems.- 3.2 Decomposition of state spaces by invariant subspaces.- 3.3 Computation of center, stable, and unstable subspaces.- 3.4 Extensions to equations with infinite delay.- 3.5 L2-stability and reduction of neutral equations.- Bibliographical Notes.- 4. Nonhomogeneous Systems and Linearized Stability.- 4.1 Dual operators and an alternative theorem.- 4.2 Variation of constants formula.- 4.3 Existence of periodic or almost periodic solutions.- 4.4 Principle of linearized stability.- 4.5 Fundamental transformations and representations of solutions.- Bibliographical Notes.- 5. Invariant Manifolds of Nonlinear Systems.- 5.1 Stable and unstable manifolds.- 5.2 Center manifolds.- 5.3 Flows on center manifolds.- 5.4 Global invariant manifolds of perturbed wave equations.- Bibliographical Notes.- 6. Hopf Bifurcations.- 6.1 Some classical Hopf bifurcation theorems for ODEs.- 6.2 Smooth local Hopf bifurcations: a special case.- 6.3 Some examples from population dynamics.- 6.4 Smooth local Hopf bifurcations: general situations.- 6.5 A topological global Hopf bifurcation theory.- 6.6 Global continuation of wave solutions.- Bibliographical Notes.- 7. Small and Large Diffusivity.- 7.1 Destablization of periodic solutions by small diffusivity.- 7.2 Large diffusivity under Neumann boundary conditions.- Bibliographical Notes.- 8. Invariance, Comparison, and Upper and Lower Solutions.- 8.1 Invariance and inequalities.- 8.2 Systems and strict inequalities.- 8.3 Applications to reaction diffusion equations with delay.- Bibliographical Notes.- 9. Convergence, Monotonicity, and Contracting Rectangles.- 9.1 Monotonicity and generic convergence.- 9.2 Stability and steady state solutions of quasimonotone systems.- 9.3 Comparison and convergence results.- 9.4 Applications to Lotka-Volterra competition models.- Bibliographical Notes.- 10. Dispativeness, Exponential Growth, and Invariance Principles.- 10.1 Point dispativeness in a scalar equation.- 10.2 Convergence in a scalar equation.- 10.3 Exponential growth in a scalar equation.- 10.4 An invariance principle.- Bibliographical Notes.- 11. Traveling Wave Solutions.- 11.1 Huxley nonlinearities and phase plane arguments.- 11.2 Delayed Fisher equation: sub-super solution method.- 11.3 Systems and monotone iteration method.- 11.4 Traveling oscillatory waves.- Bibliographical Notes.

1,747 citations


Book
06 Nov 1996
TL;DR: In this paper, the oblique derivative problem for quasilinear parabolic equations was studied and the theory of weak solutions was introduced. And the boundary gradient was used to estimate global and local gradient bounds.
Abstract: Maximum principles introduction to the theory of weak solutions Holder estimates existence, uniqueness and regularity of solutions further theory of weak solutions strong solutions fixed point theorems and their applications comparison and maximum principles boundary gradient estimates global and local gradient bounds Holder gradient estimates and existence theorems the oblique derivative problem for quasilinear parabolic equations fully nonlinear equations I - introduction fully nonlinear equations II - Monge-Ampere and Hessian equations.

1,680 citations


Book
14 Jun 1996
TL;DR: In this article, the theory of nonlinear optics frequency doubling and mixing optical parametric generation, amplification, and oscillation characterization of second order nonlinear optical materials properties of selected second-order nonlinearoptical materials is discussed.
Abstract: Elements of the theory of nonlinear optics frequency doubling and mixing optical parametric generation, amplification, and oscillation characterization of second order nonlinear optical materials properties of selected second order nonlinearoptical materials nonlinear index of refraction characterization of nonlinear refractive index materials optical properties of selected third order nonlinear optics materials nonlinear absorption experimental techniques in nonlinear absorption ultrafast characterization techniques laser flash photolysis nonlinear absorption properties of selected materials stimulated Raman scattering stimulated Brillouin scattering properties of selected stimulated light-scattering materials theelectro-optic effect.

1,563 citations


Book
20 Jun 1996
TL;DR: In this article, the authors present a mathematical model for phase transitions in Eutectoid carbon steels, based on the Caginalp model and the Penrose-Fife model.
Abstract: 1. Some Mathematical Tools.- 1.1 Measure and Integration.- 1.2 Function Spaces.- 1.3 Nonlinear Equations.- 1.4 Ordinary Differential Equations.- 2. Hysteresis Operators.- 2.1 Basic Examples.- 2.2 General Hysteresis Operators.- 2.3 The Play Operator.- 2.4 Hysteresis Operators of Preisach Type.- 2.5 Hysteresis Potentials and Energy Dissipation.- 2.6 Hysteresis Counting and Damage.- 2.7 Characterization of Preisach Type Operators.- 2.8 Hysteresis Loops in the Prandtl Model.- 2.9 Hysteresis Loops in the Preisach Model.- 2.10 Composition of Preisach Type Operators.- 2.11 Inverse and Implicit Hysteresis Operators.- 2.12 Hysteresis Count and Damage, Part II.- 3. Hysteresis and Differential Equations.- 3.1 Hysteresis in Ordinary Differential Equations.- 3.2 Auxiliary Imbedding Results.- 3.3 The Heat Equation with Hysteresis.- 3.4 A Convexity Inequality.- 3.5 The Wave Equation with Hysteresis.- 4. Phase Transitions and Hysteresis.- 4.1 Thermodynamic Notions and Relations.- 4.2 Phase Transitions and Order Parameters.- 4.3 Landau and Devonshire Free Energies.- 4.4 Ginzburg Theory and Phase Field Models.- 5. Hysteresis Effects in Shape Memory Alloys.- 5.1 Phenomenology and Falk's Model.- 5.2 Well-Posedness for Falk's Model.- 5.3 Numerical Approximation.- 5.4 Complementary Remarks.- 6. Phase Field Models With Non-Conserving Kinetics.- 6.1 Auxiliary Results from Linear Elliptic and Parabolic Theory.- 6.2 Well-Posedness of the Caginalp Model.- 6.3 Well-Posedness of the Penrose-Fife Model.- 6.4 Complementary Remarks.- 7. Phase Field Models With Conserved Order Parameters.- 7.1 Well-Posedness of the Caginalp Model.- 7.2 Well-Posedness of the Penrose-Fife Model.- 8. Phase Transitions in Eutectoid Carbon Steels.- 8.1 Phenomenology of the Phase Transitions.- 8.2 The Mathematical Model.- 8.3 Well-Posedness of the Model.- 8.4 The Jominy Test: A Numerical Study.

1,415 citations


Book
27 Sep 1996
TL;DR: In this paper, the authors define and define nonlinear hyperbolic systems in one space dimension and define finite difference schemes for one-dimensional systems in the case of multidimensional systems.
Abstract: From the contents: Introduction: Definitions and Examples.- Nonlinear hyperbolic systems in one space dimension.- Gas dynamics and reaction flows.- Finite Difference Schemes for one-dimensional systems.- The case of multidimensional systems.- An Introduction to Boundary conditions.

Book
10 Dec 1996
TL;DR: PDE examples by type linear problems as mentioned in this paper, including nonlinear stationary problems, nonlinear evolution problems, and nonlinear Cauchy problems, can be found in this paper.
Abstract: PDE examples by type Linear problems...An introduction Nonlinear stationary problems Nonlinear evolution problems Accretive operators and nonlinear Cauchy problems Appendix Bibliography Index.

Journal ArticleDOI
TL;DR: The authors believe that the H-infinity-optimal control theory is now at a stage where it can easily be incorporated into a second-level graduate course in a control curriculum, that would follow a basic course in linear control theory covering LQ and LQG designs.
Abstract: One of the major concentrated activities of the past decade in control theory has been the development of the so-called "H-infinity-optimal control theory", which addresses the issue of worst-case controller design for linear plants subject to unknown disturbances and plant uncertainties. Among the different time-domain approaches to this class of worst-case design problems, the one that uses the framework of dynamic, differential game theory stands out to be the most natural. This is so because the original H-infinity control problem (in its equivalent time-domain formulation) is in fact a minimax optimization problem, and hence a zero-sum game, where the controller can be viewed as the minimizing player and disturbance as the maximizing player. Using this framework, the authors present in this book a complete theory that encompasses continuous-time as well as discrete-time systems, finite as well as infinite horizons, and several different measurement schemes, including closed loop perfect state, delayed perfect state, samples state, closed-loop imperfect state, delayed imperfect state and sampled imperfect state information patterns. They also discuss extensions of the linear theory to nonlinear systems, and derivation of the lower dimensional controller for systems with regularly and singularly perturbed dynamics. This is the second edition of a 1991 book with the same title, which, besides featuring a more streamlined presentation of the results included in the first edition, and at places under more refined conditions, also contains substantial new material, reflecting new developments in the field since 1991. Among these are the nonlinear theory; connections between H-infinity-optimal control and risk sensitive stochastic control problems; H-infinity filtering for linear and nonlinear systems; and robustness considerations in the presence of regular and singular perturbations. Also included are a rather detailed description of the relationship between frequency-and time-domain approaches to robust controller design, and a complete set of results on the existence of value and characterization of optimal policies in finite- and infinite-horizon LQ differential games. The authors believe that the theory is now at a stage where it can easily be incorporated into a second-level graduate course in a control curriculum, that would follow a basic course in linear control theory covering LQ and LQG designs. The framework adopted in this book makes such an ambitious plan possible. For the most part, the only prerequisite for the book is a basic knowledge of linear control theory. No background in differential games, or game theory in general, is required, as the requisite concepts and results have been developed in the book at the appropriate level. The book is written in such a way that makes it possible to follow the theory for the continuous- and discrete-time systems independently (and also in parallel).

Journal ArticleDOI
TL;DR: Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.
Abstract: A general automatic approach is presented for accommodating local shape variation when mapping a two-dimensional (2-D) or three-dimensional (3-D) template image into alignment with a topologically similar target image. Local shape variability is accommodated by applying a vector-field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth (globally positive definite Jacobian). Smoothness is guaranteed without specifically penalizing large-magnitude deformations of small subvolumes by constraining the transformation on the basis of a Stokesian limit of the fluid-dynamical Navier-Stokes equations. This differs fundamentally from quadratic penalty methods, such as those based on linearized elasticity or thin-plate splines, in that stress restraining the motion relaxes over time allowing large-magnitude deformations. Kinematic nonlinearities are inherently necessary to maintain continuity of structures during large-magnitude deformations, and are included in all results. After initial global registration, final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem. Automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular. Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.

Journal ArticleDOI
TL;DR: A design methodology is developed that expands the class of nonlinear systems that adaptive neural control schemes can be applied to and relaxes some of the restrictive assumptions that are usually made.
Abstract: Based on the Lyapunov synthesis approach, several adaptive neural control schemes have been developed during the last few years. So far, these schemes have been applied only to simple classes of nonlinear systems. This paper develops a design methodology that expands the class of nonlinear systems that adaptive neural control schemes can be applied to and relaxes some of the restrictive assumptions that are usually made. One such assumption is the requirement of a known bound on the network reconstruction error. The overall adaptive scheme is shown to guarantee semiglobal uniform ultimate boundedness. The proposed feedback control law is a smooth function of the state.

Journal ArticleDOI
TL;DR: New stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization, H/sup /spl infin// control theory, and linear matrix inequalities.
Abstract: This paper presents stability analysis for a class of uncertain nonlinear systems and a method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems, First, a stability condition for Takagi and Sugeno's fuzzy model is given in terms of Lyapunov stability theory. Next, new stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization, H/sup /spl infin// control theory, and linear matrix inequalities. The derived stability conditions are used to analyze the stability of Takagi and Sugeno's fuzzy control systems with uncertainty which can be regarded as a generalized class of uncertain nonlinear systems, The design method employs the so-called parallel distributed compensation, important issues for the stability analysis and design are remarked. Finally, three design examples of fuzzy controllers for stabilizing nonlinear systems and uncertain nonlinear systems are presented.

Book
15 May 1996
TL;DR: In this article, the authors present a model for singular boundary problems with variable coefficients and a method of multiple scale expansions for Ordinary Differential Equations (ODE) in the standard form.
Abstract: 1. Introduction.- 1.1. Order Symbols, Uniformity.- 1.2. Asymptotic Expansion of a Given Function.- 1.3. Regular Expansions for Ordinary and Partial Differential Equations.- References.- 2. Limit Process Expansions for Ordinary Differential Equations.- 2.1. The Linear Oscillator.- 2.2. Linear Singular Perturbation Problems with Variable Coefficients.- 2.3. Model Nonlinear Example for Singular Perturbations.- 2.4. Singular Boundary Problems.- 2.5. Higher-Order Example: Beam String.- References.- 3. Limit Process Expansions for Partial Differential Equations.- 3.1. Limit Process Expansions for Second-Order Partial Differential Equations.- 3.2. Boundary-Layer Theory in Viscous, Incompressible Flow.- 3.3. Singular Boundary Problems.- References.- 4. The Method of Multiple Scales for Ordinary Differential Equations.- 4.1. Method of Strained Coordinates for Periodic Solutions.- 4.2. Two Scale Expansions for the Weakly Nonlinear Autonomous Oscillator.- 4.3. Multiple-Scale Expansions for General Weakly Nonlinear Oscillators.- 4.4. Two-Scale Expansions for Strictly Nonlinear Oscillators.- 4.5. Multiple-Scale Expansions for Systems of First-Order Equations in Standard Form.- References.- 5. Near-Identity Averaging Transformations: Transient and Sustained Resonance.- 5.1. General Systems in Standard Form: Nonresonant Solutions.- 5.2. Hamiltonian System in Standard Form Nonresonant Solutions.- 5.3. Order Reduction and Global Adiabatic Invariants for Solutions in Resonance.- 5.4. Prescribed Frequency Variations, Transient Resonance.- 5.5. Frequencies that Depend on the Actions, Transient or Sustained Resonance.- References.- 6. Multiple-Scale Expansions for Partial Differential Equations.- 6.1. Nearly Periodic Waves.- 6.2. Weakly Nonlinear Conservation Laws.- 6.3. Multiple-Scale Homogenization.- References.

Journal ArticleDOI
TL;DR: In this article, the solitary wave solutions of the approximate equations for long water waves, coupled KdV equations and the dispersive long wave equations in 2 + 1 dimensions are constructed by using a homogeneous balance method.

Journal Article
TL;DR: In this paper, the search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities, and the relation to frequency domain methods such as the circle and Popov criteria is explained.
Abstract: This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.

Journal ArticleDOI
TL;DR: The overall adaptive scheme is shown to guarantee global uniform ultimate boundedness.

Journal ArticleDOI
TL;DR: An inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory which leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case.
Abstract: An inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory. In the case of minimum phase systems, the trajectory produced agrees with that generated by Hirschorn's inverse dynamic system; however, the preimage trajectory is noncausal (rather than unstable) in the nonminimum phase case. In addition, the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case. With the addition of stabilizing feedback to the preimage trajectory, asymptotically exact output tracking is achieved. Tracking is demonstrated with a numerical example and compared to the well-known Byrnes-Isidori regulator. Rather than solving a partial differential equation to construct a regulator, the inverse is calculated using a Picard-like interaction. When preactuation is not possible, noncausal inverse trajectories can be truncated resulting in the tracking-error transients found in other control schemes.

Book
01 May 1996
TL;DR: In this article, a concise treatment of the theory of nonlinear evolutionary partial differential equations is provided, and a rigorous analysis of non-Newtonian fluids is provided for applications in physics, biology, and mechanical engineering.
Abstract: This book provides a concise treatment of the theory of nonlinear evolutionary partial differential equations. It provides a rigorous analysis of non-Newtonian fluids, and outlines its results for applications in physics, biology, and mechanical engineering

Journal ArticleDOI
TL;DR: A nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation.
Abstract: A nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation. The theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear, feedforward form. This result, in turn, is applied to the control of: 1) linear systems (stable and unstable) with inputs subject to magnitude and rate saturation and time delays; 2) the cascade of globally asymptotically stable nonlinear systems with certain linear systems (those that are stabilizable, right invertible, and such that all of their invariant zeros have nonpositive real part); 3) the inverted pendulum on a cart; and 4) the planar vertical takeoff and landing aircraft.

Journal ArticleDOI
TL;DR: In this article, the tanh-function method for finding explicit travelling solitary wave solutions to non-linear evolution equations is described, and a Mathematica package ATFM is presented to deal with the tedious algebra and outputs directly the required solutions.

Book
01 Jan 1996
TL;DR: Heath 2/e, presents a broad overview of numerical methods for solving all the major problems in scientific computing, including linear and nonlinear equations, least squares, eigenvalues, optimization, interpolation, integration, ordinary and partial differential equations, fast Fourier transforms, and random number generators.
Abstract: Heath 2/e, presents a broad overview of numerical methods for solving all the major problems in scientific computing, including linear and nonlinear equations, least squares, eigenvalues, optimization, interpolation, integration, ordinary and partial differential equations, fast Fourier transforms, and random number generators. The treatment is comprehensive yet concise, software-oriented yet compatible with a variety of software packages and programming languages. The book features more than 160 examples, 500 review questions, 240 exercises, and 200 computer problems. Changes for the second edition include: expanded motivational discussions and examples; formal statements of all major algorithms; expanded discussions of existence, uniqueness, and conditioning for each type of problem so that students can recognize "good" and "bad" problem formulations and understand the corresponding quality of results produced; and expanded coverage of several topics, particularly eigenvalues and constrained optimization. The book contains a wealth of material and can be used in a variety of one- or two-term courses in computer science, mathematics, or engineering. Its comprehensiveness and modern perspective, as well as the software pointers provided, also make it a highly useful reference for practicing professionals who need to solve computational problems. Table of contents 1 Scientific Computing 2 Systems of Linear Equations 3 Linear Least Squares 4 Eigenvalues Problems 5 Nonlinear Equations 6 Optimization 7 Interpolation 8 Numerical Integration and Differentiation 9 Initial Value Problems for ODEs 10 Boundary Value Problems for ODEs 11 Partial Differential Equations 12 Fast Fourier Transform 13 Random Numbers and Simulation

Journal ArticleDOI
TL;DR: A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a bi-Hamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures, and it is demonstrated how their recombination leads toIntegrable hierarchies endowed with nonlinear dispersion that supports compactons, or cusped and/or peaked solitons.
Abstract: A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a bi-Hamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures. We demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons (solitary-wave solutions having compact support), or cusped and/or peaked solitons. A general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified Korteweg--de Vries equation, the nonlinear Schr\"odinger equation, the integrable Boussinesq system used to model the two-way propagation of shallow water waves, and the Ito system of coupled nonlinear wave equations. These hierarchies include a remarkable variety of interesting integrable nonlinear differential equations. \textcopyright{} 1996 The American Physical Society.

Journal ArticleDOI
TL;DR: In this article, a transformation method is proposed to establish a relation between linear and nonlinear wave theories, which can be obtained from the sine-Gordon equation and is simpler than the hyperbolic tangent method in solving differential equations.

Journal ArticleDOI
TL;DR: A number of finite element discretization techniques based on two (or more) subspaces for nonlinear elliptic partial differential equations (PDEs) is presented and optimal error estimates are obtained.
Abstract: A number of finite element discretization techniques based on two (or more) subspaces for nonlinear elliptic partial differential equations (PDEs) is presented. Convergence estimates are derived to justify the efficiency of these algorithms. With the new proposed techniques, solving a large class of nonlinear elliptic boundary value problems will not be much more difficult than the solution of one linearized equation. Similar techniques are also used to solve nonsymmetric and/or indefinite linear systems by solving symmetric positive definite (SPD) systems. For the analysis of these two-grid or multigrid methods, optimal ${\cal L}^p$ error estimates are also obtained for the classic finite element discretizations.

Journal ArticleDOI
TL;DR: A Lyapunov statement and proof of the recent nonlinear small-gain theorem for interconnected input/state-stable (ISS) systems is provided.

Journal ArticleDOI
TL;DR: In this article, a formalism for analyzing errors in nonlinear problems is developed in the context of finite difference approximations for the Navier?Stokes equations when the flow is fully turbulent.