Techniques of Functional Analysis for Differential and Integral Equations 1st Edition by Paul Sacks 0128114576 9780128114575

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ISBN-10 :  0128114576 

ISBN-13 :  9780128114575

Author :  Paul Sacks 

Techniques of Functional Analysis for Differential and Integral Equations describes a variety of powerful and modern tools from mathematical analysis, for graduate study and further research in ordinary differential equations, integral equations and partial differential equations. Knowledge of these techniques is particularly useful as preparation for graduate courses and PhD research in differential equations and numerical analysis, and more specialized topics such as fluid dynamics and control theory. Striking a balance between mathematical depth and accessibility, proofs involving more technical aspects of measure and integration theory are avoided, but clear statements and precise alternative references are given . The work provides many examples and exercises drawn from the literature.

Techniques of Functional Analysis for Differential and Integral Equations 1st Table of contents:

Chapter 1: Some Basic Discussion of Differential and Integral Equations
Abstract
1.1 Ordinary Differential Equations
1.2 Integral Equations
1.3 Partial Differential Equations
1.4 Well-Posed and Ill-Posed Problems
Chapter 2: Vector Spaces
Abstract
2.1 Axioms of a Vector Space
2.2 Linear Independence and Bases
2.3 Linear Transformations of a Vector Space
Chapter 3: Metric Spaces
Abstract
3.1 Axioms of a Metric Space
3.2 Topological Concepts
3.3 Functions on Metric Spaces and Continuity
3.4 Compactness and Optimization
3.5 Contraction Mapping Theorem
Chapter 4: Banach Spaces
Abstract
4.1 Axioms of a Normed Linear Space
4.2 Infinite Series
4.3 Linear Operators and Functionals
4.4 Contraction Mappings in a Banach Space
Chapter 5: Hilbert Spaces
Abstract
5.1 Axioms of an Inner Product Space
5.2 Norm in a Hilbert Space
5.3 Orthogonality
5.4 Projections
5.5 Gram-Schmidt Method
5.6 Bessel’s Inequality and Infinite Orthogonal Sequences
5.7 Characterization of a Basis of a Hilbert Space
5.8 Isomorphisms of a Hilbert Space
Chapter 6: Distribution Spaces
Abstract
6.1 The Space of Test Functions
6.2 The Space of Distributions
6.3 Algebra and Calculus With Distributions
6.4 Convolution and Distributions
Chapter 7: Fourier Analysis
Abstract
7.1 Fourier Series in One Space Dimension
7.2 Alternative Forms of Fourier Series
7.3 More About Convergence of Fourier Series
7.4 The Fourier Transform on RN
7.5 Further Properties of the Fourier Transform
7.6 Fourier Series of Distributions
7.7 Fourier Transforms of Distributions
Chapter 8: Distributions and Differential Equations
Abstract
8.1 Weak Derivatives and Sobolev Spaces
8.2 Differential Equations in D′
8.3 Fundamental Solutions
8.4 Fundamental Solutions and the Fourier Transform
8.5 Fundamental Solutions for Some Important PDEs
Chapter 9: Linear Operators
Abstract
9.1 Linear Mappings Between Banach Spaces
9.2 Examples of Linear Operators
9.3 Linear Operator Equations
9.4 The Adjoint Operator
9.5 Examples of Adjoints
9.6 Conditions for Solvability of Linear Operator Equations
9.7 Fredholm Operators and the Fredholm Alternative
9.8 Convergence of Operators
Chapter 10: Unbounded Operators
Abstract
10.1 General Aspects of Unbounded Linear Operators
10.2 The Adjoint of an Unbounded Linear Operator
10.3 Extensions of Symmetric Operators
Chapter 11: Spectrum of an Operator
Abstract
11.1 Resolvent and Spectrum of a Linear Operator
11.2 Examples of Operators and Their Spectra
11.3 Properties of Spectra
Chapter 12: Compact Operators
Abstract
12.1 Compact Operators
12.2 Riesz-Schauder Theory
12.3 The Case of Self-Adjoint Compact Operators
12.4 Some Properties of Eigenvalues
12.5 Singular Value Decomposition and Normal Operators
Chapter 13: Spectra and Green’s Functions for Differential Operators
Abstract
13.1 Green’s Functions for Second-Order ODEs
13.2 Adjoint Problems
13.3 Sturm-Liouville Theory
13.4 Laplacian With Homogeneous Dirichlet Boundary Conditions
Chapter 14: Further Study of Integral Equations
Abstract
14.1 Singular Integral Operators
14.2 Layer Potentials
14.3 Convolution Equations
14.4 Wiener-Hopf Technique
Chapter 15: Variational Methods
Abstract
15.1 The Dirichlet Quotient
15.2 Eigenvalue Approximation
15.3 The Euler-Lagrange Equation
15.4 Variational Methods for Elliptic Boundary Value Problems
15.5 Other Problems in the Calculus of Variations
15.6 The Existence of Minimizers
15.7 Calculus in Banach Spaces
Chapter 16: Weak Solutions of Partial Differential Equations
Abstract
16.1 Lax-Milgram Theorem
16.2 More Function Spaces
16.3 Galerkin’s Method
16.4 Introduction to Linear Semigroup Theory

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