Showing papers on "Nonlinear system published in 1970"

Applied numerical analysis

Abstract: The fifth edition of this book continues teaching numerical analysis and techniques. Suitable for students with mathematics and engineering backgrounds, the breadth of topics (partial differential equations, systems of nonlinear equations, and matrix algebra), provide comprehensive and flexible coverage of numerical analysis.

Differential and Integral Inequalities

Abstract: In 1964 the author's mono graph "Differential- und Integral-Un gleichungen," with the subtitle "und ihre Anwendung bei Abschatzungs und Eindeutigkeitsproblemen" was published The present volume grew out of the response to the demand for an English translation of this book In the meantime the literature on differential and integral in equalities increased greatly We have tried to incorporate new results as far as possible As a matter of fact, the Bibliography has been almost doubled in size The most substantial additions are in the field of existence theory In Chapter I we have included the basic theorems on Volterra integral equations in Banach space (covering the case of ordinary differential equations in Banach space) Corresponding theorems on differential inequalities have been added in Chapter II This was done with a view to the new sections; dealing with the line method, in the chapter on parabolic differential equations Section 35 contains an exposition of this method in connection with estimation and convergence An existence theory for the general nonlinear parabolic equation in one space variable based on the line method is given in Section 36 This theory is considered by the author as one of the most significant recent applications of in equality methods We should mention that an exposition of Krzyzanski's method for solving the Cauchy problem has also been added The numerous requests that the new edition include a chapter on elliptic differential equations have been satisfied to some extent"

Advanced Calculus for Applications

Abstract: 1. Ordinary Differential Equations 1.1 Introduction 1.2 Linear Dependence 1.3 Complete Solutions of Linear Equations 1.4 The Linear Differential Equation of First Order 1.5 Linear Differential Equations with Constant Coefficients 1.6 The Equidimensional Linear Differential Equation 1.7 Properties of Linear Operators 1.8 Simultaneous Linear Differential Equations 1.9 particular Solutions by Variation of Parameters 1.10 Reduction of Order 1.11 Determination of Constants 1.12 Special Solvable Types of Nonlinear Equations 2. The Laplace Transform 2.1 An introductory Example 2.2 Definition and Existence of Laplace Transforms 2.3 Properties of Laplace Transforms 2.4 The Inverse Transform 2.5 The Convolution 2.6 Singularity Functions 2.7 Use of Table of Transforms 2.8 Applications to Linear Differential Equations with Constant Coefficients 2.9 The Gamma Function 3. Numerical Methods for Solving Ordinary Differential Equations 3.1 Introduction 3.2 Use of Taylor Series 3.3 The Adams Method 3.4 The Modified Adams Method 3.5 The Runge-Kutta Method 3.6 Picard's Method 3.7 Extrapolation with Differences 4. Series Solutions of Differential Equations: Special Functions 4.1 Properties of Power Series 4.2 Illustrative Examples 4.3 Singular Points of Linear Second-Order Differential Equations 4.4 The Method of Frobenius 4.5 Treatment of Exceptional Cases 4.6 Example of an Exceptional Case 4.7 A Particular Class of Equations 4.8 Bessel Functions 4.9 Properties of Bessel Functions 4.10 Differential Equations Satisfied by Bessel Functions 4.11 Ber and Bei Functions 4.12 Legendre Functions 4.13 The Hypergeometric Function 4.14 Series Solutions Valid for Large Values of x 5. Boundary-Value Problems and Characteristic-Function Representations 5.1 Introduction 5.2 The Rotating String 5.3 The Rotating Shaft 5.4 Buckling of Long Columns Under Axial Loads 5.5 The Method of Stodola and Vianello 5.6 Orthogonality of Characteristic Functions 5.7 Expansion of Arbitrary Functions in Series of Orthogonal Functions 5.8 Boundary-Value Problems Involving Nonhomogeneous Differential Equations 5.9 Convergence of the Method of Stodola and Vianello 5.10 Fourier Sine Series and Cosine Series 5.11 Complete Fourier Series 5.12 Term-by-Term Differentiation of Fourier Series 5.13 Fourier-Bessel Series 5.14 Legendre Series 5.15 The Fourier Integral 6. Vector Analysis 6.1 Elementary Properties of Vectors 6.2 The Scalar Product of Two Vectors 6.3 The Vector Product of Two Vectors 6.4 Multiple Products 6.5 Differentiation of Vectors 6.6 Geometry of a Space Curve 6.7 The Gradient Vector 6.8 The Vector Operator V 6.9 Differentiation Formulas 6.10 Line Integrals 6.11 The Potential Function 6.12 Surface Integrals 6.13 Interpretation of Divergence. The Divergence Theorem 6.14 Green's Theorem 6.15 Interpretation of Curl. Laplace's Equation 6.16 Stokes's Theorem 6.17 Orthogonal Curvilinear Coordinates 6.18 Special Coordinate Systems 6.19 Application to Two-Dimensional Incompressible Fluid Flow 6.20 Compressible Ideal Fluid Flow 7. Topics in Higher-Dimensional Calculus 7.1 Partial Differentiation. Chain Rules 7.2 Implicit Functions. Jacobian Determinants 7.3 Functional Dependence 7.4 Jacobians and Curvilinear Coordinates. Change of Variables in Integrals 7.5 Taylor Series 7.6 Maxima and Minima 7.7 Constraints and Lagrange Multipliers 7.8 Calculus of Variations 7.9 Differentiation of Integrals Involving a Parameter 7.10 Newton's Iterative Method 8. Partial Differential Equations 8.1 Definitions and Examples 8.2 The Quasi-Linear Equation of First Order 8.3 Special Devices. Initial Conditions 8.4 Linear and Quasi-Linear Equations of Second Order 8.5 Special Linear Equations of Second Order, with Constant Coefficients 8.6 Other Linear Equations 8.7 Characteristics of Linear First-Order Equations 8.8 Characteristics of Linear Second-Order Equations 8.9 Singular Curves on Integral Surfaces 8.10 Remarks on Linear Second-Order Initial-Value Problems 8.11 The Characteristics of a Particular Quasi-Linear Problem 9. Solutions of Partial Differential Equations of Mathematical Physics 9.1 Introduction 9.2 Heat Flow 9.3 Steady-State Temperature Distribution in a Rectangular Plate 9.4 Steady-State Temperature Distribution in a Circular Annulus 9.5 Poisson's Integral 9.6 Axisymmetrical Temperature Distribution in a Solid Sphere 9.7 Temperature Distribution in a Rectangular Parallelepiped 9.8 Ideal Fluid Flow about a Sphere 9.9 The Wave Equation. Vibration of a Circular Membrane 9.10 The Heat-Flow Equation. Heat Flow in a Rod 9.11 Duhamel's Superposition Integral 9.12 Traveling Waves 9.13 The Pulsating Cylinder 9.14 Examples of the Use of Fourier Integrals 9.15 Laplace Transform Methods 9.16 Application of the Laplace Transform to the Telegraph Equations for a Long Line 9.17 Nonhomogeneous Conditions. The Method of Variation of Parameters 9.18 Formulation of Problems 9.19 Supersonic Flow of ldeal Compressible Fluid Past an Obstacle 10. Functions of a Complex Variable 10.1 Introduction. The Complex Variable 10.2 Elementary Functions of a Complex Variable 10.3 Other Elementary Functions 10.4 Analytic Functions of a Complex Variable 10.5 Line Integrals of Complex Functions 10.6 Cauchy's Integral Formula 10.7 Taylor Series 10.8 Laurent Series 10.9 Singularities of Analytic Functions 10.10 Singularities at Infinity 10.11 Significance of Singularities 10.12 Residues 10.13 Evaluation of Real Definite Integrals 10.14 Theorems on Limiting Contours 10.15 Indented Contours 10.16 Integrals Involving Branch Points 11. Applications of Analytic Function Theory 11.1 Introduction 11.2 Inversion of Laplace Transforms 11.3 Inversion of Laplace Transforms with Branch Points. The Loop Integral 11.4 Conformal Mapping 11.5 Applications to Two-Dimensional Fluid Flow 11.6 Basic Flows 11.7 Other Applications of Conformal Mapping 11.8 The Schwarz-Christoffel Transformation 11.9 Green's Functions and the Dirichlet Problem 11.10 The Use of Conformal Mapping 11.11 Other Two-Dimensional Green's Functions Answers to Problems Index Contents

On the maximality of sums of nonlinear monotone operators

Abstract: is called the effective domain of F, and F is said to be locally bounded at a point x e D(T) if there exists a neighborhood U of x such that the set (1.4) T(U) = (J{T(u)\ueU} is a bounded subset of X. It is apparent that, given any two monotone operators Tx and T2 from X to X*, the operator F», + T2 is again monotone, where (1 5) (Ti + T2)(x) = Tx(x) + T2(x) = {*? +x% I xf e Tx(x), xt e T2(x)}. If Tx and F2 are maximal, it does not necessarily follow, however, that F», + T2 is maximal—some sort of condition is needed, since for example the graph of Tx + T2 can even be empty (as happens when D(Tx) n D(T2)= 0). The problem of determining conditions under which Tx + T2 is maximal turns out to be of fundamental importance in the theory of monotone operators. Results in this direction have been proved by Lescarret [9] and Browder [5], [6], [7]. The strongest result which is known at present is :

The use of nonlinear, noncompensatory models in decision making.

Abstract: An important problem in decision making concerns finding the utility of a multidimensional stimulus. This has traditionally been done by assuming that total utility is a linear function of the attributes of the stimulus. In clinical decision making, the linear regression model has been used to predict and diagnose on the basis of multidimensio nal information as well as to approximate the clinician's own judgment. Other nonlinear, noncompensatory models are available for combining information. These models, called conjunctive and disjunctive, are approximated here by suitable nonlinear functions of utility. They are then shown to give a better fit to certain decision data than the linear model. The factors affecting the use of these models and their implications are discussed.

Smoothing, filtering, and boundary effects

Abstract: Numerical integrations of finite-difference analogs of systems of nonlinear partial differential equations, such as those arising in atmospheric dynamics, are subject to computational instability from a variety of causes. One type of instability is produced by a spurious, nonlinear growth of high-frequency components that may be introduced by roundoff, truncation, and observational error. This type of instability, first discussed by N. A. Phillips, can be suppressed by a suitable choice of finite-difference method or by the use of a filter that selectively damps the high-frequency components. Though much effort is being devoted to the development of stable finite-difference procedures, and considerable success has been achieved, all such methods involve high-frequency smoothing either implicitly or explicitly. It is therefore important that the effects of such filtering be fully understood. Filtering and smoothing operators are developed for use in conjunction with the numerical integration of nonlinear systems and for other purposes. The general procedure is demonstrated for simple one-dimensional operators and the properties of such operators are thoroughly explored. The development is then expanded to allow for compound operators designed to suit some particular requirement and further extended to more than one dimension. Both real and complex operators are discussed. Reverse smoothers or wave amplifiers are introduced, and some of the problems associated with their use are discussed. A general procedure is outlined for the construction of an ‘ideal’ low-pass filter; that is, a filter that removes the shortest resolvable wave component (the 2-grid-interval wave) but restores all other wave components as close as is desired to their original amplitudes without amplifying or changing the phase of any wave component. Finally, the effects, sometimes disastrous, of finite domains on the properties of the smoothing operators are explored for a variety of common boundary assumptions.

Random differential equations in control theory

Abstract: Control processes and optimization problems solutions by stochastic differential equations, discussing dynamic models and programming, linear filtering and optimal feedback

Galerkin Methods for Parabolic Equations

Abstract: Galerkin-type methods, both continuous and discrete in time, are considered for approximating solutions of linear and nonlinear parabolic problems. Bounds reducing the estimation of the error to questions in approximation theory are derived for the several methods studied. These methods include procedures that lead to linear algebraic equations even for strongly nonlinear problems. A number of computational questions related to these procedures are also discussed.

Waves in Nonlinear Lattice

Abstract: In this article waves in nonlinear lattice or in nonlinear medium are studied. One of the aims is to seek for the point of view to deal with the great majority of phenomena related to nonlinear waves in general. For one dimensional nonlinear lattice analytic and computer-experimental treatments have been developed. It has been found that a certain kind of pulse-like waves (solitons) is the fundamental motion in nonlinear lattice vibration. If two or more solitons collide, they interact nonlinearly, pass through one another and, when they separate, return to their original forms. Thus solitons are conserved and behave like particles.

Nonlinear Electrodynamics: Lagrangians and Equations of Motion

Abstract: After a brief discussion of well‐known classical fields we formulate two principles: When the field equations are hyperbolic, particles move along rays like disturbances of the field; the waves associated with stable particles are exceptional. This means that these waves will not transform into shock waves. Both principles are applied to nonlinear electrodynamics. The starting point of the theory is a Lagrangian which is an arbitrary nonlinear function of the two electromagnetic invariants. We obtain the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts. The general ``exceptional'' Lagrangian is found. It reduces to the Lagrangian of Born and Infeld when some constant (probably simply connected with the Planck constant) vanishes. A nonsymmetric tensor is introduced in analogy to the Born‐Infeld theory, and finally, electromagnetic waves are compared with those of Einstein‐Schrodinger theory.

A model for inhomogeneous turbulent flow

Abstract: A set of model equations is given to describe the gross features of a statistically steady or 9slowly varying’ inhomogeneous field of turbulence and the mean velocity distribution. The equations are based on the idea that turbulence can be characterized by ‘densities’ which obey nonlinear diffusion equations. The diffusion equations contain terms to describe the convection by the mean flow, the amplification due to interaction with a mean velocity gradient, the dissipation due to the interaction of the turbulence with itself, and the dif­fusion also due to the self interaction. The equations are similar to a set proposed by Kolmo­gorov (1942). It is assumed that both an ‘energy density’ and a ‘vorticity density’ satisfy diffusion equations, and that the self diffusion is described by an eddy viscosity which is a function of the energy and vorticity densities; the eddy viscosity is also assumed to describe the diffu­sion of mean momentum by the turbulent fluctuations. It is shown that with simple and plausible assumptions about the nature of the interaction terms, the equations form a closed set. The appropriate boundary conditions at a solid wall and a turbulent interface, with and without entrainment, are discussed. It is shown that the dimensionless constants which appear in the equations can all be estimated by general arguments. The equations are then found to predict the von Karman constant in the law of the wall with reasonable accuracy. An analytical solution is given for Couette flow, and the result of a numerical study of plane Poiseuille flow is described. The equations are also applied to free turbulent flows. It is shown that the model equations completely determine the structure of the similarity solutions, with the rate of spread, for instance, determined by the solution of a nonlinear eigenvalue problem. Numerical solutions have been obtained for the two-dimensional wake and jet. The agreement with experiment is good. The solutions have a sharp interface between turbulent and non-turbulent regions and the mean velocity in the turbulent part varies linearly with distance from the interface. The equations are applied qualitatively to the accelerating boundary layer in flow towards a line sink, and the decelerating boundary layer with zero skin friction. In the latter case, the equations predict that the mean velocity should vary near the wall like the 5/3 power of the distance. It is shown that viscosity can be incorporated formally into the model equations and that a structure can be given to the interface between turbulent and non-turbulent parts of the flow.

Theory of intracavity optical second-harmonic generation

Abstract: An analysis of optical second-harmonic generation internal to the laser cavity is presented. It is shown that the maximum second-harmonic power generated in this way is equal to the maximum fundamental power available from the laser. Further, it is found that there exists a value of nonlinearity that optimally couples the harmonic out for all power levels of the laser. The magnitude of the nonlinearity required for optimum coupling is shown to be proportional to the linear losses at the fundamental and inversely proportional to the saturation parameter for the laser transition. For the YAlG:Nd laser at 1.06 μ using Ba 2 NaNb 5 O 15 as the nonlinear material, the required crystal length for optimum coupling is given by l\min{c}\max{2}(cm)\simeq 2.7 \times 10^{2}L/f where L is the linear round-trip loss and f is the ratio of the fundamental power density in the nonlinear crystal to that in the laser medium. For low-loss cavities, optimum coupling can thus be achieved for crystal lengths of 1 cm or less. The use of a mirror or mirrors within the cavity, reflecting at the harmonic, is considered as a means to couple out the total harmonic in one direction. Considerations of temperature stability and the finite oscillating linewidth of the laser are shown to favor a configuration with a single harmonic mirror located on the same surface as the fundamental mirror.

Nonlinear optical properties of periodic laminar structures

Abstract: Some novel phase‐matching schemes are suggested for nonlinear processes occurring in a composite layered structure of GaP and GaAs.

A method of unconstrained global optimization

Abstract: The method that is defined in the following finds the maximum or minimum of a real-valued function of many variables even if the function has local maxima or minima. The methods is iterative and guaranteed to converge for polynomials in several variables up to fourth degree. It can also be used successfully for other types of functions. The method approximates a function automatically if it is not a polynomial of degree four or less. It is shown that a global optimization method for fourth degree polynomials can solve systems of polynomial equations in many variables of any degree. The speed of convergence is analysed theoretically and empirically. The method has been applied by the author and collaborators to the solution of systems of nonlinear equations in many variables (up to 100 variables), determination of rate constants in nonlinear differential equations (systems identification), chemical equilibrium equations, curve fitting of sums of exponentials, pattern recognition, and analysis of spectra with nonlinear superposition (Bremermann and Lam [11]). The applications will be reported elsewhere.

Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian

Abstract: For solving large systems of nonlinear equations by quasi-Newton methods it may often be preferable to store an approximation to the Jacobian rather than an approximation to the inverse Jacobian The main reason is that when the Jacobian is sparse and the locations of the zeroes are known, the updating procedure can be made more efficient for the approximate Jacobian than for the approximate inverse Jacobian

Nonlinear Controllability via Lie Theory

Abstract: Complete system associated with given control, discussing trajectories uniform approximation and nonlinear controllability conditions based on linear partial differential equation

Implicit flood routing in natural channels

Abstract: Flood routing in natural channels and many other applications in hydraulic engineering based on the solution of the equations of unsteady flow require fast and accurate numerical methods. Numerical methods which are successful in other applications prove to be inefficient when used for flood routing. An implicit numerical method which is both fast and accurate can be established on the basis of: (1) a centered difference scheme to represent the primary differential equations in finite difference form; and (2) the simultaneous solution of the finite difference equations for each time step. The difference equations constitute a system of nonlinear algebraic equations which can be solved on a digital computer by Newton iteration method. The computational scheme becomes very efficient when advantage is taken of the sparseness of the matrix of coefficients of the linear systems employed in the iteration. Applications of the implicit method show that it can be conveniently used for highly irregular channels.

Construction of minimal linear state-variable models from finite input-output data

Abstract: An algorithm for constructing minimal linear finite-dimensional realizations (a minimal partial realization) of an unknown (possibly infinite-dimensional) system from an external description as given by its Markov parameters is presented. It is shown that the resulting realization in essence models the transient response of the unknown system. If the unknown system is linear, this technique can be used to find a smaller dimensional linear system having the same transient characteristics. If the unknown system is nonlinear, the technique can be used either 1) to determine a useful nonlinear model, or 2) te determine a linear model, both of which approximate the transient response of the nonlinear system.

Lumped-circuit models for nonlinear inductors exhibiting hysteresis loops

Abstract: A new mathematical model of dynamic hysteresis loops is presented. The model is completely specified by two strictly monotonically increasing functions: a restoring function f(.) and a dissipation function g(.). Simple procedures are given for constructing these two functions so that the resulting model will simulate a given hysteresis loop exactly. The model is shown to exhibit many important hysteretic properties commonly observed in practice such as the presence of minor loops and an increase in area of the loop with frequency. In the case of an iron-core inductor, the mathematical model is shown to be equivalent to a lumped-circuit model, consisting of a nonlinear inductor in parallel with a nonlinear resistor. Extensive experimental investigations using different types of cores show remarkable agreement between results predicted by the model with those actually measured. The most serious limitation of this dynamic model is its inability to predict dc behaviors. For the class of switching circuits where dc solutions are important, a special dc lumped-circuit model is also presented.

Nonlinear Stabilization of High‐Frequency Instabilities in a Magnetic Field

Abstract: It is shown that the broadening of wave‐particle resonances by the random motion of particles in a turbulent electric field may determine the saturation level of a variety of high‐frequency instabilities. Secular changes of the guiding center positions, cyclotron radii, and phase angles give rise to resonance broadening and diffusion, similar to that produced by collisional scattering. The field dependent broadening is expressed in terms of resonance functions which replace the familiar resonant denominators of the linear theory. Resonance functions are derived in a simple manner from the solution of a Brownian motion problem, leading to an expression in terms of diffusion coefficients. The close resemblance of the theory to quasilinear theory and the linear theory including collisions allows one to start from a linear stability analysis and then assess the importance of nonlinear effects. This method is illustrated by the determination of the saturation level of cyclotron instabilities from the condition of vanishing nonlinear growth rate.