工程科學中的套用分析

內容簡介

叢書名：中外物理學精品書系

教材主要介紹非線性問題的一些基本數學分析方法和理論，具體包括四部分：<br><br>常微分方程定性理論（包括不動點理論、混沌的簡介）;反應擴散方程（包括攝動法簡介）;橢圓型方程;雙曲型守恆率;套用分析個例（包括孤立子、斑圖選擇簡介）。適合高年級本科生以及研究生閱讀。

1 ODE 10

1.1 Basic notions

1.2 Local existence

1.3 Critical point

1.4 Plane analysis for the Duffing equation

1.5 Homoclinic orbit and limit cycle

1.6 Stability and Lyapunov function

1.7 Bifurcation

1.8 Chaos: Lorenz equations and logistic map

2 Parabolic Equations 53

2.1 Introduction: BVP and IBVP, equilibrium

2.2 Dispersion relation, linear and nonlinear stability

2.3 Invariant domain

2.4 Perturbation method

2.5 Traveling waves

2.6 Burgers' equation and Cole-Hopf transform

2.7 Evolutionary Duffing equation

3 Elliptic Equations 85

3.1 Sobolev spaces

3.2 Variational Formulation

3.3 Neumann boundary value problem

4 Hyperbolic Equations 93

4.1 Linear advection equation, characteristics method

4.2 Nonlinear hyperbolic equations

4.3 Discontinuities in inviscid Burgers' equation

4.4 Elementary waves in inviscid Burgers' equation

4.5 Wave interactions in inviscid Burgers' equation

4.6 Elementary waves in a polytropic gas

4.7 Riemann problem in a polytropic gas

4.8 Elementary waves in a polytropic ideal gas