Sturm-Liouville Theory and its Applications

verfasst von: M. A. Al-Gwaiz

Verlag: Springer London

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Developed from a course taught to senior undergraduates, this book provides a unified introduction to Fourier analysis and special functions based on the Sturm-Liouville theory in L2. The treatment relies heavily on the convergence properties of sequences and series of numbers as well as functions, and assumes a solid background in advanced calculus and an acquaintance with ordinary differential equations and linear algebra. Familiarity with the relevant theorems of real analysis, such as the Ascoli–Arzelà theorem, is also useful for following the proofs.

The presentation follows a clear and rigorous mathematical style that is both readable and well motivated, with many examples and applications used to illustrate the theory. Although addressed primarily to undergraduate students of mathematics, the book will also be of interest to students in related disciplines, such as physics and engineering, where Fourier analysis and special functions are used extensively for solving linear differential equations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Inner Product Space

An inner product space is the natural generalization of the Euclidean space Rⁿ, with its well-known topological and geometric properties. It constitutes the framework, or setting, for much of our work in this book, as it provides the appropriate mathematical structure that we need.

Chapter 2. The Sturm–Liouville Theory

Complete orthogonal sets of functions in L2 arise naturally as solutions of certain second-order linear differential equations under appropriate boundary conditions, commonly referred to as Sturm-Liouville boundary-value problems, after the Swiss mathematician Jacques Sturm (1803-1855) and the French mathematician Joseph Liouville (1809-1882), who studied these problems and the properties of their solutions. The differential equations considered here arise directly as mathematical models of motion according to Newton’s law, but more often as a result of using the method of separation of variables to solve the classical partial differential equations of physics, such as Laplace’s equation, the heat equation, and the wave equation.

Chapter 3. Fourier Series

This chapter deals with the theory and applications of Fourier series, named after Joseph Fourier (1768-1830), the French physicist who developed the series in his investigation of the transfer of heat. His results were later refined by others, especially the German mathematician Gustav Lejeune Dirichlet (1805- 1859), who made important contributions to the convergence properties of the series.

Chapter 4. Orthogonal Polynomials

In this chapter we consider three typical examples of singular SL problems whose eigenfunctions are real polynomials.

Chapter 5. Bessel Functions

We start by presenting the gamma function and some of its properties. This function is used to define the Bessel functions, hence its relevance to the subject of this chapter.

Chapter 6. The Fourier Transformation

The underlying theme of the previous chapters was the Sturm-Liouville theory. The last three chapters show how the eigenfunctions of various SL problems serve as bases for L², either through conventional Fourier series or its generalized version. In this chapter we introduce the Fourier integral as a limiting case of the classical Fourier series, and show how it serves, under certain conditions, as a method for representing nonperiodic functions on R where the series approach does not apply. This chapter and the next are therefore concerned with extending the theory of Fourier series to nonperiodic functions.

Chapter 7. The Laplace Transformation

Backmatter

Titel: Sturm-Liouville Theory and its Applications
verfasst von: M. A. Al-Gwaiz
Verlag: Springer London
Electronic ISBN: 978-1-84628-972-9
Print ISBN: 978-1-84628-971-2
DOI: https://doi.org/10.1007/978-1-84628-972-9

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