Introduction to Partial Differential Equations

ISBN-10: 052161323X

ISBN-13: 9780521613231

Edition: 2005

List price: $93.99 Buy it from $77.33
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Description:

A complete introduction to partial differential equations, this textbook provides a rigorous yet accessible guide to students in mathematics, physics and engineering. The presentation is lively and up to date, paying particular emphasis to developing an appreciation of underlying mathematical theory. Beginning with basic definitions, properties and derivations of some basic equations of mathematical physics from basic principles, the book studies first order equations, classification of second order equations, and the one-dimensional wave equation. Two chapters are devoted to the separation of variables, whilst others concentrate on a wide range of topics including elliptic theory, Green's functions, variational and numerical methods. A rich collection of worked examples and exercises accompany the text, along with a large number of illustrations and graphs to provide insight into the numerical examples. Solutions to selected exercises are included for students and extended solution sets are available to lecturers from solutions@cambridge.org.
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Book details

List price: $93.99
Copyright year: 2005
Publisher: Cambridge University Press
Publication date: 5/12/2005
Binding: Paperback
Pages: 384
Size: 6.75" wide x 9.75" long x 0.50" tall
Weight: 1.694
Language: English

Preface
Introduction
Preliminaries
Classification
Differential operators and the superposition principle
Differential equations as mathematical models
Associated conditions
Simple examples
Exercises
First-order equations
Introduction
Quasilinear equations
The method of characteristics
Examples of the characteristics method
The existence and uniqueness theorem
The Lagrange method
Conservation laws and shock waves
The eikonal equation
General nonlinear equations
Exercises
Second-order linear equations in two indenpendent variables
Introduction
Classification
Canonical form of hyperbolic equations
Canonical form of parabolic equations
Canonical form of elliptic equations
Exercises
The one-dimensional wave equation
Introduction
Canonical form and general solution
The Cauchy problem and d' Alembert's formula
Domain of dependence and region of influence
The Cauchy problem for the nonhomogeneous wave equation
Exercises
The method of separation of variables
Introduction
Heat equation: homogeneous boundary condition
Separation of variables for the wave equation
Separation of variables for nonhomogeneous equations
The energy method and uniqueness
Further applications of the heat equation
Exercises
Sturm-Liouville problems and eigenfunction expansions
Introduction
The Sturm-Liouville problem
Inner product spaces and orthonormal systems
The basic properties of Sturm-Liouville eigenfunctions and eigenvalues
Nonhomogeneous equations
Nonhomogeneous boundary conditions
Exercises
Elliptic equations
Introduction
Basic properties of elliptic problems
The maximum principle
Applications of the maximum principle
Green's identities
The maximum principle for the heat equation
Separation of variables for elliptic problems
Poisson's formula
Exercises
Green's functions and integral representations
Introduction
Green's function for Dirichlet problem in the plane
Neumann's function in the plane
The heat kernel
Exercises
Equations in high dimensions
Introduction
First-order equations
Classification of second-order equations
The wave equation in R[superscript 2] and R[superscript 3]
The eigenvalue problem for the Laplace equation
Separation of variables for the heat equation
Separation of variables for the wave equation
Separation of variables for the Laplace equation
Schrodinger equation for the hydrogen atom
Musical instruments
Green's functions in higher dimensions
Heat kernel in higher dimensions
Exercises
Variational methods
Calculus of variations
Function spaces and weak formulation
Exercises
Numerical methods
Introduction
Finite differences
The heat equation: explicit and implicit schemes, stability, consistency and convergence
Laplace equation
The wave equation
Numerical solutions of large linear algebraic systems
The finite elements method
Exercises
Solutions of odd-numbered problems
Trigonometric formulas
Integration formulas
Elementary ODEs
Differential operators in polar coordinates
Differential operators in spherical coordinates
References
Index
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