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1.

図書

図書
Ernst Hairer, Christian Lubich, Gerhard Wanner
出版情報: Berlin ; Tokyo : Springer, c2002  xiii, 515 p. ; 24 cm
シリーズ名: Springer series in computational mathematics ; v. 31
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2.

図書

図書
Y. Eliashberg, N. Mishachev
出版情報: Providence, R.I. : American Mathematical Society, c2002  xvii, 206 p. ; 26 cm
シリーズ名: Graduate studies in mathematics ; v. 48
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目次情報: 続きを見る
Intrigue Holonomic approximation: Jets and holonomy
Thom transversality theorem Holonomic approximation
Applications Differential relations and Gromov's $h$-principle: Differential relations
Homotopy principle Open Diff $V$-invariant differential relations
Applications to closed manifolds
The homotopy principle in symplectic geometry: Symplectic and contact basics
Symplectic and contact structures on open manifolds
Symplectic and contact structures on closed manifolds
Embeddings into symplectic and contact manifolds
Microflexibility and holonomic $\mathcal{R}$-approximation
First applications of microflexibility
Microflexible $\mathfrak{U}$-invariant differential relations
Further applications to symplectic geometry
Convex integration: One-dimensional convex integration
Homotopy principle for ample differential relations
Directed immersions and embeddings
First order linear differential operators Nash-Kuiper theorem
Bibliography
Index
Intrigue Holonomic approximation: Jets and holonomy
Thom transversality theorem Holonomic approximation
Applications Differential relations and Gromov's $h$-principle: Differential relations
3.

図書

図書
Franz J. Vesely
出版情報: New York : Kluwer Academic/Plenum Publishers, c2001  xvi, 259 p. ; 26 cm
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The Three Pillars of Computational Physics / I:
Finite Differences / 1:
Interpolation Formulae / 1.1:
NGF Interpolation / 1.1.1:
NGB Interpolation / 1.1.2:
ST Interpolation / 1.1.3:
Difference Quotients / 1.2:
DNGF Formulae / 1.2.1:
DNGB Formulae / 1.2.2:
DST Formulae / 1.2.3:
Finite Differences in Two Dimensions / 1.3:
Sample Applications / 1.4:
Classical Point Mechanics / 1.4.1:
Diffusion and Thermal Conduction / 1.4.2:
Linear Algebra / 2:
Exact Methods / 2.1:
Gauss Elimination and Back Substitution / 2.1.1:
Simplifying Matrices: The Householder Transformation / 2.1.2:
LU Decomposition / 2.1.3:
Tridiagonal Matrices: Recursion Method / 2.1.4:
Iterative Methods / 2.2:
Jacobi Relaxation / 2.2.1:
Gauss-Seidel Relaxation (GSR) / 2.2.2:
Successive Over-Relaxation (SOR) / 2.2.3:
Alternating Direction Implicit Method (ADI) / 2.2.4:
Conjugate Gradient Method (CG) / 2.2.5:
Eigenvalues and Eigenvectors / 2.3:
Largest Eigenvalue and Related Eigenvector / 2.3.1:
Arbitrary Eigenvalue/-vector: Inverse Iteration / 2.3.2:
Potential Equation / 2.4:
Electronic Orbitals / 2.4.3:
Stochastics / 3:
Equidistributed Random Variates / 3.1:
Linear Congruential Generators / 3.1.1:
Shift Register Generators / 3.1.2:
Other Distributions / 3.2:
Fundamentals / 3.2.1:
Transformation Method / 3.2.2:
Generalized Transformation Method / 3.2.3:
Rejection Method / 3.2.4:
Multivariate Gaussian Distribution / 3.2.5:
Equidistribution in Orientation Space / 3.2.6:
Random Sequences / 3.3:
Markov Processes / 3.3.1:
Autoregressive Processes / 3.3.3:
Wiener-Levy Process / 3.3.4:
Markov Chains and the Monte Carlo method / 3.3.5:
Stochastic Optimization / 3.4:
Simulated Annealing / 3.4.1:
Genetic Algorithms / 3.4.2:
Everything Flows / II:
Ordinary Differential Equations / 4:
Initial Value Problems of First Order / 4.1:
Euler-Cauchy Algorithm / 4.1.1:
Stability and Accuracy of Difference Schemes / 4.1.2:
Explicit Methods / 4.1.3:
Implicit Methods / 4.1.4:
Predictor-Corrector Method / 4.1.5:
Runge-Kutta Method / 4.1.6:
Extrapolation Method / 4.1.7:
Initial Value Problems of Second Order / 4.2:
Verlet Method / 4.2.1:
Nordsieck Formulation of the PC Method / 4.2.2:
Symplectic Algorithms / 4.2.4:
Numerov's Method / 4.2.6:
Boundary Value Problems / 4.3:
Shooting Method / 4.3.1:
Relaxation Method / 4.3.2:
Partial Differential Equations / 5:
Initial Value Problems I (Hyperbolic) / 5.1:
FTCS Scheme; Stability Analysis / 5.1.1:
Lax Scheme / 5.1.2:
Leapfrog Scheme (LF) / 5.1.3:
Lax-Wendroff Scheme (LW) / 5.1.4:
Lax and Lax-Wendroff in Two Dimensions / 5.1.5:
Initial Value Problems II (Parabolic) / 5.2:
FTCS Scheme / 5.2.1:
Implicit Scheme of First Order / 5.2.2:
Crank-Nicholson Scheme (CN) / 5.2.3:
Dufort-Frankel Scheme (DF) / 5.2.4:
Boundary Value Problems: Elliptic DE / 5.3:
Relaxation and Multigrid Techniques / 5.3.1:
ADI Method for the Potential Equation / 5.3.2:
Fourier Transform Method (FT) / 5.3.3:
Cyclic Reduction (CR) / 5.3.4:
Anchors Aweigh / III:
Simulation and Statistical Mechanics / 6:
Model Systems of Statistical Mechanics / 6.1:
A Nutshellfull of Fluids and Solids / 6.1.1:
Tricks of the Trade / 6.1.2:
Monte Carlo Method / 6.2:
Molecular Dynamics Simulation / 6.3:
Hard Spheres / 6.3.1:
Continuous Potentials / 6.3.2:
Beyond Basic Molecular Dynamics / 6.3.3:
Evaluation of Simulation Experiments / 6.4:
Pair Correlation Function / 6.4.1:
Autocorrelation Functions / 6.4.2:
Particles and Fields / 6.5:
Ewald summation / 6.5.1:
Particle-Mesh Methods (PM and P3M) / 6.5.2:
Stochastic Dynamics / 6.6:
Quantum Mechanical Simulation / 7:
Diffusion Monte Carlo (DMC) / 7.1:
Path Integral Monte Carlo (PIMC) / 7.2:
Wave Packet Dynamics (WPD) / 7.3:
Density Functional Molecular Dynamics (DFMD) / 7.4:
Hydrodynamics / 8:
Compressible Flow without Viscosity / 8.1:
Explicit Eulerian Methods / 8.1.1:
Particle-in-Cell Method (PIC) / 8.1.2:
Smoothed Particle Hydrodynamics (SPH) / 8.1.3:
Incompressible Flow with Viscosity / 8.2:
Vorticity Method / 8.2.1:
Pressure Method / 8.2.2:
Free Surfaces: Marker-and-Cell Method (MAC) / 8.2.3:
Lattice Gas Models for Hydrodynamics / 8.3:
Lattice Gas Cellular Automata / 8.3.1:
The Lattice Boltzmann Method / 8.3.2:
Direct Simulation Monte Carlo / Bird method / 8.4:
Appendixes
Machine Errors / A:
Discrete Fourier Transformation / B:
Fast Fourier Transform (FFT) / B.1:
Bibliography
Index
The Three Pillars of Computational Physics / I:
Finite Differences / 1:
Interpolation Formulae / 1.1:
4.

図書

図書
by Maria do Rosário Grossinho and Stepan Agop Tersian
出版情報: Dordrecht : Kluwer Academic Publishers, c2001  xii, 269 p. ; 25 cm
シリーズ名: Nonconvex optimization and its applications ; v. 52
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5.

図書

図書
Eusebius Doedel, Laurette S. Tuckerman, editors
出版情報: New York : Springer, c2000  x, 471 p. ; 25 cm
シリーズ名: The IMA volumes in mathematics and its applications ; v. 119
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6.

図書

図書
George Emanuel
出版情報: Boca Raton, FL : Chapman & Hall/CRC, 2001  220 p.
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目次情報: 続きを見る
Preface
Background / I:
Introduction / 1:
Continuous One-Parameter Groups-I / 2:
Group Concept / 2.1:
Infinitesimal Transformation / 2.2:
Global Group Equations / 2.3:
Problems / 2.4:
Method of Characteristics / 3:
Theory / 3.1:
Examples / 3.2:
Continuous One-Parameter Groups-II / 3.3:
Invariance / 4.1:
The Once-Extended Group / 4.2:
Higher-Order Extended Groups / 4.3:
Ordinary Differential Equations / 4.4:
First-Order ODEs / 5:
Invariance Under a One-Parameter Group / 5.1:
Canonical Coordinates / 5.2:
Special Procedures / 5.3:
Compendium / 5.4:
Higher-Order ODEs / 5.5:
Invariant Equations / 6.1:
Finding the Groups / 6.2:
System of First-Order ODEs / 6.3:
Second-Order ODEs / 6.4:
Classification of Two-Parameter Groups / 7.1:
Invariance and Canonical Coordinates / 7.2:
Appendices / 7.3:
Bibliography and References / A:
The Rotation Group / B:
Basic Relations / C:
Tables / D:
Answers to Selected Problems / E:
Preface
Background / I:
Introduction / 1:
7.

図書

図書
Arieh Iserles
出版情報: Cambridge : Cambridge University Press, 2009  xviii, 459 p. ; 25 cm
シリーズ名: Cambridge texts in applied mathematics
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Preface to the second edition
Preface to the first edition
Flowchart of contents
Ordinary differential equations / I:
Euler's method and beyond / 1:
Ordinary differential equations and the Lipschitz condition / 1.1:
Euler's method / 1.2:
The trapezoidal rule / 1.3:
The theta method / 1.4:
Comments and bibliography
Exercises
Multistep methods / 2:
The Adams method / 2.1:
Order and convergence of multistep methods / 2.2:
Backward differentiation formulae / 2.3:
Runge-Kutta methods / 3:
Gaussian quadrature / 3.1:
Explicit Runge-Kutta schemes / 3.2:
Implicit Runge-Kutta schemes / 3.3:
Collocation and IRK methods / 3.4:
Stiff equations / 4:
What are stiff ODEs? / 4.1:
The linear stability domain and A-stability / 4.2:
A-stability of Runge-Kutta methods / 4.3:
A-stability of multistep methods / 4.4:
Geometric numerical integration / 5:
Between quality and quantity / 5.1:
Monotone equations and algebraic stability / 5.2:
From quadratic invariants to orthogonal flows / 5.3:
Hamiltonian systems / 5.4:
Error control / 6:
Numerical software vs. numerical mathematics / 6.1:
The Milne device / 6.2:
Embedded Runge-Kutta methods / 6.3:
Nonlinear algebraic systems / 7:
Functional iteration / 7.1:
The Newton-Raphson algorithm and its modification / 7.2:
Starting and stopping the iteration / 7.3:
The Poisson equation / II:
Finite difference schemes / 8:
Finite differences / 8.1:
The finite element method / 8.2:
Two-point boundary value problems / 9.1:
A synopsis of FEM theory / 9.2:
Spectral methods / 9.3:
Sparse matrices vs. small matrices / 10.1:
The algebra of Fourier expansions / 10.2:
The fast Fourier transform / 10.3:
Second-order elliptic PDEs / 10.4:
Chebyshev methods / 10.5:
Gaussian elimination for sparse linear equations / 11:
Banded systems / 11.1:
Graphs of matrices and perfect Cholesky factorization / 11.2:
Classical iterative methods for sparse linear equations / 12:
Linear one-step stationary schemes / 12.1:
Classical iterative methods / 12.2:
Convergence of successive over-relaxation / 12.3:
Multigrid techniques / 12.4:
In lieu of a justification / 13.1:
The basic multigrid technique / 13.2:
The full multigrid technique / 13.3:
Poisson by multigrid / 13.4:
Conjugate gradients / 14:
Steepest, but slow, descent / 14.1:
The method of conjugate gradients / 14.2:
Krylov subspaces and preconditioners / 14.3:
Poisson by conjugate gradients / 14.4:
Fast Poisson solvers / 15:
TST matrices and the Hockney method / 15.1:
Fast Poisson solver in a disc / 15.2:
Partial differential equations of evolution / III:
The diffusion equation / 16:
A simple numerical method / 16.1:
Order, stability and convergence / 16.2:
Numerical schemes for the diffusion equation / 16.3:
Stability analysis I: Eigenvalue techniques / 16.4:
Stability analysis II: Fourier techniques / 16.5:
Splitting / 16.6:
Hyperbolic equations / 17:
Why the advection equation? / 17.1:
Finite differences for the advection equation / 17.2:
The energy method / 17.3:
The wave equation / 17.4:
The Burgers equation / 17.5:
Appendix Bluffer's guide to useful mathematics
Linear algebra / A.1:
Vector spaces / A.1.1:
Matrices / A.1.2:
Inner products and norms / A.1.3:
Linear systems / A.1.4:
Eigenvalues and eigenvectors / A.1.5:
Bibliography
Analysis / A.2:
Introduction to functional analysis / A.2.1:
Approximation theory / A.2.2:
Index / A.2.3:
Preface to the second edition
Preface to the first edition
Flowchart of contents
8.

図書

図書
Ole Christensen
出版情報: Boston : Birkhäuser, c2003  xx, 440 p ; 24 cm
シリーズ名: Applied and numerical harmonic analysis / series editor, John J. Benedetto
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Preface
Frames in Finite-dimensional Inner Product Spaces / 1:
Some basic facts about frames / 1.1:
Frame bounds and frame algorithms / 1.2:
Frames in C[superscript n] / 1.3:
The discrete Fourier transform / 1.4:
Pseudo-inverses and the singular value decomposition / 1.5:
Finite-dimensional function spaces / 1.6:
Exercises / 1.7:
Infinite-dimensional Vector Spaces and Sequences / 2:
Sequences / 2.1:
Banach spaces and Hilbert spaces / 2.2:
L[superscript 2] (R) and l[superscript 2] (N) / 2.3:
The Fourier transform / 2.4:
Operators on L[superscript 2] (R) / 2.5:
Bases / 2.6:
Bases in Banach spaces / 3.1:
Bessel sequences in Hilbert spaces / 3.2:
Bases and biorthogonal systems in H / 3.3:
Orthonormal bases / 3.4:
The Gram matrix / 3.5:
Riesz bases / 3.6:
Fourier series and Gabor bases / 3.7:
Wavelet bases / 3.8:
Bases and their Limitations / 3.9:
Gabor systems and the Balian-Low Theorem / 4.1:
Bases and wavelets / 4.2:
General shortcomings / 4.3:
Frames in Hilbert Spaces / 5:
Frames and their properties / 5.1:
Frame sequences / 5.2:
Frames and operators / 5.3:
Frames and bases / 5.4:
Characterization of frames / 5.5:
The dual frames / 5.6:
Tight frames / 5.7:
Continuous frames / 5.8:
Frames and signal processing / 5.9:
Frames versus Riesz Bases / 5.10:
Conditions for a frame being a Riesz basis / 6.1:
Riesz frames and near-Riesz bases / 6.2:
Frames containing a Riesz basis / 6.3:
A frame which does not contain a basis / 6.4:
A moment problem / 6.5:
Exercise / 6.6:
Frames of Translates / 7:
Sequences in R[superscript d] / 7.1:
Frames of translates / 7.2:
Frames of integer-translates / 7.3:
Irregular frames of translates / 7.4:
The sampling problem / 7.5:
Frames of exponentials / 7.6:
Gabor Frames in L[superscript 2] (R) / 7.7:
Continuous representations / 8.1:
Gabor frames / 8.2:
Necessary conditions / 8.3:
Sufficient conditions / 8.4:
The Wiener space W / 8.5:
Special functions / 8.6:
General shift-invariant systems / 8.7:
Selected Topics on Gabor Frames / 8.8:
Popular Gabor conditions / 9.1:
Representations of the Gabor frame operator and duality / 9.2:
The duals of a Gabor frame / 9.3:
The Zak transform / 9.4:
Tight Gabor frames / 9.5:
The lattice parameters / 9.6:
Irregular Gabor systems / 9.7:
Applications of Gabor frames / 9.8:
Wilson bases / 9.9:
Gabor Frames in l[superscript 2] (Z) / 9.10:
Translation and modulation on l[superscript 2] (Z) / 10.1:
Discrete Gabor systems through sampling / 10.2:
Gabor frames in C[superscript L] / 10.3:
Shift-invariant systems / 10.4:
Frames in l[superscript 2] (Z) and filter banks / 10.5:
General Wavelet Frames / 10.6:
The continuous wavelet transform / 11.1:
Sufficient and necessary conditions / 11.2:
Irregular wavelet frames / 11.3:
Oversampling of wavelet frames / 11.4:
Dyadic Wavelet Frames / 11.5:
Wavelet frames and their duals / 12.1:
Tight wavelet frames / 12.2:
Wavelet frame sets / 12.3:
Frames and multiresolution analysis / 12.4:
Frame Multiresolution Analysis / 12.5:
Frame multiresolution analysis / 13.1:
Relaxing the conditions / 13.2:
Construction of frames / 13.4:
Frames with two generators / 13.5:
Some limitations / 13.6:
Wavelet Frames via Extension Principles / 13.7:
The general setup / 14.1:
The unitary extension principle / 14.2:
Applications to B-splines I / 14.3:
The oblique extension principle / 14.4:
Fewer generators / 14.5:
Applications to B-splines II / 14.6:
Approximation orders / 14.7:
Construction of pairs of dual wavelet frames / 14.8:
Applications to B-splines III / 14.9:
Perturbation of Frames / 14.10:
A Paley-Wiener Theorem for frames / 15.1:
Compact perturbation / 15.2:
Perturbation of frame sequences / 15.3:
Perturbation of Gabor frames / 15.4:
Perturbation of wavelet frames / 15.5:
Perturbation of the Haar wavelet / 15.6:
Approximation of the Inverse Frame Operator / 15.7:
The first approach / 16.1:
A general method / 16.2:
Applications to Gabor frames / 16.3:
Integer oversampled Gabor frames / 16.4:
The finite section method / 16.5:
Expansions in Banach Spaces / 16.6:
Representations of locally compact groups / 17.1:
Feichtinger-Grochenig theory / 17.2:
Banach frames / 17.3:
p-frames / 17.4:
Gabor systems and wavelets in L[superscript p] (R) and related spaces / 17.5:
Appendix A / 17.6:
Normed vector spaces and inner product spaces / A.1:
Linear algebra / A.2:
Integration / A.3:
Some special normed vector spaces / A.4:
Operators on Banach spaces / A.5:
Operators on Hilbert spaces / A.6:
The pseudo-inverse / A.7:
Some special functions / A.8:
B-splines / A.9:
Notes / A.10:
List of symbols
References
Index
Preface
Frames in Finite-dimensional Inner Product Spaces / 1:
Some basic facts about frames / 1.1:
9.

図書

図書
Bhimsen K. Shivamoggi
出版情報: Boston : Birkhäuser, c2003  xiv, 354 p. ; 24 cm
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10.

図書

図書
Andrei D. Polyanin, Valentin F. Zaitsev
出版情報: Boca Raton ; London : Chapman & Hall/CRC, c2003  xxvi, 787 p. ; 27 cm
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Authors
Foreword
Notations and Some Remarks
Introduction: Some Definitions, Formulas, Methods, and Transformations
First-Order Differential Equations / 0.1.:
General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems / 0.1.1.:
Equations Solved for the Derivative. Simplest Techniques of Integration / 0.1.2.:
Exact Differential Equations. Integrating Factor / 0.1.3.:
Riccati Equation / 0.1.4.:
Abel Equations of the First Kind / 0.1.5.:
Abel Equations of the Second Kind / 0.1.6.:
Equations Not Solved for the Derivative / 0.1.7.:
Contact Transformations / 0.1.8.:
Approximate Analytic Methods for Solution of Equations / 0.1.9.:
Numerical Integration of Differential Equations / 0.1.10.:
Second-Order Linear Differential Equations / 0.2.:
Formulas for the General Solution. Some Transformations / 0.2.1.:
Representation of Solutions as a Series in the Independent Variable / 0.2.2.:
Asymptotic Solutions / 0.2.3.:
Boundary Value Problems / 0.2.4.:
Eigenvalue Problems / 0.2.5.:
Second-Order Nonlinear Differential Equations / 0.3.:
Form of the General Solution. Cauchy Problem / 0.3.1.:
Equations Admitting Reduction of Order / 0.3.2.:
Methods of Regular Series Expansions with Respect to the Independent Variable or Small Parameter / 0.3.3.:
Perturbation Methods of Mechanics and Physics / 0.3.4.:
Galerkin Method and Its Modifications (Projection Methods) / 0.3.5.:
Iteration and Numerical Methods / 0.3.6.:
Linear Equations of Arbitrary Order / 0.4.:
Linear Equations with Constant Coefficients / 0.4.1.:
Linear Equations with Variable Coefficients / 0.4.2.:
Asymptotic Solutions of Linear Equations / 0.4.3.:
Nonlinear Equations of Arbitrary Order / 0.5.:
Structure of the General Solution. Cauchy Problem / 0.5.1.:
A Method for Construction of Solvable Equations of General Form / 0.5.2.:
Lie Group and Discrete-Group Methods / 0.6.:
Lie Group Method. Point Transformations / 0.6.1.:
Contact Transformations. Backlund Transformations. Formal Operators. Factorization Principle / 0.6.2.:
First Integrals (Conservation Laws) / 0.6.3.:
Discrete-Group Method. Point Transformations / 0.6.4.:
Discrete-Group Method. The Method of RF-Pairs / 0.6.5.:
Simplest Equations with Arbitrary Functions Integrable in Closed Form / 1.:
Equations of the Form y'[subscript x] = f(x) / 1.1.1.:
Equations of the Form y'[subscript x] = f(y) / 1.1.2.:
Separable Equations y'[subscript x] = f(x)g(y) / 1.1.3.:
Linear Equation g(x)y'[subscript x] = f[subscript 1](x)y + f[subscript 0](x) / 1.1.4.:
Bernoulli Equation g(x)y'[subscript x] = f[subscript 1](x)y + f[subscript n](x)y[superscript n] / 1.1.5.:
Homogeneous Equation y'[subscript x] = f(y/x) / 1.1.6.:
Riccati Equation g(x)y'[subscript x] = f[subscript 2](x)y[superscript 2] + f[subscript 1](x)y + f[subscript 0](x) / 1.2.:
Preliminary Remarks / 1.2.1.:
Equations Containing Power Functions / 1.2.2.:
Equations Containing Exponential Functions / 1.2.3.:
Equations Containing Hyperbolic Functions / 1.2.4.:
Equations Containing Logarithmic Functions / 1.2.5.:
Equations Containing Trigonometric Functions / 1.2.6.:
Equations Containing Inverse Trigonometric Functions / 1.2.7.:
Equations with Arbitrary Functions / 1.2.8.:
Some Transformations / 1.2.9.:
Equations of the Form yy'[subscript x] - y = f(x) / 1.3.:
Equations of the Form yy'[subscript x] = f(x)y + 1 / 1.3.2.:
Equations of the Form yy'[subscript x] = f[subscript 1](x)y + f[subscript 0](x) / 1.3.3.:
Equations of the Form [g[subscript 1](x)y + g[subscript 0](x)]y'[subscript x] = f[subscript 2](x)y[superscript 2] + f[subscript 1](x)y + f[subscript 0](x) / 1.3.4.:
Some Types of First- and Second-Order Equations Reducible to Abel Equations of the Second Kind / 1.3.5.:
Equations Containing Polynomial Functions of y / 1.4.:
Abel Equations of the First Kind y'[subscript x] = f[subscript 3](x)y[superscript 3] + f[subscript 2](x)y[superscript 2] + f[subscript 1](x)y + f[subscript 0](x) / 1.4.1.:
Equations of the Form (A[subscript 22]y[superscript 2] + A[subscript 12]xy + A[subscript 11]x[superscript 2] + A[subscript 0])y'[subscript x] = B[subscript 22]y[superscript 2] + B[subscript 12]xy + B[subscript 11]x[superscript 2] + B[subscript 0] / 1.4.2.:
Equations of the Form (A[subscript 22]y[superscript 2] + A[subscript 12]xy + A[subscript 11]x[superscript 2] + A[subscript 2]y + A[subscript 1]x)y'[subscript x] = B[subscript 22]y[superscript 2] + B[subscript 12]xy + B[subscript 11]x[superscript 2] + B[subscript 2]y + B[subscript 1]x / 1.4.3.:
Equations of the Form (A[subscript 22]y[superscript 2] + A[subscript 12]xy + A[subscript 11]x[superscript 2] + A[subscript 2]y + A[subscript 1]x + A[subscript 0])y'[subscript x] = B[subscript 22]y[superscript 2] + B[subscript 12]xy + B[subscript 11]x[superscript 2] + B[subscript 2]y + B[subscript 1]x + B[subscript 0] / 1.4.4.:
Equations of the Form (A[subscript 3]y[superscript 3] + A[subscript 2]xy[superscript 2] + A[subscript 1]x[superscript 2]y + A[subscript 0]x[superscript 3] + a[subscript 1]y + a[subscript 0]x)y'[subscript x] = B[subscript 3]y[superscript 3] + B[subscript 2]xy[superscript 2] + B[subscript 1]x[superscript 2]y + B[subscript 0]x[superscript 3] + b[subscript 1]y + b[subscript 0]x / 1.4.5.:
Equations of the Form f(x, y)y'[subscript x] = g(x, y) Containing Arbitrary Parameters / 1.5.:
Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions / 1.5.1.:
Equations of the Form F(x, y, y'[subscript x]) = 0 Containing Arbitrary Parameters / 1.6.:
Equations of the Second Degree in y'[subscript x] / 1.6.1.:
Equations of the Third Degree in y'[subscript x] / 1.6.2.:
Equations of the Form (y'[subscript x])[superscript k] = f(y) + g(x) / 1.6.3.:
Other Equations / 1.6.4.:
Equations of the Form f(x, y)y'[subscript x] = g(x, y) Containing Arbitrary Functions / 1.7.:
Equations Containing Exponential and Hyperbolic Functions / 1.7.1.:
Equations Containing Combinations of Exponential, Logarithmic, and Trigonometric Functions / 1.7.3.:
Equations of the Form F(x, y, y'[subscript x]) = 0 Containing Arbitrary Functions / 1.8.:
Some Equations / 1.8.1.:
Second-Order Differential Equations / 1.8.2.:
Linear Equations / 2.1.:
Representation of the General Solution Through a Particular Solution / 2.1.1.:
Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions / 2.1.2.:
Autonomous Equations y"[subscript x x] = F(y, y'[subscript x]) / 2.1.9.:
Equations of the Form y"[subscript x x] - y'[subscript x] = f(y) / 2.2.1.:
Equations of the Form y"[subscript x x] + f(y)y'[subscript x] + y = 0 / 2.2.2.:
Lienard Equations y"[subscript x x] + f(y)y'[subscript x] + g(y) = 0 / 2.2.3.:
Rayleigh Equations y"[subscript x x] + f(y'[subscript x]) + g(y) = 0 / 2.2.4.:
Emden-Fowler Equation y"[subscript x x] = Ax[superscript n]y[superscript m] / 2.3.:
Exact Solutions / 2.3.1.:
Some Formulas and Transformations / 2.3.2.:
Equations of the Form y"[subscript x x] = A[subscript 1]x[superscript n[subscript 1]y[superscript m[subscript 1] + A[subscript 2]x[superscript n[subscript 2]y[superscript m[subscript 2] / 2.4.:
Classification Table / 2.4.1.:
Generalized Emden-Fowler Equation y"[subscript x x] = Ax[superscript n]y[superscript m](y'[subscript x])[superscript l] / 2.4.2.:
Equations of the Form y"[subscript x x] = A[subscript 1]x[superscript n[subscript 1]y[superscript m[subscript 1](y'[subscript x])[superscript l[subscript 1] + A[subscript 2]x[superscript n[subscript 2]y[superscript m[subscript 2](y'[subscript x])[superscript l[subscript 2] / 2.5.1.:
Modified Emden-Fowler Equation y"[subscript x x] = A[subscript 1]x[superscript -1]y'[subscript x] + A[subscript 2]x[superscript n]y[superscript m] / 2.6.1.:
Equations of the Form y"[subscript x x] = (A[subscript 1]x[superscript n[subscript 1]y[superscript m[subscript 1] + A[subscript 2]x[superscript n[subscript 2]y[superscript m[subscript 2])(y'[subscript x])[superscript l] / 2.6.2.:
Equations of the Form y"[subscript x x] = [sigma] Ax[superscript n]y[superscript m](y'[subscript x])[superscript l] + Ax[superscript n-1]y[superscript m+1](y'[subscript x])[superscript l-1] / 2.6.3.:
Other Equations (l[subscript 1] [not equal] l[subscript 2]) / 2.6.4.:
Equations of the Form y"[subscript x x] = f(x)g(y)h(y'[subscript x]) / 2.7.:
Equations of the Form y"[subscript x x] = f(x)g(y) / 2.7.1.:
Equations Containing Power Functions (h [characters not reproducible] const) / 2.7.2.:
Equations Containing Exponential Functions (h [characters not reproducible] const) / 2.7.3.:
Equations Containing Hyperbolic Functions (h [characters not reproducible] const) / 2.7.4.:
Equations Containing Trigonometric Functions (h [characters not reproducible] const) / 2.7.5.:
Some Nonlinear Equations with Arbitrary Parameters / 2.7.6.:
Painleve Transcendents / 2.8.1.:
Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions / 2.8.3.:
Equations Containing Arbitrary Functions / 2.9.:
Equations of the Form F(x, y)y"[subscript xx] + G(x, y) = 0 / 2.9.1.:
Equations of the Form F(x, y)y"[subscript xx] + G(x, y)y'[subscript x] + H(x, y) = 0 / 2.9.2.:
Equations of the Form F(x, y)y"[subscript xx] + [characters not reproducible] G[subscript m](x, y)(y'[subscript x])[superscript m] = 0 (M = 2, 3, 4) / 2.9.3.:
Equations of the Form F(x, y, y'[subscript x])y"[subscript xx] + G(x, y, y'[subscript x]) = 0 / 2.9.4.:
Equations Not Solved for Second Derivative / 2.9.5.:
Equations of General Form / 2.9.6.:
Third-Order Differential Equations / 2.9.7.:
Equations of the Form y'"[subscript xxx] = Ax[superscript [alpha]y[superscript [beta](y'[subscript x])[superscript gamma](y"[subscript xx])[superscript [delta ] / 3.1.:
Equations of the Form y'"[subscript xxx] = Ay[superscript beta] / 3.2.1.:
Equations of the Form y'''[subscript xxx] = Ax[superscript alpha]y[superscript beta] / 3.2.3.:
Equations with |[gamma]| + |[delta]| [not equal] 0 / 3.2.4.:
Equations of the Form y'''[subscript xxx] = f(y)g(y'[subscript x])h(y''[subscript xx]) / 3.2.5.:
Nonlinear Equations with Arbitrary Parameters / 3.3.1.:
Nonlinear Equations Containing Arbitrary Functions / 3.4.1.:
Equations of the Form F(x, y)y'''[subscript xxx] + G(x, y) = 0 / 3.5.1.:
Equations of the Form F(x, y, y'[subscript x])y'''[subscript xxx] + G(x, y, y'[subscript x]) = 0 / 3.5.2.:
Equations of the Form F(x, y, y'[subscript x])y'''[subscript xxx] + G(x, y, y'[subscript x])y''[subscript xx] + H(x, y, y'[subscript x]) = 0 / 3.5.3.:
Equations of the Form F(x, y, y'[subscript x])y'''[subscript xxx] + [characters not reproducible]G[subscript alpha](x, y, y'[subscript x])(y''[subscript xx])[superscript alpha] = 0 / 3.5.4.:
Fourth-Order Differential Equations / 3.5.5.:
Nonlinear Equations / 4.1.:
Higher-Order Differential Equations / 4.2.1.:
Supplements / 5.1.:
Elementary Functions and Their Properties / S.1.:
Trigonometric Functions / S.1.1.:
Hyperbolic Functions / S.1.2.:
Inverse Trigonometric Functions / S.1.3.:
Inverse Hyperbolic Functions / S.1.4.:
Special Functions and Their Properties / S.2.:
Some Symbols and Coefficients / S.2.1.:
Error Functions and Exponential Integral / S.2.2.:
Gamma and Beta Functions / S.2.3.:
Incomplete Gamma and Beta Functions / S.2.4.:
Bessel Functions / S.2.5.:
Modified Bessel Functions / S.2.6.:
Degenerate Hypergeometric Functions / S.2.7.:
Hypergeometric Functions / S.2.8.:
Legendre Functions and Legendre Polynomials / S.2.9.:
Parabolic Cylinder Functions / S.2.10.:
Orthogonal Polynomials / S.2.11.:
The Weierstrass Function / S.2.12.:
Tables of Indefinite Integrals / S.3.:
Integrals Containing Rational Functions / S.3.1.:
Integrals Containing Irrational Functions / S.3.2.:
Integrals Containing Exponential Functions / S.3.3.:
Integrals Containing Hyperbolic Functions / S.3.4.:
Integrals Containing Logarithmic Functions / S.3.5.:
Integrals Containing Trigonometric Functions / S.3.6.:
Integrals Containing Inverse Trigonometric Functions / S.3.7.:
References
Index
Authors
Foreword
Notations and Some Remarks
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