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内容简介
Thisbookhasevolvedfromatwo-termgraduatecourseinpartialdifferentialequationswhichIhavetaughtatNortheasternUniversitymanytimessince1980.Thefirsttermisintendedtogivethestudentabasicandclassicalintroductiontothesubject,includingfirst-orderequationsbythemethodofcharacteristicsandlinearsecond-orderequationswhichariseinmathematicalphysics:thewaveequation,Laplaceequation,andheatequation.Allofthismaterialismorethanadequatelycoveredbymanytextbookswhicharereadilyavailable.Thesecondterm,however,isintendedtointroducethestudenttoawidevarietyofmoremodernmethods,especiallytheuseoffunctionalanalysis,whichhascharacterizedmuchoftherecentdevelopmentofpartialdifferentialequations.Thislattermaterialisnotasreadilyavailable,exceptinanumberofspecializedreferencebooks.Thistextbookisintendedtobridgethisgapbyprovidingthestudentwithabasicintroductiontothesubjectandanexposuretosomeofthemoremodernmethods.
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目录
Chapter 1 First\|Order Equations
1.1 The Cauchy Problem for Quasilinear Equations
1.2 Weak Solutions for Quasilinear Equations
1.3 General Nonlinear Equations
1.4 Concluding remarks on First-Order Equations
Chapter 2 Principles for Higher\|Order Equations
2.1 Trhe Cauchy Problem
2.2 Second-Order Equations in Two Varibales
2.3 Linear Equations and Generalized Solutions
Chapter 3 The Wave Equation
3.1 The One-Dimensional Wave Equation
3.2 Higher Dimensions
3.3 Energy Methods
3.4 Lower-order Terms
Chapter 4 The Laplace Equation
4.1 Introudiction to the Laplace Equation
4.2 Potential Theory and Green''''''''s Functions
4.3 General Existence Theory
4.4 Eigenvalues of the Laplacian
Chapter 5 The Heat Equation
5.1 The Heat Equation in a Bounded Domain
5.2 The Pure Initial Value Problem
5.3 Regularity and Similarity
Chapter 6 Linear Functional Analysis
6.1 Function Spaces and Linear Operators
6.2 Application to the Dirichlet Problem
6.3 Duality and Compactness
6.4 Sobolev Imbedding Theorems
6.5 Generalizations and Refinements
Chapter 7 Differential Calculus Methods
……
Chapter 8 Linear Elliptic Theory
Chapter 9 Two Additional Methods
Chapter10 Systems of Conservation Laws
Chapter11 Linear and Nonlinear Diffusion
Chapter12 Linear and Nonlinear Waves
Chapter13 Nonlinear Elliptic Equations
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