ALBERT

Note: Contents: Preface. - PART I FUNDAMENTALS. - 1 Ordinary Differential Equations. - 1.1 Ordinary Differential Equations (definitions; introductory examples). - 1.2 Initial-Value and Boundary-Value Problems (definitions; comparison of local and global analysis; examples of initialvalue problems). - 1.3 Theory of Homogeneous Linear Equations (linear dependence and independence; Wronskians; well-posed and ill-posed initial-value and boundary-value problems). - 1.4 Solutions of Homogeneous Linear Equations (how to solve constant-coefficient, equidimensional, and exact equations; reduction of order). - 1.5 Inhomogeneous Linear Equations (first-order equations; Variation of Parameters; Green's functions; delta function; reduction of order; method of undetermined coefficients). - 1.6 First-Order Nonlinear Differential Equations (methods for solving Bernoulli, Riccati, and exact equations; factoring; integrating factors; substitutions). - 1.7 Higher-Order Nonlinear Differential Equations (methods to reduce the order of autonomous, equidimensional, and scale-invariant equations). - 1.8 Eigenvalue Problems (examples of eigenvalue problems on finite and infinite domains). - 1.9 Differential Equations in the Complex Plane (comparison of real and complex differential equations). - Problems for Chapter 1. - 2 Difference Equations. - 2.1 The Calculus of Differences (definitions; parallels between derivatives and differences, integrals, and sums). - 2.2 Elementary Difference Equations (examples of simple linear and nonlinear difference equations; gamma function; general first-order linear homogeneous and inhomogeneous equations). - 2.3 Homogeneous Linear Difference Equations (constant-coefficient equations; linear dependence and independence; Wronskians; initial-value and boundary-value problems; reduction of order; Euler equations; generating functions; eigenvalue problems). - 2.4 Inhomogeneous Linear Difference Equations (Variation of parameters; reduction of order; method of undetermined coefficients). - 2.5 Nonlinear Difference Equations (elementary examples). - Problems for Chapter 2. - PART II LOCAL ANALYSIS. - 3 Approximate Solution of Linear Differential Equations. - 3.1 Classification of Singular Points of Homogeneous Linear Equations (ordinary, regulär singular, and irregulär singular points; survey of the possible kinds of behaviors of Solutions). - 3.2 Local Behavior Near Ordinary Points of Homogeneous Linear Equations (Taylor series Solution of first- and second-order equations; Airy equation). - 3.3 Local Series Expansions About Regulär Singular Points of Homogeneous Linear Equations (methods of Fuchs and Frobenius; modified Bessel equation). - 3.4 Local Behavior at Irregulär Singular Points of Homogeneous Linear Equations (failure of Taylor and Frobenius series; asymptotic relations; Controlling factor and leading behavior; method of dominant balance; asymptotic series expansion of Solutions at irregular singular points). - 3.5 Irregulär Singular Point at Infinity (theory of asymptotic power series; optimal asymptotic approximation; behavior of modified Bessel, parabolic cylinder, and Airy functions for large positive x). - 3.6 Local Analysis of Inhomogeneous Linear Equations (illustrative examples). - 3.7 Asymptotic Relations (asymptotic relations for oscillatory functions; Airy functions and Bessel functions; asymptotic relations in the complex plane; Stokes phenomenon; subdominance). - 3.8 Asymptotic Series (formal theory of asymptotic power series; Stieltjes series and integrals; optimal asymptotic approximations; error estimates; outline of a rigorous theory of the asymptotic behavior of Solutions to differential equations). - Problems for Chapter 3. - 4 Approximate Solution of Noniinear Differential Equations. - 4.1 Spontaneous Singularities (comparison of the behaviors of Solutions to linear and nonlinear equations). - 4.2 Approximate Solutions of First-Order Nonlinear Differential Equations (several examples analyzed in depth). - 4.3 Approximate Solutions to Higher-Order Nonlinear Differential Equations (Thomas-Fermi equation; first Painleve transcendent; other examples). - 4.4 Nonlinear Autonomous Systems (phase-space interpretation; Classification of critical points; one- and two-dimensional phase space). - 4.5 Higher-Order Nonlinear Autonomous Systems (brief, nontechnical survey of properties of higher-order Systems; periodic, almost periodic, and random behavior; Toda lattice, Lorenz model, and other Systems). - Problems for Chapter 4. - 5 Approximate Solution of Difference Equations. - 5.1 Introductory Comments (comparison of the behavior of differential and difference equations). - 5.2 Ordinary and Regular Singular Points of Linear Difference Equations (Classification of n = ∞ as an ordinary, a regular Singular, or an irregular singular point; Taylor and Frobenius series at ∞). - 5.3 Local Behavior Near an Irregulär Singular Point at Infinity: Determination of Controlling Factors (three general methods). - 5.4 Asymptotic Behavior of n! as n → ∞: The Stirling Series [asymptotic behavior of the gamma function Γ(x) as x → ∞ obtained from the difference equations Γ(x + 1) = xΓ(x)]. - 5.5 Local Behavior Near an Irregular Singular Point at Infinity: Full Asymptotic Series (Bessel functions of large order; Legendre polynomials of large degree). - 5.6 Local Behavior of Nonlinear Difference Equations / (Newton's method and other nonlinear difference equations; Statistical analysis of an unstable difference equation). - Problems for Chapter 5. - 6 Asymptotic Expansion of Integrals. - 6.1 Introduction (integral representations of Solutions to difference and differential equations). - 6.2 Elementary Examples (incomplete gamma function; exponential integral; other examples). - 6.3 Integration by Parts (many examples including some where the method fails). - 6.4 Laplace's Method and Watson's Lemma (modified Bessel, parabolic cylinder, and gamma functions; many other illustrative examples). - 6.5 Method of Stationary Phase (leading behavior of integrals with rapidly oscillating integrands). - 6.6 Method of Steepest Descents (steepest ascent and descent paths in the complex plane; saddle points; Stokes phenomenon). - 6.7 Asymptotic Evaluation of Sums (approximation of sums by integrals; Laplace's method for sums; Euler-Maclaurin sum formula). - Problems for Chapter 6. - PART III PERTURBATION METHODS. - 7 Perturbation Series. - 7.1 Perturbation Theory (elementary introduction; application to polynomial equations and initialvalue problems for differential equations). - 7.2 Regular and Singular Perturbation Theory (Classification of perturbation problems as regulär or Singular; introductory examples of boundary-layer, WKB, and multiple-scale problems). - 7.3 Perturbation Methods for Linear Eigenvalue Problems (Rayleigh-Schrödinger perturbation theory). - 7.4 Asymptotic Matching (matched asymptotic expansions; applications to differential equations, eigenvalue problems and integrals). - 7.5 Mathematical Structure of Perturbative Eigenvalue Problems (singularity structure of eigenvalues as functions of complex perturbing Parameter; level crossing). - Problems for Chapter 7. - 8 Summation of Series. - 8.1 Improvement of Convergence (Shanks transformation; Richardson extrapolation; Riemann zeta function). - 8.2 Summation of Divergent Series (Euler, Borel, and generalized Borel summation). - 8.3 Pade Summation (one- and two-point Pade summation; generalized Shanks transformation; many numerical examples). - 8.4 Continued Fractions and Pade Approximants (efficient methods for obtaining and evaluating Pade approximants). - 8.5 Convergence of Pade Approximants (asymptotic analysis of the rate of convergence of Pade approximants). - 8.6 Pade Sequences for Stieltjes Functions (monotonicity; convergence theory; moment problem; Carleman's condition). - Problems for Chapter 8. - PART IV GLOBAL ANALYSIS. - 9 Boundary Layer Theory. - 9.1 Introduction to Boundary-Layer Theo