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| PART 1 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 initial-value 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, equidimansional, 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 seale-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 2 LOCAL ANALYSIS 3 Approximate Solution of Linear Differential Equations 3.1 Classification of Singular Points of Homogeneous Linear Equations(ordinary, regular singular, and irresular singular points; survey of the possible kinds of behaviors of solutions) 3.2 Local Behavior Near Ordinary Points of Momogeneous Linear Equations (Taylor series solution of first- and second-order equations; Airy equation) 3.3 Local Series Expansions About Regular Singular Points of Homogeneous Linear Equations (methods of Fuchs and Frobenius; modified Bessel equation) 3.4 Local Behavior at Irregular Singular Points of Homogeneous Linear Equations(failure of Taylor and Frobemus series; asymptotic relations; controllingfactor and leading behavior; method of dominant balance; asymptotic series expansion of solutions at irregular singular points) …… PART 3 PERTURBATION METHODS PART 4 GLPBAL ANALYSIS Appendix-Useful Formulas References Index |
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