Analytic Methods for Partial Differential Equations

G. Evans

ISBN 10: 3540761241 / ISBN 13: 9783540761242

Editore: Springer London, 1999

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Druck auf Anfrage Neuware - Printed after ordering - The subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation. Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. J ames Clerk Maxwell, for example, put electricity and magnetism into a unified theory by estab lishing Maxwell's equations for electromagnetic theory, which gave solutions for problems in radio wave propagation, the diffraction of light and X-ray developments. Schrodinger's equation for quantum mechankal processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. In fluid mechanics, the Navier-Stokes' equations form a basis for huge number-crunching activities associated with such widely disparate topics as weather forcasting and the design of supersonic aircraft. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics. Codice articolo 9783540761242

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Titolo: Analytic Methods for Partial Differential ...

Casa editrice: Springer London

Data di pubblicazione: 1999

Legatura: Taschenbuch

Condizione: Neu

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Riassunto:

This is the practical introduction to the analytical approach taken in Volume 2. Based upon courses in partial differential equations over the last two decades, the text covers the classic canonical equations, with the method of separation of variables introduced at an early stage. The characteristic method for first order equations acts as an introduction to the classification of second order quasi-linear problems by characteristics. Attention then moves to different co-ordinate systems, primarily those with cylindrical or spherical symmetry. Hence a discussion of special functions arises quite naturally, and in each case the major properties are derived. The next section deals with the use of integral transforms and extensive methods for inverting them, and concludes with links to the use of Fourier series.

Contenuti:

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.