Partial Differential Equations
Theory and Technique
- 1st Edition - September 23, 2014
- Authors: George F. Carrier, Carl E. Pearson
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 2 - 3 6 9 9 - 5
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 5 9 1 6 - 1
Partial Differential Equations: Theory and Technique provides formal definitions, notational conventions, and a systematic discussion of partial differential equations. The text… Read more
Partial Differential Equations: Theory and Technique provides formal definitions, notational conventions, and a systematic discussion of partial differential equations. The text emphasizes the acquisition of practical technique in the use of partial differential equations. The book contains discussions on classical second-order equations of diffusion, wave motion, first-order linear and quasi-linear equations, and potential theory. Certain chapters elaborate Green's functions, eigenvalue problems, practical approximation techniques, perturbations (regular and singular), difference equations, and numerical methods. Students of mathematics will find the book very useful.
Preface
Introduction
1.1 Definitions and Examples
Chapter 1 The Diffusion Equation
1.1 Derivation
1.2 Problems
1.3 Simple Solutions
1.4 Problems
1.5 Series Solutions
1.6 Problems
1.7 Nonhomogeneous End Conditions
1.8 Problems
1.9 The Maximum Principle
1.10 Problems
Chapter 2 Laplace Transform Methods
2.1 Introductory Example
2.2 Problems
2.3 A Finite Interval Problem
2.4 Problems
2.5 Delta Function
2.6 Problems
2.7 Supplementary Problems
Chapter 3 The Wave Equation
3.1 Derivation
3.2 Problems
3.3 An Infinite-Interval Problem
3.4 Problems
3.5 Series Solutions
3.6 Problems
3.7 A Problem with Radial Symmetry
3.8 Problems
3.9 Transforms
3.10 Problems
3.11 Uniqueness
3.12 Supplementary Problems
Chapter 4 The Potential Equation
4.1 Laplace's and Poisson's Equations
4.2 Problems
4.3 Simple Properties of Harmonic Functions
4.4 Some Special Solutions—Series
4.5 Problems
4.6 Discontinuous Boundary Data
4.7 Complex Variables and Conformai Mapping
4.8 Problems
Chapter 5 Classification of Second-Order Equations
5.1 Cauchy Data on y-Axis
5.2 Cauchy Data on Arbitrary Curve
5.3 Problems
5.4 Case I: B2 - AC > 0
5.5 Case II: B2 - AC = 0
5.6 Case III: J52 - AC < 0
5.7 Problems
5.8 Discontinuities; Signal Propagation
5.9 Problems
5.10 Some Remarks
Chapter 6 First-Order Equations
6.1 Linear Equation Examples
6.2 Problems
6.3 Quasi-Linear Case
6.4 Problems
6.5 Further Properties of Characteristics
6.6 Problems
6.7 More Variables
Chapter 7 Extensions
7.1 More Variables
7.2 Problems
7.3 Series and Transforms
7.4 Problems
7.5 Legendre Functions
7.6 Problems
7.7 Spherical Harmonics
7.8 Problems
Chapter 8 Perturbations
8.1 A Nonlinear Problem
8.2 Problems
8.3 Two Examples from Fluid Mechanics
8.4 Boundary Perturbations
8.5 Problems
Chapter 9 Green's Functions
9.1 Some Consequences of the Divergence Theorem
9.2 The Laplacian Operator
9.3 Problems
9.4 Potentials of Volume and Surface Distributions
9.5 Problems
9.6 Modified Laplacian
9.7 Problems
9.8 Wave Equation
9.9 Problems
Chapter 10 Variational Methods
10.1 A Minimization Problem
10.2 Problems
10.3 Natural Boundary Conditions
10.4 Subsidiary Conditions
10.5 Problems
10.6 Approximate Methods
10.7 Problems
10.8 Finite-Element Method
10.9 Supplementary Problems
Chapter 11 Eigenvalue Problems
11.1 A Prototype Problem
11.2 Some Eigenvalue Properties
11.3 Problems
11.4 Perturbations
11.5 Approximations
11.6 Problems
Chapter 12 More on First-Order Equations
12.1 Envelopes
12.2 Characteristic Strips
12.3 Complete Integral
12.4 Problems
12.5 Legendre Transformation
12.6 Problems
12.7 Propagation of a Disturbance
12.8 Complete Integral and Eikonal Function
12.9 Hamilton-Jacobi Equation
12.10 Problems
Chapter 13 More on Characteristics
13.1 Discontinuities—A Preliminary Example
13.2 Weak Solutions
13.3 Burgers' Equation
13.4 Problems
13.5 A Compressible Flow Problem
13.6 A Numerical Approach
13.7 Problems
13.8 More Dependent Variables
13.9 More Independent Variables
13.10 Problems
Chapter 14 Finite-Difference Equations and Numerical Methods
14.1 Accuracy and Stability; A Diffusion Equation Example
14.2 Error Analysis
14.3 Problems
14.4 More Dimensions, or Other Complications
14.5 Series Expansions
14.6 Problems
14.7 Wave Equation
14.8 A Nonlinear Equation
14.9 Problems
14.10 Boundary Value Problems
14.11 Problems
14.12 Series; Fast Fourier Transform
14.13 Problems
Chapter 15 Singular Perturbation Methods
15.1 A Boundary Layer Problem
15.2 A More General Procedure
15.3 Problems
15.4 A Transition Situation
15.5 Problems
15.6 Asymptotic Analysis of Wave Motion
15.7 Boundary Layer near a Caustic
15.8 Problems
15.9 Multiple Scaling
15.10 Problems
References
Index
- Edition: 1
- Published: September 23, 2014
- Language: English
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