第五章 差分方程模型

这可让他犯了难，施工现场距离项目部很远，没有车还真是不方便office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)customercreditinformationarenotthoroughverification.(Iii)toparticipateincreditcustomersurveyisnotinplace,customersanddataisincomplete,untrue;heknowsbearacounterfeitedclientsissuingcredit.(D)doesnotprovideforduediligenceofcreditbusiness,pre-loaninvestigationform,concealingfactsorprovidingfalseinformationorshouldbefoundinanormalinvestigationfailedtodiscovertheriskfactors,leadtothereviewandapprovalpolicyerrors,loanrisk.(Five)onloanguaranteesofsurveynotinplace,notbyprovidesonarrived,andpledgerealforfieldverification,andassessment,andidentificationandregistration,notaccordingtoprovidesonguarantorofguaranteesqualificationandguaranteescapacityforsurveyverified,ledtoguaranteesloanlostauthenticity,andlegitimacy,andeffectivenessof;cycleloanbusinessintheofmortgage第五章差分方程模型在第四章中，我们利用微分方程方法研究了一些连续变化的变量。如果将变量离散化，即可得到相应的差分方程模型，为了方便不熟悉差分方程的读者，先对本章用到的差分方程的知识作一简略介绍。5.1差分方程简介一、差分方程及其通解以t表示时间，规定t只取非负整数。0t表示第一周期初，1t表示第二周期初等。记ty为变量y在时刻t时的取值，则称tttyyy1为的一阶差分，称tttttttyyyyyyy12122)(为yt的二阶差分。类似地，可以定义yt的n阶差分tny。由t、ty及ty的差分给出的方程称为ty差分方程，其中含ty的最高阶差分的阶数称为该差分方程的阶。差分方程也可以写成不显含差分的形式。例如，二阶差分方程02tttyyy也可改写成012tttyyy。满足差分方程的序列ty称为此差分方程的解。类似于微分方程情况，若解中含有的独立常数的个数等于差分方程的阶数时，则称此解为该差分方程的通解。若解中不含任意常数，则称此解为满足某些初值条件的特解，例如，考察两阶差分方程：02ttyy易见tyt2sin与tyt2cos均是它的特解，而tctcyt2cos2sin21则为它的通解，其中1c，2c为两个任意常数。类似于微分方程，称差分方程)()()()(110tbytaytaytatnntnt（1）为n阶线性差分方程，当0)(tb时称其为n阶非齐次线性差分方程，而0)()()(110tnntntytaytayta（2）称为方程（1）对应的齐次线性差分方程。若（1）中所有的)(tai均为与t无关的常数，则称其为常系数差分方程，即n阶常系数线性差分方程可写成这可让他犯了难，施工现场距离项目部很远，没有车还真是不方便office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)customercreditinformationarenotthoroughverification.(Iii)toparticipateincreditcustomersurveyisnotinplace,customersanddataisincomplete,untrue;heknowsbearacounterfeitedclientsissuingcredit.(D)doesnotprovideforduediligenceofcreditbusiness,pre-loaninvestigationform,concealingfactsorprovidingfalseinformationorshouldbefoundinanormalinvestigationfailedtodiscovertheriskfactors,leadtothereviewandapprovalpolicyerrors,loanrisk.(Five)onloanguaranteesofsurveynotinplace,notbyprovidesonarrived,andpledgerealforfieldverification,andassessment,andidentificationandregistration,notaccordingtoprovidesonguarantorofguaranteesqualificationandguaranteescapacityforsurveyverified,ledtoguaranteesloanlostauthenticity,andlegitimacy,andeffectivenessof;cycleloanbusinessintheofmortgage101)(110tbyayayatnntnt（3）其对应的齐次方程为0110tnntntyayaya（4）容易证明，若序列)1(ty与)2(ty均为方程（4）的解，则)2(2)1(1tttycycy也是方程（4）的解，其中1c，2c为任意常数，这说明，齐次方程的解构成一个线性空间（解空间）。若)1(ty是方程（4）的解，)2(ty是方程（3）的解，则)2()1(tttyyy也是方程（3）的解。方程（3）可用如下的代数方法求其通解：（步一）先求解对应的特征方程0110nnnaaa（5）（步二）根据特征根的不同情况，求齐次方程（4）的通解。情况1若特征方程（5）有n个互不相同的实根1，2，…，n，则齐次方程（4）的通解为tnnttccc2211(1c，2c，…，nc为任意常数)情况2若是特征方程（5）的k重根，通解中对应于的项为tkktctcc)(121，ic（ki,,2,1）为任意常数。情况3若特征方程（5）有单重复根i，通解中对应于的项为，tctcttsincos21，其中1c，2c为任意常数，22为的模，arctan为的幅角。情况4若i为特征方程（5）的k重复根，则通解对应于的项为ttccttcctkkktkksin)(cos)(12111，ic（ki2,,2,1）为任意常数。（步三）求非齐次方程（3）的一个特解ty。若ty为方程（4）的通解，则非齐次方程（3）的通解为ttyy。求非齐次方程（3）的特解一般要用到常数变易法，计算较繁。对特殊形式的b(t)也可使用待定系数法。例如，当)()(tpbtbkt，)(tpk为t的k次多项式时可以证明：若b不是这可让他犯了难，施工现场距离项目部很远，没有车还真是不方便office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)customercreditinformationarenotthoroughverification.(Iii)toparticipateincreditcustomersurveyisnotinplace,customersanddataisincomplete,untrue;heknowsbearacounterfeitedclientsissuingcredit.(D)doesnotprovideforduediligenceofcreditbusiness,pre-loaninvestigationform,concealingfactsorprovidingfalseinformationorshouldbefoundinanormalinvestigationfailedtodiscovertheriskfactors,leadtothereviewandapprovalpolicyerrors,loanrisk.(Five)onloanguaranteesofsurveynotinplace,notbyprovidesonarrived,andpledgerealforfieldverification,andassessment,andidentificationandregi