Partial Differential Equations through Examples and Exercises

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