Differential and Integral Equations

Name: Differential and Integral Equations
Brand: Oxford University Press
Price: 85.59 EUR
Availability: OnlineOnly

Collins Peter(Autor*in)

Oxford University Press

1. Auflage

Erschienen am 3. August 2006

387 Seiten

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978-0-19-152400-4 (ISBN)

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Inhalt

Intro
Contents
Preface
How to use this book
Prerequisites
0 Some Preliminaries
1 Integral Equations and Picard's Method
1.1 Integral equations and their relationship to differential equations
1.2 Picard's method
2 Existence and Uniqueness
2.1 First-order differential equations in a single independent variable
2.2 Two simultaneous equations in a single variable
2.3 A second-order equation
3 The Homogeneous Linear Equation and Wronskians
3.1 Some linear algebra
3.2 Wronskians and the linear independence of solutions of the second-order homogeneous linear equation
4 The Non-Homogeneous Linear Equation
4.1 The method of variation of parameters
4.2 Green's functions
5 First-Order Partial Differential Equations
5.1 Characteristics and some geometrical considerations
5.2 Solving characteristic equations
5.3 General solutions
5.4 Fitting boundary conditions to general solutions
5.5 Parametric solutions and domains of definition
5.6 A geometric interpretation of an analytic condition
6 Second-Order Partial Differential Equations
6.1 Characteristics
6.2 Reduction to canonical form
6.3 General solutions
6.4 Problems involving boundary conditions
6.5 Appendix: technique in the use of the chain rule
7 The Diffusion and Wave Equations and the Equation of Laplace
7.1 The equations to be considered
7.2 One-dimensional heat conduction
7.3 Transverse waves in a finite string
7.4 Separated solutions of Laplace's equation in polar co-ordinates and Legendre's equation
7.5 The Dirichlet problem and its solution for the disc
7.6 Radially symmetric solutions of the two-dimensional wave equation and Bessel's equation
7.7 Existence and uniqueness of solutions, well-posed problems
7.8 Appendix: proof of the Mean Value Theorem for harmonic functions
8 The Fredholm Alternative
8.1 A simple case
8.2 Some algebraic preliminaries
8.3 The Fredholm Alternative Theorem
8.4 A worked example
9 Hilbert-Schmidt Theory
9.1 Eigenvalues are real and eigenfunctions corresponding to distinct eigenvalues are orthogonal
9.2 Orthonormal families of functions and Bessel's inequality
9.3 Some results about eigenvalues deducible from Bessel's inequality
9.4 Description of the sets of all eigenvalues and all eigenfunctions
9.5 The Expansion Theorem
10 Iterative Methods and Neumann Series
10.1 An example of Picard's method
10.2 Powers of an integral operator
10.3 Iterated kernels
10.4 Neumann series
10.5 A remark on the convergence of iterative methods
11 The Calculus of Variations
11.1 The fundamental problem
11.2 Some classical examples from mechanics and geometry
11.3 The derivation of Euler's equation for the fundamental problem
11.4 The special case F = F(y, y')
11.5 When F contains higher derivatives of y
11.6 When F contains more dependent functions
11.7 When F contains more independent variables
11.8 Integral constraints
11.9 Non-integral constraints
11.10 Varying boundary conditions
12 The Sturm-Liouville Equation
12.1 Some elementary results on eigenfunctions and eigenvalues
12.2 The Sturm-Liouville Theorem
12.3 Derivation from a variational principle
12.4 Some singular equations
12.5 The Rayleigh-Ritz method
13 Series Solutions
13.1 Power series and analytic functions
13.2 Ordinary and regular singular points
13.3 Power series solutions near an ordinary point
13.4 Extended power series solutions near a regular singular point: theory
13.5 Extended power series solutions near a regular singular point: practice
13.6 The method of Frobenius
13.7 Summary
13.8 Appendix: the use of complex variables
14 Transform Methods
14.1 The Fourier transform
14.2 Applications of the Fourier transform
14.3 The Laplace transform
14.4 Applications of the Laplace transform
14.5 Applications involving complex analysis
14.6 Appendix: similarity solutions
15 Phase-Plane Analysis
15.1 The phase-plane and stability
15.2 Linear theory
15.3 Some non-linear systems
15.4 Linearisation
Appendix: the solution of some elementary ordinary differential equations
Bibliography
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W

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