| 《偏微分方程与数值方法》的作者Stig Larss《常微分方程的解法I：非刚性问题(第二版)》主要论述非刚性常微分方程。第一章介绍自牛顿、莱布尼兹、欧拉和哈密尔顿以来经典理论的历史发展，极限环及奇异吸引子。第二章用现代观念阐述龙格库塔方法和外插法，并讨论稠密输出的连续方法、并行龙格库塔方法、哈密尔顿系统的特殊方法、二阶常微分方程和时滞方程。第三章从多步方法的古典理论开始，论述变步长方法和Nordsieck方法及一般线性方法的理论。 本书包括非刚性问题在物理、化学、生物和天文中的应用，计算机程序及数值比较。 第二版中重写了某些章节，增加了新的内容。 on现任瑞典ChaImers大学数学系教授、瑞典科学院院士。本书将微分方程的数学分析及有限差分理论和有限元方法结合起来，讲述线性偏微分方程的基本理论及其常用的数值解法。分别用三章阐述椭圆型、抛物型及双曲型偏微分方程，一章关于其数学理论，一章关于其有限差分方法，一章关于其有限元方法。在论述椭圆型方程之前，讲述常微分方程的两点边值问题；类似地，在论述抛物型和双曲型发展问题之前，讲述常微分方程的初值问题。另有一章研究椭圆型特征值问题和特征函数的展开。附录提供了阅读本书所要求的线性泛函分析及索伯列夫空间的背景知识。阅读本书不需要高深的数学分析和泛函分析知识。本书适用于应用数学专业和工程专业的高年级本科生和低年级研究生。 |
| Chapter I. Classical Mathematical Theory I.1 Terminology I.2 The Oldest Differential Equations I.3 Elementary Integration Methods I.4 Linear Differential Equations I.5 Equations with Weak Singularities I.6 Systme of Equations I.7 A General Existence Theorem I.8 Existence Theory using Iteration Methods and Taylor Series I.9 Existence Theory for Systems of Equations I.10 Differential Inequalities I.11 Systems of Linear Differential Equations I.12 Systmes with Constant Coefficients I.13 Stability I.14 Derivatives with Respect ot Parameters and Initial Values I.15 boundary Value and Eigenvalue Problems I.16 Periodic Solutions， Limit Cycles， Strange Attractors Chapter II. Runge-Kutta and Extrapolation Methods II.1 The First Runge-Kutta Methods II.2 Order Conditions for Runge-Kutta Methods II.3 Error Estimation and Convergence for RK Methods II.4 Practical Error Estimation and Step Size Selection II.5 Explicit Runge-Kutta Methods of Higher Order II.6 Dense Output， Discontinuities， Derviatives II.7 Implicit Runge-Kutta Methods II.8 Asymptotic Expansion of the Golbal Error II.9 Extrapolation Methods II.10 Numerical Comparisons II.11 Parallel Methods II.12 Composition of B-Series II.13 Higher Derivative Methods II.14 Numerical Methods for Second Order Differential Equations II.15 P-Series for Partitioned Differential Equations II.16 Symplectic Integration Methods II.17 Delay Differential Equations Chapter III. Multistep Methods and General Linear Methods III.1 Classical Linear Multistep Formulas III.2 Local Error and Order Conditions III.3 Stability and the First Dahlquist Barrier III.4 Convergence of Multistep Methods III.5 Variable Step Size Multistep Muthods III.6 Nordisieck Methods III.7 Implementation and Numerical Comparisons III.8 General Linear Methods III.9 Asymptotic Expansion of the Global Error III.10 Multistep Methods for Second Order Differential Equations Appendix. Fortran Codes Bibliography Symbol Index Subject Index |
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