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1.

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:
2.

図書

図書
Bhimsen K. Shivamoggi
出版情報: Boston : Birkhäuser, c2003  xiv, 354 p. ; 24 cm
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3.

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
4.

図書

図書
J.C. Butcher
出版情報: Chichester : Wiley, c2003  xiv, 425 p. ; 24 cm
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Preface
Differential and Difference Equations / 1:
Differential Equation Problems / 10:
Differential Equation Theory / 11:
Difference Equation Problems / 100:
Introduction to differential equations
Difference Equation Theory / 13:
The Kepler problem / 2:
Numerical Differential Equation Methods
The Euler Method / 102:
Many-body gravitational problems
Analysis of the Euler Method / 21:
A problem arising from the method of lines / 22:
Generalizations of the Euler Method
Runge-Kutta Methods / 104:
The simple pendulum
Linear Multistep Methods / 24:
A chemical kinetics problem / 25:
Taylor Series Methods
Hybrid Methods / 106:
The van der Pol equation and limit cycles
The Lotka-Volterra problem and periodic orbits / 3:
Preliminaries
Order Conditions / 31:
Low Order Explicit Methods / 32:
Existence and uniqueness of solutions / 33:
Runge-Kutta Methods with Error Estimates
Implicit Runge-Kutta Methods / 111:
Linear systems of differential equations
Stability of Implicit Runge-Kutta Methods / 35:
Stiff differential equations / 36:
Implementable Implicit Runge-Kutta Methods
Order Barriers / 113:
Laplace transforms
Algebraic Properties of Runge-Kutta Methods / 38:
Implementation Issues / 39:
Introduction to difference equations / 120:
A linear problem / 40:
The Order of Linear Multistep Methods
Errors and Error Growth / 122:
The Fibonacci difference equation
Stability Characteristics / 43:
Three quadratic problems / 44:
Order and Stability Barriers
One-leg Methods and G-stability / 124:
Iterative solutions of a polynomial equation
The arithmetic-geometric mean / 46:
General Linear Methods
Representing Methods in General Linear Form / 50:
Consistency, Stability and Convergence / 51:
Linear difference equations / 52:
The Stability of General Linear Methods
The Order of General Linear Methods / 131:
Constant coefficients
Methods with Runge-Kutta Stabiulity / 54:
References
Powers of matrices
Index
The Z-transform / 133:
Introduction to the Euler method / 200:
Some numerical experiments / 201:
Calculations with stepsize control / 202:
Calculations with mildly stiff problems / 203:
Calculations with the implicit Euler method / 204:
Formulation of the Euler method / 210:
Local truncation error / 211:
Global truncation error / 212:
Convergence of the Euler method / 213:
Order of convergence / 214:
Asymptotic error formula / 215:
Stability characteristics / 216:
Local truncation error estimation / 217:
Rounding error / 218:
Introduction / 220:
More computations in a step / 221:
Greater dependence on previous values / 222:
Use of higher derivatives / 223:
Multistep-multistage-multiderivative methods / 224:
Implicit methods / 225:
Local error estimates / 226:
Historical introduction / 230:
Second order methods / 231:
The coefficient tableau / 232:
Third order methods / 233:
Introduction to order conditions / 234:
Fourth order methods / 235:
Higher orders / 236:
Implicit Runge-Kutta methods / 237:
Numerical examples / 238:
Adams methods / 240:
General form of linear multistep methods / 242:
Consistency, stability and convergence / 243:
Predictor-corrector Adams methods / 244:
The Milne device / 245:
Starting methods / 246:
Introduction to Taylor series methods / 247:
Manipulation of power series / 251:
An example of a Taylor series solution / 252:
Other methods using higher derivatives / 253:
The use of f derivatives / 254:
Further numerical examples / 255:
Pseudo Runge-Kutta methods / 260:
Generalized linear multistep methods / 262:
General linear methods / 263:
Rooted trees / 264:
Functions on trees / 301:
Some combinatorial questions / 302:
The use of labelled trees / 303:
Differentiation / 304:
Taylor's theorem / 305:
Elementary differentials / 310:
The Taylor expansion of the exact solution / 311:
Elementary weights / 312:
The Taylor expansion of the approximate solution / 313:
Independence of the elementary differentials / 314:
Conditions for order / 315:
Order conditions for scalar problems / 316:
Independence of elementary weights / 317:
Methods of orders less than four / 318:
Simplifying assumptions / 321:
Methods of order four / 322:
New methods from old / 323:
Methods of order five / 324:
Methods of order six / 325:
Methods of orders greater than six / 326:
Richardson error estimates / 330:
Methods with built-in estimates / 332:
A class of error-estimating methods / 333:
The methods of Fehlberg / 334:
The methods of Verner / 335:
The methods of Dormand and Prince / 336:
Solvability of implicit equations / 340:
Methods based on Gaussian quadrature / 342:
Reflected methods / 343:
Methods based on Radau and Lobatto quadrature / 344:
A-stability, A([alpha])-stability and L-stability / 350:
Criteria for A-stability / 351:
Pade approximations to the exponential function / 352:
A-stability of Gauss and related methods / 353:
Order stars / 354:
Order arrows and the Ehle barrier / 355:
AN-stability / 356:
Non-linear stability / 357:
BN-stability of collocation methods / 358:
The V and W transformations / 359:
Implementation of implicit Runge-Kutta methods / 360:
Diagonally-implicit Runge-Kutta methods / 361:
The importance of high stage order / 362:
Singly implicit methods / 363:
Generalizations of singly-implicit methods / 364:
Effective order and DESIRE methods / 365:
Explicit barriers / 370:
An upper bound on the required number of stages / 371:
Motivation / 380:
Equivalence classes of Runge-Kutta methods / 381:
The group of Runge-Kutta methods / 382:
The Runge-Kutta group / 383:
A homomorphism between two groups / 384:
A generalization of G[subscript 1] / 385:
Recursive formula for the product / 386:
Some special elements of G / 387:
Some subgroups and quotient groups / 388:
An algebraic interpretation of effective order / 389:
Optimal sequences / 390:
Acceptance and rejection of steps / 392:
Error per step versus error per unit step / 393:
Control theoretic considerations / 394:
Solving the implicit equations / 395:
Fundamentals / 400:
Convergence / 401:
Stability / 403:
Consistency / 404:
Necessity of conditions for convergence / 405:
Sufficiency of conditions for convergence / 406:
Criteria for order / 410:
Derivation of methods / 411:
Backward difference methods / 412:
Further remarks on error growth / 420:
The underlying one-step method / 422:
Weakly stable methods / 423:
Variable stepsize / 424:
Stability regions / 430:
Examples of the boundary locus method / 432:
Examples of the Schur criterion / 433:
Stability of predictor-corrector methods / 434:
Survey of barrier results / 440:
Maximum order for a convergent k step method / 441:
Order stars for linear multistep methods / 442:
Order arrows for linear multistep methods / 443:
The one-leg counterpart to a linear multistep method / 450:
The concept of G-stability / 451:
Transformations relating one-leg and linear multistep methods / 452:
Effective order interpretation / 453:
Concluding remarks on G-stability / 454:
Survey of implementation considerations / 460:
Representation of data / 461:
Variable stepsize for Nordsieck methods / 462:
Local error estimation / 463:
Multivalue multistage methods / 500:
Transformations of methods / 501:
Runge-Kutta methods as general linear methods / 502:
Linear multistep methods as general linear methods / 503:
Some known unconventional methods / 504:
Some recently discovered general linear methods / 505:
Definitions of consistency and stability / 510:
Covariance of methods / 511:
Definition of convergence / 512:
The necessity of stability / 513:
The necessity of consistency / 514:
Stability and consistency imply convergence / 515:
Methods with maximal stability order / 520:
Possible definitions of order / 530:
Algebraic analysis of order / 531:
An example of the algebraic approach to order / 532:
Methods with Runge-Kutta stability / 533:
Design criteria for general linear methods / 540:
The types of DIMSIM methods / 541:
Runge-Kutta stability / 542:
Almost Runge-Kutta methods / 543:
Fourth order, four stage ARK methods / 544:
Doubly companion matrices / 545:
Inherent Runge-Kutta stability / 546:
Derivation of methods with IRK stability / 547:
Some nonstiff methods / 548:
Some stiff methods / 549:
Preface
Differential and Difference Equations / 1:
Differential Equation Problems / 10:
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