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Analytic Methods for Partial Differential Equations

'Springer Undergraduate Mathematics Series'. 1999. Auflage. Book. Sprache: Englisch.
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The subject of partial differential equations holds an exciting place in mathematics. At one extreme the interest lies in the existence and uniqueness of solutions, and the functional analysis of the proofs of these properties. At the other extreme l... weiterlesen
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Produktdetails
Titel: Analytic Methods for Partial Differential Equations
Autor/en: J. Blackledge, G. Evans, P. Yardley

ISBN: 3540761241
EAN: 9783540761242
'Springer Undergraduate Mathematics Series'.
1999. Auflage.
Book.
Sprache: Englisch.
Springer

November 1999 - kartoniert - 316 Seiten

Beschreibung

The subject of partial differential equations holds an exciting place in mathematics. At one extreme the interest lies in the existence and uniqueness of solutions, and the functional analysis of the proofs of these properties. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, either analytically or numerically, to these important equations which can be used in design and construction. The objective of this book is to actually solve equations rather than discuss the theoretical properties of their solutions. The topics are approached practically, without losing track of the underlying mathematical foundations of the subject. The topics covered include the separation of variables, the characteristic method, D'Alembert's method, integral transforms and Green's functions. Numerous exercises are provided as an integral part of the learning process, with solutions provided in a substantial appendix.

Inhaltsverzeichnis

1. Mathematical Preliminaries.
- 1.1 Introduction.
- 1.2 Characteristics and Classification.
- 1.3 Orthogonal Functions.
- 1.4 Sturm-Liouville Boundary Value Problems.
- 1.5 Legendre Polynomials.
- 1.6 Bessel Functions.
- 1.7 Results from Complex Analysis.
- 1.8 Generalised Functions and the Delta Function.
- 1.8.1 Definition and Properties of a Generalised Function.
- 1.8.2 Differentiation Across Discontinuities.
- 1.8.3 The Fourier Transform of Generalised Functions.
- 1.8.4 Convolution of Generalised Functions.
- 1.8.5 The Discrete Representation of the Delta Function.
- 2. Separation of the Variables.
- 2.1 Introduction.
- 2.2 The Wave Equation.
- 2.3 The Heat Equation.
- 2.4 Laplace's Equation.
- 2.5 Homogeneous and Non-homogeneous Boundary Conditions.
- 2.6 Separation of variables in other coordinate systems.
- 3. First-order Equations and Hyperbolic Second-order Equations.
- 3.1 Introduction.
- 3.2 First-order equations.
- 3.3 Introduction to d'Alembert's Method.
- 3.4 d'Alembert's General Solution.
- 3.5 Characteristics.
- 3.6 Semi-infinite Strings.
- 4. Integral Transforms.
- 4.1 Introduction.
- 4.2 Fourier Integrals.
- 4.3 Application to the Heat Equation.
- 4.4 Fourier Sine and Cosine Transforms.
- 4.5 General Fourier Transforms.
- 4.6 Laplace transform.
- 4.7 Inverting Laplace Transforms.
- 4.8 Standard Transforms.
- 4.9 Use of Laplace Transforms to Solve Partial Differential Equations.
- 5. Green's Functions.
- 5.1 Introduction.
- 5.2 Green's Functions for the Time-independent Wave Equation.
- 5.3 Green's Function Solution to the Three-dimensional Inhomogeneous Wave Equation.
- 5.4 Green's Function Solutions to the Inhomogeneous Helmholtz and Schrödinger Equations: An Introduction to Scattering Theory.
- 5.5 Green's Function Solution to Maxwell's Equations and Time-dependent Problems.
- 5.6 Green's Functions and Optics: Kirchhoff Diffraction Theory.
- 5.7 Approximation Methods and the Born Series.
- 5.8 Green's Function Solution to the Diffusion Equation.
- 5.9 Green's Function Solution to the Laplace and Poisson Equations.
- 5.10 Discussion.- A. Solutions of Exercises.

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