常微分方程

內(nèi)容概要

本書是在1996年出版的《常微分方程》（德文）一書的基礎(chǔ)上編寫而成的，書中主要介紹常微分方程的基礎(chǔ)理論。內(nèi)容包括：可積一階微分方程，微分方程解的存在性和唯一性，微分方程的初極值問(wèn)題，邊值問(wèn)題和特征值問(wèn)題，穩(wěn)定性與漸進(jìn)穩(wěn)定性理論。閱讀本書需要具備一定的計(jì)算代數(shù)、線性代數(shù)及泛函分析的基礎(chǔ)知識(shí)。適用于高校數(shù)學(xué)專業(yè)、牧業(yè)專業(yè)和計(jì)算機(jī)科學(xué)等相關(guān)專業(yè)的本科生和研究生。

書籍目錄

Preface Note to the Reader Introduction Chapter I. First Order Equations: Some Integrable Cases 　1. Explicit First Order Equations 　2. The Linear Differential Equation. Related Equations 　　　Supplement: The Generalized Logistic Equation 　3. Dif　ferential Equations for Families of Curves. Exact Equations 　4. Implicit First Order Differential Equations Chapter II: Theory of First Order Differential Equations 　5. Tools from Functional Analysis 　6. An Existence and Uniqueness Theorem 　　Supplement: Singular Initial Value Problems 　7. The Peano Existence Theorem 　　Supplement: Methods of Functional Analysis 　8. Complex Differential Equations. Power Series Expansions 　9. Upper and Lower Solutions. Maximal and Minimal Integrals 　　Supplement: The Separatrix Chapter III: First Order Systems. Equations of Higher Order 　10. The Initial Value Problem for a System of First Order　　Supplement I: Differential Inequalities and Invarian e　　Supplement II: Differential Equations in the Senseof  aratheodory　11. Initial Value Problems for Equations of Higher Order　　Supplement: Second Order Differential Inequalities　12. Continuous Dependence of Solutions　　Supplement: General Uniqueness and Dependence Theorems　13. Dependence of Solutions on Initial Values and ParametersChapter IV: Linear Differential Equations　14. Linear Systems　15. Homogeneous Linear Systems　16. Inhomogeneous Systems　　Supplement: L1-Estimation of  C-Solutions　17. Systems with  Constant  Coefficients　18. Matrix Functions. Inhomogeneous Systems　　Supplement: Floquet Theory　19. Linear Differential Equations of Order n　20. Linear Equations of Order n with  Constant  Coefficients　　Supplement: Linear Differential Equations with　Periodic  CoefficientsChapter V:  Complex Linear Systems　21. Homogeneous Linear Systems in the Regular  Case　22. Isolated Singularities　23. Weakly Singular Points. Equations of Fuchsian Type　24. Series Expansion of Solutions　25. Second Order Linear EquationsChapter VI: Boundary Value and Eigenvalue Problems　26. Boundary Value Problems　　Supplement I: Maximum and Minimum Prin iples　　Supplement II: Nonlinear Boundary Value Problems　27. The Sturm-Liouville Eigenvalue Problem　　Supplement: Rotation-Symmetri  Ellipti  Problems　28. Compact Self-Adjoint Operators in Hilbert SpaceChapter VII: Stability and Asymptotic Behavior　29. Stability　30. The Method of LyapunovAppendix　A. Topology　B. Real Analysis　C. Complex Analysis　D. Fun tional AnalysisSolutions and Hints for Selected ExercisesLiteratureIndexNotation

媒體關(guān)注與評(píng)論

書評(píng)The author's book on GewShnliche Differentialgleichungen (Ordinary Differential Equations) was published in 1972. The present book is based on a translation of the latest, 6th, edition, which appeared in 1996, but it also treats some important subjects that are not found there. The German book is widely used as a textbook for a first course in ordinary differential equations. This is a rigorous course, and it contains some material that is more difficult than that usually found in a first course textbook; such as, for example, Peano's existence theorem. It is addressed to students of mathematics, physics, and computer science and is usually taken in the third semester. Let me remark here that in the German system the student learns calculus of one variable at the gymnasium1 and begins at the university with a two-semester course on real analysis which is usually followed by ordinary differential equations.

編輯推薦

Theauthor'sbookonGewShnlicheDifferentialgleichungen(OrdinaryDifferentialEquations)waspublishedin1972.Thepresentbookisbasedonatranslationofthelatest,6th,edition,whichappearedin1996,butitalsotreatssomeimportantsubjectsthatarenotfoundthere.TheGermanbookiswidelyusedasatextbookforafirstcourseinordinarydifferentialequations.Thisisarigorouscourse,anditcontainssomematerialthatismoredifficultthanthatusuallyfoundinafirstcoursetextbook;suchas,forexample,Peano'sexistencetheorem.Itisaddressedtostudentsofmathematics,physics,andcomputerscienceandisusuallytakeninthethirdsemester.LetmeremarkherethatintheGermansystemthestudentlearnscalculusofonevariableatthegymnasium1andbeginsattheuniversitywithatwo-semestercourseonrealanalysiswhichisusuallyfollowedbyordinarydifferentialequations.

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