本书是一部给工程人员和科技工作者介绍数学工具和方法的书(全英文版),其直接目的是讲述对问题的深层次的认识或者解决问题的能力，这些技巧对于解决那些在实践科研的学习过程中所遇到的疑难数学问题是很有用的。

本书是从对微分方程和差分方程预习的讲述开始的，层层深入，进一步讲述了微分方程和差分方程的局部逼近方法，重点结合实际详细介绍了渐近方法和扰动法理论。并且运用这些方法获得了微分方程或者差分方程的近似解析解。这些方法可以很好的帮助读者解决物理和工程中的不可能用其它的方法解决的实际问题或者那些数值方法不能够收敛于真正的方程的。以全局逼近方法的介绍而结束，包括边界层理论，WKb理论和多尺度分析。书中的600多道练习题以及用于对特殊函数性质的总结的附录都是很值得认真阅读的。

Preface

PART Ⅰ FUNDAMENTALS

Ordinary Differential Equations

 1.1 Ordinary Differential Equations

(definitions; introductory examples)

 1.2 Initial-Value and Boundary-Value Problems

Idefinitions; 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 iil-po~ed

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, regular singular, and irregular 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 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 Frobenius series; asymptotic relations; controlling

factor and leading behavior; method of dominant balance; asymptotic

series expansion of solutions at irregular singular points)

 ……

4 Approximate Solution of Nonlinear Differential Equations

5 Approximate Solution of Difference Equations

6 Asymptotic Expansion of Integrals

PART Ⅲ PERTURBATION METHODS

7 Perturbation Series

8 Summation of Series

PART Ⅳ GLOBAL ANALYSIS

9 Boundary Layer Theory

10 WKB Theory

11 Multiple-Scale Analysis

Appendix--Useful Formulas

References

Index