Daniel Zwillinger

CRC standard mathematical tables and formulae; 30th edition

Numbers and Elementary Mathematics Proofs without words Constants Special numbers Number theory Series and products Algebra Elementary algebra Polynomials Vector algebra Linear and matrix algebra Abstract algebra Discrete Mathematics Set theory Combinatorics Graphs Combinatorial design theory Difference equations Geometry Euclidean geometry Coordinate systems in the plane Plane symmetries or isometries Other transformations of the plane Lines Polygons Surfaces of revolution: the torus Quadrics Spherical geometry and trigonometry Conics Special plane curves Coordinate systems in space Space symmetries or isometries Other transformations of space Direction angles and direction cosines Planes Lines in space Polyhedra Cylinders Cones Differential geometry Analysis Differential calculus Differential forms Integration Table of indefinite integrals Table of definite integrals Ordinary differential equations Partial differential equations Integral equations Tensor analysis Orthogonal coordinate systems Interval analysis Real analysis Generalized functions Complex analysis Special Functions Ceiling and floor functions Exponentiation Logarithmic functions Exponential function Trigonometric functions Circular functions and planar triangles Tables of trigonometric functions Angle conversion Inverse circular functions Hyperbolic functions Inverse hyperbolic functions Gudermannian function Orthogonal polynomials Gamma function Beta function Error functions Fresnel integrals Sine, cosine, and exponential integrals Polylogarithms Hypergeometric functions Legendre functions Bessel functions Elliptic integrals Jacobian elliptic functions Clebsch-Gordan coefficients Integral transforms: Preliminaries Fourier integral transform Discrete Fourier transform (DFT) Fast Fourier transform (FFT) Multidimensional Fourier transforms Laplace transform Hankel transform Hartley transform Mellin transform Hilbert transform Z-Transform Tables of transforms Probability and Statistics Probability theory Classical probability problems Probability distributions Queuing theory Markov chains Random number generation Control charts and reliability Statistics Confidence intervals Tests of hypotheses Linear regression Analysis of variance (ANOVA) Sample size Contingency tables Probability tables Scientific Computing Basic numerical analysis Numerical linear algebra Numerical integration and differentiation Mathematical Formulae from the Sciences Acoustics Astrophysics Atmospheric physics Atomic physics Basic mechanics Beam dynamics Classical mechanics Coordinate systems - Astronomical Coordinate systems - Terrestrial Earthquake engineering Electromagnetic transmission Electrostatics and magnetism Electronic circuits Epidemiology Finance Fluid mechanics Fuzzy logic Human body Image processing matrices Macroeconomics Modeling physical systems Optics Population genetics Quantum mechanics Quaternions Relativistic mechanics Solid mechanics Statistical mechanics Thermodynamics Miscellaneous Calendar computations Cellular automata Communication theory Control theory Computer languages Cryptography Discrete dynamical systems and chaos Electronic resources Elliptic curves Financial formulae Game theory Knot theory Lattices Moments of inertia Music Operations research Recreational mathematics Risk analysis and decision rules Signal processing Symbolic logic Units Voting power Greek alphabet Braille code Morse code List of References List of Figures List of Notation Index

Matrix Riccati Equations

This chapter presents matrix Riccati equations applicable to systems of quadratic ordinary differential equations. It yields an exact solution. The chapter highlights that there is an exact solution available for matrix Riccati equations. If the given ordinary differential equations can be put in the form of a matrix Riccati equation, the solution can be found. Matrix Riccati equations arise naturally in a number of physical settings. The gains in a Kalman–Bucy filter satisfy a matrix Riccati equation. Also, the deflection of a beam can be described by such equations. They also appear quite often in the context of control theory and invariant embedding solutions. Kerner has also shown that nonlinear differential systems of arbitrary order, ζ i = X i (ζ i , ζ 2 ,…. ζ k , t ) for t = 1,2,…, k , may often be reduced to Riccati systems.

Numerical Methods for PDEs

This chapter reviews the numerical methods for finding solutions to partial differential equations (PDEs). Boundary element method is applicable to most often linear elliptic partial differential equations, often Laplaces equation, and sometimes to parabolic, hyperbolic, or nonlinear elliptic equations. This yields an integral equation. The solution of the integral equation can be used in an integral representation of the solution. The problem of solving a partial differential equation within a given domain can be transformed into one solving an equivalent integral equation on the boundary of the domain. The unknown in the integral equation will be the charge density on the boundary of the domain.

Differential PPM has a higher throughput than PPM for the band-limited and average-power-limited optical channel

Because of the optimality of the pulse position modulation (PPM) technique for the infinite-bandwidth channel, it has been suggested as a technique for the bandwidth-limited channel. The use of differential pulse position modulation (dPPM) as an alternative coding scheme to PPM is proposed. In dPPM each new word begins at the end of the previous word. To avoid synchronization problems, a simple scheme that decreases the throughput of the dPPM channel but is easy to analyze is chosen. For some parameter values this restricted dPPM scheme has greater throughput (in terms of capacity and cutoff rate) than PPM. >

Method of Images

This chapter focuses on method of images applicable to differential equations with homogeneous boundary conditions and sources present. If the solution to a free space problem is known, then superposition can be used to find a solution in a finite domain with homogeneous boundary conditions. Given a problem with a source present, the free space problem is solved (that is, disregarding the boundary conditions). By superposition, the solution is determined when there are sources at different points, of different strengths. The position and strengths of these sources are chosen so as to obtain the desired boundary conditions. The added sources cannot appear in the physical domain of the problem. Symmetry considerations tend to simplify the process of determining where the sources should go. The chapter also highlights that the method of images is often used to solve Laplaces equation in hydrodynamics and electrostatics.

Numerical Methods for ODEs

This chapter reviews the numerical methods used for finding solutions to ordinary differential equations (ODEs). The analytical continuation method is applicable to initial value ordinary differential equations, a single equation or a system to yield a numerical approximation in the form of a Taylor series. If the Taylor series of a function is known at a single point, then the Taylor series of that function may be found at another (nearby) point. This process may be repeated until a particular point is reached.

Stochastic Limit Theorems

This chapter elaborates the procedure, and application of stochastic limit theorems. It is applicable to linear differential equations that contain a small parameter, and a random forcing term of a certain form. It yields a Fokker–Planck equation. Some equations do not have a white noise forcing term, and so a Fokker–Planck equation cannot be directly constructed. It is often true that random forcing terms behave like white noise in some asymptotic limit. In this limit, a Fokker–Planck equation can be constructed. Using the geometric optics approximation to the wave equation, the scaled position, and velocity of a ray in a weakly random medium satisfy dx / dt = v after a ray has traveled a long distance in the random medium.

Predictor—Corrector Methods

This chapter elaborates about predictor–corrector methods applicable to ordinary differential equations of the form y′ = f ( x,y ). A sequence of numerical approximations is obtained. One set of predictor–corrector equations is the Adams–Bashforth predictor formula and the Adams–Moulton corrector formula. The corrector formula could be iterated as many times as is necessary to insure convergence; this is called correcting to convergence. In general, if more than two iterations are required, then the step size h is probably too large. The predictor–corrector method is a finite difference scheme that is not a linear multistep method. To obtain the starting values so that the predictor-corrector pair can be used, Runge–Kutta methods can be used.

117 – Floquet Theory

Publisher Summary This chapter describes the yield and application of Floquet theory. It is applicable to linear ordinary differential equations with periodic coefficients, and periodic boundary conditions. It yields knowledge of whether all solutions are stable. It is found that if a linear differential equation has periodic coefficients, and periodic boundary conditions, then the solutions will generally be a periodic function times an exponentially increasing, or an exponentially decreasing function. Floquet theory will determine if the solution is exponentially increasing, or exponentially decreasing. An nth order linear ordinary differential equation whose coefficients are periodic with common period T is assumed. The general technique is to write the ordinary differential equation as a first-order vector system, of dimension n, and then the vector ordinary differential equation for any set of n linearly independent conditions is solved.

Lattice Gas Dynamics

This chapter discusses the lattice gas dynamics applicable to partial differential equations that physically arise from the motion of particles. A numerical approximation methodology is obtained. Partial differential equations are usually derived from some microscopic dynamical system. It may be possible to simulate the dynamical system directly, without first formulating differential equations. By considering the interacting particles that make up a fluid, and using continuum theory, the usual Navier–Stokes equation can be derived. This equation describes the evolution of the fluid. To numerically approximate the solution to this equation, the equation is discretized, and the resulting algebraic equations are solved on a computer. Methods have been found for constructing cellular automata that are microscopically reversible; obey exact conservation laws and model continuum phenomena.