分段連續(xù)型延遲微分方程的數(shù)值穩(wěn)定性.pdf

1、本文討論了自變量分段連續(xù)型延遲微分方程的收斂性與數(shù)值穩(wěn)定性這類方程有著極為廣泛的應(yīng)用背景，控制理論、生物醫(yī)學(xué)及物理學(xué)等領(lǐng)域中許多問題都可由它來描述，因此，對這類方程的研究具有重要的意義一類具有代表性的自變量分段連續(xù)型延遲微分方程包含在某些區(qū)間上為常數(shù)的自變量，其解是連續(xù)的、局部光滑的函數(shù)，在這些區(qū)間上滿足方程，在連接任意兩個相鄰區(qū)間的端點上，解的連續(xù)性使它在這些點上存在某種遞推關(guān)系所以，此類方程的解是由一個初始數(shù)據(jù)的有限集來確定，而不像

2、一般的泛函微分方程那樣由初始函數(shù)確定本文研究了EulerMaclaurin方法作用于超前型自變量分段連續(xù)的延遲微分方程的數(shù)值解的收斂階及穩(wěn)定性證明了對于n級EulerMaclaurin方法，超前型自變量分段連續(xù)的延遲微分方程的收斂階為2n2，并得到了數(shù)值解的穩(wěn)定區(qū)域包含解析解穩(wěn)定區(qū)域的條件文中研究了滯后型自變量分段連續(xù)的延遲微分方程的數(shù)值解的收斂性，并證明了數(shù)值方法穩(wěn)定區(qū)域包含解析解穩(wěn)定區(qū)域當(dāng)且僅當(dāng)n為奇數(shù)本文分析了帶有忙期項的超前滯后

3、交替混合型自變量分段連續(xù)的延遲微分方程的數(shù)值穩(wěn)定性，并給出數(shù)值解保持解析解漸近穩(wěn)定的充分必要條件，這與超前型的條件完全不同數(shù)值算例表明本文所給出結(jié)論的正確性關(guān)鍵詞：延遲微分方程；自變量分段連續(xù)；EulerMaclaurin9Yi去；收斂性；穩(wěn)定性AbstractThi8paperdiscussestheconvergenceandthenumericalstabilityofthedelaydifferentialequationswi

4、thpiecewisecontinuousargumentsThi8kindofequationshasextremelyextensiveapplicationbackgroundInsomefieldssuch嬲controltheorybiomedicalsciencesandphysics，manyproblemsCanbedescribedbyitTherefore，Itisimportanttostudythiskindof

5、equationsAtypicaldelaydifferentialequationswithpiecewisecontinuousargumentscontain8argumentsthatareconstantsoncertainintervalsAsolutionisdefinedasacontinuous，sectionallysmoothfunctionthatsatisfiestheequationwithinthesein

6、tervalsContinuityofasolutionatapointjoininganytwoconsecutiveintervalsleadstosomerecursionrelationforthesolutionsatsuchpointsHence，thesolutionsofthiskindofequationsaredeterminedbyafinitesetofinitialdataratherthanbyaniniti

7、alfunctionasinthecaseofgeneralfunctionaldifferentialequationsThethesisinvestigatestheconvergenceandstabiUtyofthenumericalsolutionoftheEulerMaclaurinmethodfortheadvanceddelaydifferentialequationswithpiecewisecontinuousarg

8、umentsItisprovedthattheconvergenceorderofthe址vanceddelaydifferentialequationswithpiecewisecontinuousargumentsisof2n2forthenth—BulerMaclaurinmethod，theconditionsthatthestabilityregionofthenumericalsolutionscontainsthestab

9、ilityreg。onoftheanalyticsolutionsareobtainedTheconvergenceofthenumericalSolutionsoftheretardeddelaydifferentialequationswithpiecewisecontinuousargumentsisdiscussedinthispaperItisprovedthatthestabilityregionofthenumerical

10、methodcontainsthestabilityregionoftheanalyticsolutionsifandonlyif他isoddThethesisanalysesthenumericalstabilityofthealternativelyadvancedandretardedequationswithpiecewisecontinuousargumentswiththetermIt翔TheSU毋cientandneces

11、saryconditionsthatthenumericalsolutionspreservetheasymptoticstabilityoftheanalyticsolutionsaregiven，whichiscompletelydifferenttotheadvancedtypeThecorrectnessoftheconclusionsgiveninthisthesisaredemonstratedbynumericalexam