Książki

Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs, the wave, heat and Lapace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

Partial Differential Equations: An Introduction

Chapter 1: Where PDEs Come From
1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions
1.5 Well-Posed Problems
1.6 Types of Second-Order Equations
Chapter 2: Waves and Diffusions
2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions

Chapter 3: Reflections and Sources
3.1 Diffusion on the Half-Line
3.2 Reflections of Waves
3.3 Diffusion with a Source
3.4 Waves with a Source
3.5 Diffusion Revisited

Chapter 4: Boundary Problems
4.1 Separation of Variables, The Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition

Chapter 5: Fourier Series
5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and the General Fourier Series
5.4 Completeness
5.5 Completeness and the Gibbs Phenomenon
5.6 Inhomogeneous Boundary Conditions

Chapter 6: Harmonic Functions
6.1 Laplace's Equation
6.2 Rectangles and Cubes
6.3 Poisson's Formula
6.4 Circles, Wedges, and Annuli

Chapter 7: Green's Identities and Green's Functions
7.1 Green's First Identity
7.2 Green's Second Identity
7.3 Green's Functions
7.4 Half-Space and Sphere

Chapter 8: Computation of Solutions
8.1 Opportunities and Dangers
8.2 Approximations of Diffusions
8.3 Approximations of Waves
8.4 Approximations of Laplace's Equation
8.5 Finite Element Method

Chapter 9: Waves in Space
9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources
9.4 The Diffusion and Schrodinger Equations
9.5 The Hydrogen Atom

Chapter 10: Boundaries in the Plane and in Space
10.1 Fourier's Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball
10.4 Nodes
10.5 Bessel Functions
10.6 Legendre Functions
10.7 Angular Momentum in Quantum Mechanics

Chapter 11: General Eigenvalue Problems
11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.3 Completeness
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues

Chapter 12: Distributions and Transforms
12.1 Distributions
12.2 Green's Functions, Revisited
12.3 Fourier Transforms
12.4 Source Functions
12.5 Laplace Transform Techniques

Chapter 13: PDE Problems for Physics
13.1 Electromagnetism
13.2 Fluids and Acoustics
13.3 Scattering
13.4 Continuous Spectrum
13.5 Equations of Elementary Particles

Chapter 14: Nonlinear PDEs
14.1 Shock Waves
14.2 Solitions
14.3 Calculus of Variations
14.4 Bifurcation Theory
14.5 Water Waves

Appendix
A.1 Continuous and Differentiable Functions
A.2 Infinite Sets of Functions
A.3 Differentiation and Integration
A.4 Differential Equations
A.5 The Gamma Function

References

Answers and Hints to Selected Exercises

Index