微分方程及边值问题：计算与模型

Application Modules PrefaceCHAPTER 1 First-Order Differential Equations 1.1 Differential Equations and Mathematical Models 1.2 Integrals as General and Paticular Solutions 1.3 Slope Fields and Solution Curves 1.4 Separable Equations and Applications 1.5 Linear First-Order Equations 1.6 Substifution Methods and Exact EquationsCHAPTER 2 Mathematical Models and Numercal Methods 2.1 Population Models 2.2 Equlilibrium Solutions and Stablility 2.3 Acceleration-Velocity Models 2.4 Numerical Approzimation:Euler's Method 2.5 A Closer Look at the Euler Method

Application Modules PrefaceCHAPTER 1 First-Order Differential Equations 1.1 Differential Equations and Mathematical Models 1.2 Integrals as General and Paticular Solutions 1.3 Slope Fields and Solution Curves 1.4 Separable Equations and Applications 1.5 Linear First-Order Equations 1.6 Substifution Methods and Exact EquationsCHAPTER 2 Mathematical Models and Numercal Methods 2.1 Population Models 2.2 Equlilibrium Solutions and Stablility 2.3 Acceleration-Velocity Models 2.4 Numerical Approzimation:Euler's Method 2.5 A Closer Look at the Euler Method 2.6 The Runge-Kutta MethodCHAPTER 3 Linear Equations of Higher Order 3.1 Introduction:Second-Order Linear Equations 3.2 General Solutions of Linear Equations 3.3 Homogeneous Equations with Constant Coelficients 3.4 Mechanical Vibrations 3.5 Nonhomogeneous Equations and Undetermined Coefficients 3.6 Forced Oscillations and Resonance 3.7 Electrical Ciruits 3.8 Endpoint Problems and EigenvaluesCHAPTER 4 Introduction to Systems of Differential Equations 4.1 First-Order Systems and Applications 4.2 The Method of Elimination 4.3 Numerical Methods for SystemsCHAPTER 5 Linear Systems of Differential Equtions 5.1 Matrices and Linear Systems 5.2 The Eigenvalue Method for Homogeneous Systems 5.3 Second-Order Systems and Mechanical Applications 5.4 Multiple Eigenvaluve Solutions 5.5 Matrix Exponentials and Linear Systems 5.6 Nonhomogeneous Linear SystemsCHAPTER 6 Nonlinear Systems and Phenomena ……CHATPER 7 Laplace Transform MethodsCHAPTER 8 Power Series MethodsCHAPTER 9 Fourier Series MethodsCHATPER 10 Eigenvalues and Boundary Value ProblemsReferences for Futher StudyAppendix:Existence and Uniqueness of SolutionsAnswers to Selected ProblemsIndex

本书以一些模型问题为背景，借助于数学软件Maple，Mathematica 及MATLAB，利用符号运算、图像表示和数值解法等手段，系统地介绍了（线性与非线性）微分方程的基本概念和基本方法。通过40多个实际模型的讨论，使读者对建模、求解、分析解所反映的性质这一过程进行全面的了解。利用Maple，Mathematica及MATLAB在图形显示、符号计算、数值计算方面的功能，定性地分析了微分方程解的性质，700余幅图将方向场、解曲线、相平面等概念形象直观地表示出来。另外，书中选配了1900余道习题供读者使用。 本书可供学习数学建模或微分方程的学生作为参考书，对于从事计算与建模的科技人员，本书也具有很高的价值。