Partial Differential Equations through Examples and Exercises

✍ Scribed by Endre Pap, Arpad Takaci, Djurdjica Takaci

• Publisher: Springer
• Year: 1997
• Tongue: English
• Leaves: 418
• Series: Kluwer Texts in the Mathematical Sciences
• Category: Library

✦ Synopsis

The book Partial Differential Equations through Examples and Exercises has evolved from the lectures and exercises that the authors have given for more than fifteen years, mostly for mathematics, computer science, physics and chemistry students. By our best knowledge, the book is a first attempt to present the rather complex subject of partial differential equations (PDEs for short) through active reader-participation. Thus this book is a combination of theory and examples. In the theory of PDEs, on one hand, one has an interplay of several mathematical disciplines, including the theories of analytical functions, harmonic analysis, ODEs, topology and last, but not least, functional analysis, while on the other hand there are various methods, tools and approaches. In view of that, the exposition of new notions and methods in our book is "step by step". A minimal amount of expository theory is included at the beginning of each section Preliminaries with maximum emphasis placed on well selected examples and exercises capturing the essence of the material. Actually, we have divided the problems into two classes termed Examples and Exercises (often containing proofs of the statements from Preliminaries). The examples contain complete solutions, and also serve as a model for solving similar problems, given in the exercises. The readers are left to find the solution in the exercises; the answers, and occasionally, some hints, are still given. The book is implicitly divided in two parts, classical and abstract.

Topics
--Partial Differential Equations
--Functional Analysis
--Operator Theory
--Computational Mathematics and Numerical Analysis
--Mathematical Modeling and Industrial Mathematics

✦ Table of Contents

Preface
List of Symbols
Chapter 1 Introduction
1.1 Basic Notions
1.1.1 Preliminaries
1.1.2 Examples and Exercises
1.2 The Cauchy-Kowalevskaya Theorem
1.2.1 Preliminaries
1.2.2 Examples and Exercises
1.3 Equations of Mathematical Physics
Chapter 2 First Order PDEs
2.1 Quasi-linear PDEs
2.1.1 Preliminaries
2.1.2 Examples and Exercises
2.2 Pfaff's Equations
2.2.1 Preliminaries
2.2.2 Examples and Exercises
2.3 Nonlinear First Order PDEs
2.3.1 Preliminaries
The Lagrange-Charpite Method
2.3.2 Examples and Exercises
Chapter 3 Classification of the Second Order PDEs
3.1 Two Independent Variables
3.1.1 Preliminaries
Cauchy's Problem
3.1.2 Examples and Exercises
3.2 n Independent Variables
3.2.1 Preliminaries
3.2.2 Examples and Exercises
3.3 Wave, Potential and Heat Equation
Chapter 4 Hyperbolic Equations
4.1 Cauchy Problem for the One-dimensional Wave Equation
4.1.1 Preliminaries
4.1.2 Examples and Exercises
4.2 Cauchy Problem for the n-dimensional Wave Equation
4.2.1 Preliminaries
4.2.2 Examples and Exercises
4.3 The Fourier Method of Separation Variables
4.3.1 Preliminaries
Fourier Series
The Fourier Method of Separation of Variables
The Mixed type Problem
4.3.2 Examples and Exercises
4.4 The Sturm-Liouville Problem
4.4.1 Preliminaries
Special Functions
4.4.2 Examples and Exercises
4.5 Miscellaneous Problems
4.6 The Vibrating String
Chapter 5 Elliptic Equations
5.1 Dirichlet Problem
5.1.1 Preliminaries
5.1.2 Examples and Exercises
5.2 The Maximum Principle
5.2.1 Preliminaries
5.2.2 Examples and Exercises
5.3 The Green Function
5.3.1 Preliminaries
5.3.2 Examples and Exercises
5.4 The Harmonic Functions
5.4.1 Examples and Exercises
5.5 Gravitational Potential
Chapter 6 Parabolic Equations
6.1 Cauchy Problem
6.1.1 Preliminaries
The Maximum Principle
6.1.2 Examples and Exercise
6.2 Mixed Type Problem
6.2.1 Preliminaries
6.2.2 Examples and Exercises
6.3 Heat conduction
Chapter 7 Numerical Methods
7.0.1 Preliminaries
The Error of Approximation
7.0.2 Examples and Exercises
Chapter 8 Lebesgue's Integral and the Fourier Transform
8.1 Lebesgue's Integral and the L2(Q) Space
8.1.1 Preliminaries
8.1.2 Examples and Exercises
8.2 Delta Nets
8.2.1 Preliminaries
8.2.2 Examples and Exercises
8.3 The Surface Integrals
8.3.1 Preliminaries
8.3.2 Examples and Exercises
8.4 The Fourier Transform
8.4.1 Preliminaries
8.4.2 Examples and Exercises
Chapter 9 Generalized Derivative and Sobolev Spaces
9.1 Generalized Derivative
9.1.1 Preliminaries
9.1.2 Examples and Exercises
9.2 Sobolev Spaces
9.2.1 Preliminaries
9.2.2 Examples and Exercises
Chapter 10 Some Elements from Functional Analysis
10.1 Hilbert Space
10.1.1 Preliminaries
10.1.2 Examples and Exercises
10.2 The Fredholm Alternatives
10.2.1 Preliminaries
10.2.2 Examples and Exercises
10.3 Normed Vector Spaces
10.3.1 Preliminaries
10.3.2 Examples and Exercises
Chapter 11 Functional Analysis Methods PDEs
11.1 Generalized Dirichlet Problem
11.1.1 Preliminaries
11.1.2 Examples and Exercises
11.2 The Generalized Mixed Problems
11.2.1 Examples and Exercises
11.3 Numerical Solutions of PDEs in the Frame workof Functional Analysis
11.3.1 Preliminaries
11.3.2 Examples and Exercises
11.4 Miscellaneous
11.4.1 Preliminaries
11.4.2 Examples and Exercises
Chapter 12 Distributions in the theory of PDEs
12.1 Basic Properties
12.1.1 Preliminaries
12.1.2 Examples and Exercises
12.2 Fundamental Solutions
12.2.1 Preliminaries
12.2.2 Examples and Exercises
Bibliography
Index