差分方程导论

出版社:世界图书出版公司
出版日期:2011-4
ISBN:9787510033070
作者:埃莱迪
页数:539页

书籍目录

preface to the third edition
preface to the second edition
preface to the first edition
list of symbols
1 dynamics of first-order difference equations
1.1 introduction
1.2 linear first-order difference equations
1.2.1 important special cases
1.3 equilibrium points
1.3.1 the stair step (cobweb) diagrams
1.3.2 the cobweb theorem of economics
1.4 numerical solutions of differential equations
1.4.1 euler's method
1.4.2 a nonstandard scheme
1.5 criterion for the asymptotic stability of equilibrium
points
1.6 periodic points and cycles
1.7 the logistic equation and bifurcation
1.7.1 equilibrium points
1.7.2 2-cycles
1.7.3 22-cycles
1.7.4 the bifurcation diagram
1.8 basin of attraction and global stability (optional)
2 linear difference equations of higher order
2.1 difference calculus
2.1.1 the power shift
2.1.2 factorial polynomials
2.1.3 the antidifference operator
2.2 general theory of linear difference equations
2.3 linear homogeneous equations with constant coefficients
2.4 nonhomogeneous equations: methods of undetermind
coefficeints
2.4.1 the method of variation of constants (parameters)
2.5 limiting behavior of solutions
2.6 nonlinear equations transformable to linear equations
2.7 applications
2.7.1 propagation of annual plants
2.7.2 gambler's ruin
2.7.3 national income
2.7.4 the transmission of information
3 systems of linear difference equations
3.1 autonomous (time-invariant) systems
3.1.1 the discrete analogue of the putzer algorithm.
3.1.2 the development of the algorithm for an
3.2 the basic theory
3.3 the jordan form: autonomous (time-invariant) systems
revisited
3.3.1 diagonalizable matrices
3.3.2 the jordan form
3.3.3 block-diagonal matrices
3.4 linear periodic systems
3.5 applications
3.5.1 markov chains
3.5.2 regular markov chains
3.5.3 absorbing markov chains
3.5.4 a trade model
3.5.5 the heat equation
4 stability theory
4.1 a norm of a matrix
4.2 notions of stability
4.3 stability of linear systems
4.3.1 nonautonomous linear systems
4.3.2 autonomous linear systems
4.4 phase space analysis
4.5 liapunov's direct, or second, method
4.6 stability by linear approximation
4.7 applications
4.7.1 one species with two age classes
4.7.2 host-parasitoid systems
4.7.3 a business cycle model
4.7.4 the nicholson-bailey model
4.7.5 the flour beetle case study
5 higher-order scalar difference equations
5.1 linear scalar equations
5.2 sufficient conditions for stability
5.3 stability via linearization
5.4 global stability of nonlinear equations
5.5 applications
5.5.1 flour beetles
5.5.2 a mosquito model
6 the z-transform method and volterra difference equations
6.1 definitions and examples
6.1.1 properties of the z-transform
6.2 the inverse z-transform and solutions of difference
equations
6.2.1 the power series method
6.2.2 the partial fractions method
6.2.3 the inversion integral method
6.3 volterra difference equations of convolution type: the scalar
case
6.4 explicit criteria for stability of volterra equations
6.5 volterra systems
6.6 a variation of constants formula
6.7 the z-transform versus the laplace transform
7 oscillation theory
7.1 three-term difference equations
7.2 self-adjoint second-order equations
7.3 nonlinear difference equations
8 asymptotic behavior of difference equations
8.1 tools of approximation
8.2 poincare's theorem
8.2.1 infinite products and perron's example
8.3 asymptotically diagonal systems
8.4. high-order difference equations
8.5 second-order difference equations
8.5.1 a generalization of the poincare-perron theorem.
8.6 birkhoff's theorem
8.7 nonlinear difference equations
8.8 extensions of the poincare and perron theorems
8.8.1 an extension of perron's second theorem
8.8.2 poincare's theorem revisited
9 applications to continued fractions and orthogonal
polynomials
9.1 continued fractions: fundamental recurrence formula
9.2 convergence of continued fractions
9.3 continued fractions and infinite series
9.4 classical orthogonal polynomials
9.5 the fundamental recurrence formula for orthogonal
polynomials
9.6 minimal solutions, continued fractions, and orthogonal
polynomials
10 control theory
10.1 introduction
10.1.1 discrete equivalents for continuous systems
10.2 controllability
10.2.1 controllability canonical forms
10.30bservability
10.3.10bservability canonical forms
10.4 stabilization by state feedback (design via pole
placement)
10.4.1 stabilization of nonlinear systems by feedback
10.5 observers
10.5.1 eigenvalue separation theorem
a stability of nonhyperboli fixed points of maps on the real
line
a.1 local stability of nonoscillatory nonhyperbolic maps
a.2 local stability of oscillatory nonhyperbolic maps
a.2.1 results with g(x)
b the vandermonde matrix
c stability of nondifferentiable maps
d stable manifold and the hartman-grobman-cushing theorems
d.1 the stable manifold theorem
d.2 the hartman-grobman-cushing theorem
e the levin-may theorem
f classical orthogonal polynomials
g identities and formulas
answers and hints to selected problems
maple programs
references
index

作者简介

《差分方程导论(英文版)(第3版)》是一本学习差分方程的本科生教程。书中将差分方程的经典方法和现代方法有机结合,包括了最新最权威的一手材料,并且在表述上足够简洁明了,适合高年级的本科生和研究生使用。《差分方程导论(英文版)(第3版)》是第三版,这版中包括了更多的证明,图表和应用,增加了许多新的内容,如,讲述高阶尺度差分方程的一章;有关一维映射的局部稳定性和全局稳定性的内容;介绍解的渐进思想的一节;levin-may定理的详细证明以及lap flour-beetle模型的最新结果。
读者对象:数学专业的本科生,研究生和相关的科研人员。

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精彩书评 (总计1条)

  •     在学习ARIMA模型时,发现几乎所有的教材都只是简略的介绍了差分算子、延时算子,以及两者之间的联系。而后在推导稳定域的时候,只是给出了充分条件,并没有证明充要性。接着在导出Green函数时,读者便无法理解它的由来。这时,回归到最开始的两个算子,发现有必要系统的了解一下差分方程的知识。该书第一章讲述了稳定性理论,与微分方程中稳定性理论,以及经济学的蛛网模型,数值分析的收敛性的判断有联系。我主要参考了第二章的内容。第二章是讲高阶差分方程,对于时间序列的初学者,会颇有感觉。差分算子可以类比微分算子,逆差分算子就是积分,进而类比着建立起一套运算规则。书中的例题让人觉得很亲切,当意识到ARIMA模型和高中的递推数列在模式上是一样的,(只不过增加了递推阶数,变量变成了随机变量,因而在处理方法上发生了改变),那些晦涩的符号就不再晦涩。

精彩短评 (总计12条)

  •     内容不错,浅显易懂,不过这排版真难看,是不是世界图书出版的都这样,之前一本微分几何也是……
  •     没仔细看,代替饭了一下,感觉还可以吧
  •     有利于学习经济或金融
  •     很不错,期待还有这样的好书
  •     比较全面系统地介绍差分方程,不过需要一定英语基础的人才能弄懂
  •     因为偶尔发现在这方面还有很多地方可以做,所以买一本补充基本知识。
  •     数学越是到了高深,就越要多看看国外的著作,较至国内水平绝对一流,学习差分值得一看
  •     书挺不错的,在国内很难看到这样的书
  •     非常经典的差分方程教材,英文版的
  •     关于差分的入门书不多,这本很不错。
  •     专门介绍差分方程的书很少,本书是本很好的教材,对差分方程做了比较全面深入的介绍,习题数量较多,属于入门级别的书。
  •     书大概看了十几页,感觉还可以,外国人写书与中国人的思维不一样
 

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