Techniques of Functional Analysis for Differential and Integral Equations

✍ Scribed by Paul Sacks

• Publisher: Academic Press;Elsevier
• Year: 2017
• Tongue: English
• Leaves: 310
• Series: Mathematics in Science and Engineering
• Edition: 1
• Category: Library

✦ Synopsis

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.

• Provides an introduction to mathematical techniques widely used in applied mathematics and needed for advanced research in ordinary and partial differential equations, integral equations, numerical analysis, fluid dynamics and other areas
• Establishes the advanced background needed for sophisticated literature review and research in differential equations and integral equations
• Suitable for use as a textbook for a two semester graduate level course for M.S. and Ph.D. students in Mathematics and Applied Mathematics

✦ Table of Contents

Content: Front Cover
Techniques of Functional Analysis for Differential and Integral Equations
Copyright
Contents
Preface
Chapter 1: Some Basic Discussion of Differential and Integral Equations
1.1 Ordinary Differential Equations
1.1.1 Initial Value Problems
1.1.2 Boundary Value Problems
1.1.3 Some Exactly Solvable Cases
1.2 Integral Equations
1.3 Partial Differential Equations
1.3.1 First Order PDEs and the Method of Characteristics
1.3.2 Second Order Problems in R2
1.3.3 Further Discussion of Model Problems
Wave Equation
Heat Equation
Laplace Equation 1.3.4 Standard Problems and Side Conditions1.4 Well-Posed and Ill-Posed Problems
1.5 Exercises
Chapter 2: Vector Spaces
2.1 Axioms of a Vector Space
2.2 Linear Independence and Bases
2.3 Linear Transformations of a Vector Space
2.4 Exercises
Chapter 3: Metric Spaces
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
3.6 Exercises
Chapter 4: Banach Spaces
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 Space4.5 Exercises
Chapter 5: Hilbert Spaces
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
5.9 Exercises
Chapter 6: Distribution Spaces
6.1 The Space of Test Functions
6.2 The Space of Distributions
6.3 Algebra and Calculus With Distributions
6.3.1 Multiplication of Distributions
6.3.2 Convergence of Distributions 6.3.3 Derivative of a Distribution6.4 Convolution and Distributions
6.5 Exercises
Chapter 7: Fourier Analysis
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
7.8 Exercises
Chapter 8: Distributions and Differential Equations
8.1 Weak Derivatives and Sobolev Spaces
8.2 Differential Equations in D'
8.3 Fundamental Solutions 8.4 Fundamental Solutions and the Fourier Transform8.5 Fundamental Solutions for Some Important PDEs
Laplace Operator
Heat Operator
Wave Operator
Schrödinger Operator
Helmholtz Operator
Klein-Gordon Operator
Biharmonic Operator
8.6 Exercises
Chapter 9: Linear Operators
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
9.9 Exercises

✦ Subjects

Functional analysis;Differential equations;Integral equations;MATHEMATICS / Calculus;MATHEMATICS / Mathematical Analysis