微分方程及邊值問題

內(nèi)容概要

本書以一些模型問題為背景，借助于數(shù)學(xué)軟件Maple，Mathematica 及MATLAB，利用符號運算、圖像表示和數(shù)值解法等手段，系統(tǒng)地介紹了（線性與非線性）微分方程的基本概念和基本方法。通過40多個實際模型的討論，使讀者對建模、求解、分析解所反映的性質(zhì)這一過程進(jìn)行全面的了解。利用Maple，Mathematica及MATLAB在圖形顯示、符號計算、數(shù)值計算方面的功能，定性地分析了微分方程解的性質(zhì)，700余幅圖將方向場、解曲線、相平面等概念形象直觀地表示出來。另外，書中選配了1900余道習(xí)題供讀者使用。    本書可供學(xué)習(xí)數(shù)學(xué)建模或微分方程的學(xué)生作為參考書，對于從事計算與建模的科技人員，本書也具有很高的價值。

書籍目錄

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

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