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Über dieses Buch

This textbook, now in its second edition, provides students with a firm grasp of the fundamental notions and techniques of applied mathematics as well as the software skills to implement them. The text emphasizes the computational aspects of problem solving as well as the limitations and implicit assumptions inherent in the formal methods. Readers are also given a sense of the wide variety of problems in which the presented techniques are useful.

Broadly organized around the theme of applied Fourier analysis, the treatment covers classical applications in partial differential equations and boundary value problems, and a substantial number of topics associated with Laplace, Fourier, and discrete transform theories. Some advanced topics are explored in the final chapters such as short-time Fourier analysis and geometrically based transforms applicable to boundary value problems. The topics covered are useful in a variety of applied fields such as continuum mechanics, mathematical physics, control theory, and signal processing.

Replete with helpful examples, illustrations, and exercises of varying difficulty, this text can be used for a one- or two-semester course and is ideal for students in pure and applied mathematics, physics, and engineering.

Key features of the software overview:

Now relies solely on the free software tools Octave, Maxima, and Python.

Appendix introduces all of these tools at a level suitable to those with some programming experience

Provides references to sources of further learning.

Code snippets incorporated throughout the text.

All graphics and illustrations generated using these tools.

Praise for the first edition:

“The author mixed in a remarkable way theoretical results and applications illustrating the results. Flexibility of presentation (increasing and decreasing level of rigor, accessibility) is a key feature...The book contains extensive examples, presented in an intuitive way with high quality figures (some of them quite spectacular)…” – Mathematica

“...Davis's book has many novel features being quite different from most other textbooks on applied mathematics.... Mainly it has a clear and consistent exposition with a strong focus on mathematical fundamentals and useful techniques. It has numerous extensive examples, illustrations, comments, and a very modern graphical presentation of results.

“…The book has style. Every theorem and mathematical result has a wonderful appealing comment.” – Studies in Informatics and Control

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This text is intended to provide an introduction to certain methods of applied mathematics, especially those arising from the area of Fourier analysis.
Jon H. Davis

Chapter 2. Fourier Series

Abstract
A common requirement in various applied mathematical problems is to express a given function as a linear combination of other known functions. The reasons for this vary from analytical necessity to computational convenience.
Jon H. Davis

Chapter 3. Elementary Boundary Value Problems

Abstract
The basic partial differential equation types are introduced as models for diffusion, wave, and equilibrium phenomena. The models are treated by the method of separation of variables in the simplest case of rectangular coordinates. Connections with discrete variable models are noted, and some of the mathematical properties of the formal series solutions are discussed.
Jon H. Davis

Chapter 4. Sturm–Liouville Theory and Boundary Value Problems

Abstract
Partial differential equation models with non-rectangular geometries, more independent variables, and variable material properties are introduced. Many examples of such problems have separated equations of Sturm-Liouville type, in turn leading to power series solutions. These topics are discussed with Bessel functions as a detailed example. More complicated problems involve inhomogeneous and forced system models. Many other examples have complicated domain shapes which make simple separation of variables methods inapplicable. For such cases, we introduce some numerical methods and emphasize the finite element approach.
Jon H. Davis

Chapter 5. Functions of a Complex Variable

Abstract
In earlier chapters, complex-valued functions appeared in connection with Fourier series expansions. In this context, while the function assumes complex values, the argument of the function is real-valued. There is a highly developed theory of (complex-valued) functions of a complex-valued argument. This theory contains some remarkably powerful results which are applicable to a variety of problems.
Jon H. Davis

Chapter 6. Laplace Transforms

Abstract
Laplace transforms associate a function of a complex variable with a function defined on a half-axis in “time.” The formal properties of the Laplace transform lead to its use in solution of initial value problems for differential equations. The definition of convolution introduces input-output models and linear system responses. Impedance analysis for linear electrical circuits actually arises from the Laplace transform analysis of the circuit element governing equations.
Jon H. Davis

Chapter 7. Fourier Transforms

Abstract
Fourier transform methods find application in problems formally similar to those for which Laplace transform techniques are a suitable tool. Such applications include integral equations and partial and ordinary differential equations. The formal difference between the two classes of problems is that Laplace transforms are applied to functions defined on a half-line, while Fourier transforms apply to functions whose domain is the entire real axis. As a consequence, Laplace transforms are associated with initial value problems (transient responses), while Fourier transforms find more common application in input-output (forced response) models.
Jon H. Davis

Chapter 8. Discrete Variable Transforms

Abstract
Functions of a discrete variable, more commonly known as sequences, have associated transform theories. There are discrete variable analogues of Laplace Transforms, Fourier Transforms, and Fourier Series. The function domain in the latter case is discrete and finite: a divide-and-conquer algorithm in this results in the ubiquitous Fast Fourier Transform.
Jon H. Davis

Chapter 9. Additional Topics

Abstract
Fourier methods broadly construed have applications beyond the problems discussed in previous chapters. All of these consist, in a sense, of different decompositions for functions. The motivation for the decomposition varies from a need for efficient storage, shifted point of view, to geometrically motivated adaptations of standard transforms.
Jon H. Davis

Erratum to: Fourier Series

Without Abstract
Jon H. Davis

Backmatter

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