Partial Differential Equations: an Introduction
Authors
Description
The experience of the authors in courses taught at the Institute of Pure and Applied Mathematics (IMPA), the Pontifical Catholic University of Rio de Janeiro (PUC-Rio) and the University of Brasilia enabled them to write and complete this book. The first five chapters introduce the topics of classical and modern Fourier analysis, which are fundamental to the study of partial differential equations. This part can be used in a course at the end of an undergraduate degree or at the beginning of a master’s degree.
The remaining chapters are more advanced. They look at the generalization and extension of ideas introduced for R n. We present a relatively extensive treatment of the eigenvalue problem for the Laplacian in limited domains of Rn using the method of integral equations. Also discussed are the Fredholm alternative, the spectral theorem for self-adjoint compact operators and the theory of Fourier integrals in R n. Topics that are accompanied by various problems and additional information are in the exercises.
Target audience
Higher education
Name: Partial Differential Equations: an Introduction
Author(s): Valéria de Magalhães Iório e Rafael Iório Júnior
Pages: 343
Publication: IMPA, 2018
ISBN: 978-85-244-0456-6
Edition: 3
I Preliminaries
1 Basic Definitions
2 Classification into Types
3 Boundary Conditions and Initial Values
4 Exercises
II The Separation of Variables Method
1 The Problem of Heat Conduction in a Bar
2 Other Examples and Comments
3 Exercises
III Fourier Series: Basic Theory
1 Normed Vector Spaces
2 Fourier Series
3 Geometric Interpretation
4 Decay Properties of f̂
5 Point Convergence
6 The Féjer, Poisson and Dirichlet Kernels
7 Applications
8 The Dirichlet Problem on the Unit Disk
9 Exercises
IV Fourier Series: Periodic Distributions and Applications
1 Periodic Functions of Class C ∞
2 Periodic Distributions
3 Fourier series in P‘
4 Convolution in P‘
5 The L2 space ([-π, π])
6 The D2 operator in L2 ([-π, π])
7 Applications
8 Exercises
V The Fourier Transform on the Straight Line
1 The Heat Equation Strikes Again
2 The Fourier Transform on the Straight Line
3 The Fourier Transform in Schwartz Space
4 Convolution Approximation
5 Temperate Distributions
6 The L2( R) Space
7 The Operator (- d2/dx2) in L2 ( R)
8 Exercises
VI Elements of Functional Analysis
1 Bounded Operators and Compact Operators
2 The Spaces Lp (X, M, μ)
3 The Fredholm Alternative
4 The Spectral Theorem
5 Exercises
VII An Eigenvalue Problem for the Laplacian
1 Preliminaries
2 Green’s Identities
3 The Maximum Principle for Harmonic Functions
4 The Green’s Function
5 Properties of the Green’s Function
6 The Eigenvalue Problem
7 Exercises
VIII The Classical Dirichlet Problem
1 Single and Double Layer Potentials
2 The Solution to the Classical Dirichlet Problem
3 Exercises
IX The Fourier Transform in Rn
1 The Fourier Transform in L1( Rn)
2 The Fourier Transform in Schwartz Space
3 The Fourier Transform in L2 ( Rn)
4 The Laplacian in L2 ( Rn)
5 Temperate Distributions
6 A Topological Parenthesis
7 The Derivative and the Fourier Transform in S‘ ( Rn)
8 Sobolev Spaces in Rn
9 Convolutions, Fundamental Solutions
10 Exercises
References
Index