科技人员用的高等数学方法

Preface

PARTⅠFUNDAMENTALS

OrdinaryDifferentialEquations

1.1OrdinaryDifferentialEquations

(definitions;introductoryexamples)

1.2Initial-ValueandBoundary-ValueProblems

Idefinitions;comparisonoflocalandglobalanalysis;examplesofinitial-

valueproblems)

1.3TheoryofHomogeneousLinearEquations

(lineardependenceandindependence;Wronskians;well-posedandiil-po~ed

initial-valueandboundary-valueproblems)

1.4SolutionsofHomogeneousLinearEquations

(howtosolveconstant-coefficient,equidimensional,andexactequations;

reductionoforder)

1.5InhomogeneousLinearEquations

(first-orderequations;variationofparameters;Greensfunctions;delta

function;reductionoforder;methodofundeterminedcoefficients)

1.6First-OrderNonlinearDifferentialEquations

(methodsforsolvingBernoulli,Riccati,andexactequations;factoring;

integratingfactors;substitutions)

1.7Higher-OrderNonlinearDifferentialEquations

(methodstoreducetheorderofautonomous,equidimensionaL

andscale-invariantequations)

1.8EigenvalueProblems

(examplesofeigenvalueproblemsonfiniteandinfinitedomains)

1.9DifferentialEquationsintheComplexPlane

(comparisonofrealandcomplexdifferentialequations)

ProblemsforChapter1

2DifferenceEquations

2.1TheCalculusofDifferences

(definitions;parallelsbetweenderivativesanddifferences,integrals,andsums)

2.2ElementaryDifferenceEquations

(examplesofsimplelinearandnonlineardifferenceequations;gamma

function;generalfirst-orderlinearhomogeneousandinhomogeneous

equations)

2.3HomogeneousLinearDifferenceEquations

(constant-coefficientequations;lineardependenceandindependence;

Wronskians;initial-valueandboundary-valueproblems;reductionoforder;

Eulerequations;generatingfunctions;eigenvalueproblems)

2.4InhomogeneousLinearDifferenceEquations

(variationofparameters;reductionoforder;methodofundetermined

coefficients)

2.5NonlinearDifferenceEquations

(elementaryexamples)

ProblemsforChapter2

PARTⅡLOCALANALYSIS

3ApproximateSolutionofLinearDifferentialEquations

3.1ClassificationofSingularPointsofHomogeneousLinearEquations

(ordinary,regularsingular,andirregularsingularpoints;surveyofthe

possiblekindsofbehaviorsofsolutions)

3.2LocalBehaviorNearOrdinaryPointsofHomogeneousLinear

Equations

(Taylorseriessolutionoffirst-andsecond-orderequations;Airyequation)

3.3LocalSeriesExpansionsAboutRegularSingularPointsof

HomogeneousLinearEquations

(methodsofFuchsandFrobenius;modifiedBesselequation)

3.4LocalBehavioratIrregularSingularPointsofHomogeneous

LinearEquations

(failureofTaylorandFrobeniusseries;asymptoticrelations;controlling

factorandleadingbehavior;methodofdominantbalance;asymptotic

seriesexpansionofsolutionsatirregularsingularpoints)

……

4ApproximateSolutionofNonlinearDifferentialEquations

5ApproximateSolutionofDifferenceEquations

6AsymptoticExpansionofIntegrals

PARTⅢPERTURBATIONMETHODS

7PerturbationSeries

8SummationofSeries

PARTⅣGLOBALANALYSIS

9BoundaryLayerTheory

10WKBTheory

11Multiple-ScaleAnalysis

Appendix--UsefulFormulas

References

Index

　　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。