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2004 | Buch

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Exponential Fitting

verfasst von: Liviu Gr. Ixaru, Guido Vanden Berghe

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

Exponential Fitting is a procedure for an efficient numerical approach of functions consisting of weighted sums of exponential, trigonometric or hyperbolic functions with slowly varying weight functions. This book is the first one devoted to this subject. Operations on the functions described above like numerical differentiation, quadrature, interpolation or solving ordinary differential equations whose solution is of this type, are of real interest nowadays in many phenomena as oscillations, vibrations, rotations, or wave propagation.
The authors studied the field for many years and contributed to it. Since the total number of papers accumulated so far in this field exceeds 200 and the fact that these papers are spread over journals with various profiles (such as applied mathematics, computer science, computational physics and chemistry) it was time to compact and to systematically present this vast material.
In this book, a series of aspects is covered, ranging from the theory of the procedure up to direct applications and sometimes including ready to use programs. The book can also be used as a textbook for graduate students.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The simple approximate formula for the computation of the first derivative of a function y(x),

$$ y'\left( x \right) \approx \frac{1}{{2h}}\left[ {y\left( {x + h} \right) - y\left( {x - h} \right)} \right], $$

(1.1)

is known to work well when y(x) is smooth enough. However, if y(x) is an oscillatory function of the form

$$ y'\left( x \right) = {f_1}\left( x \right)\sin \left( {\omega x} \right) + {f_2}\left( x \right)\cos \left( {\omega x} \right) $$

(1.2)

with smooth f ₁(x) and f ₂(x), the slightly modified formula

$$ y'\left( x \right) \approx \frac{1}{{2h}} \cdot \frac{\theta }{{\sin \theta }} \cdot \left[ {y\left( {x + h} \right) - y\left( {x - h} \right)} \right], $$

(1.3)

whereθ=ωh, becomes appropriate.

Liviu Gr. Ixaru, Guido Vanden Berghe

Chapter 2. Mathematical Properties

Abstract

In this chapter we present the main mathematical elements of the exponential fitting procedure. It will be seen that this procedure is rather general. However, later on in this book the procedure will be mainly applied in the restricted area of the generation of formulae and algorithms for functions with oscillatory or hyperbolic variation.

Liviu Gr. Ixaru, Guido Vanden Berghe

Chapter 3. Construction of EF Formulae for Functions with Oscillatory or Hyperbolic Variation

Abstract

The considerations of the previous chapter referred to the general features of the exponential fitting technique. When the functions to be approximated are oscillatory or with a variation well described by hyperbolic functions the technique exhibits some helpful features. This chapter aims at presenting these features and at formulating a simple algorithm-like flow chart to be followed in the current practice.

Liviu Gr. Ixaru, Guido Vanden Berghe

Chapter 4. Numerical Differentiation, Quadrature and Interpolation

Abstract

A series of ef formulae tuned on functions of the form (3.38) or (3.39) are derived here by the procedure described in the previous chapter. We construct the ef coefficients for approximations of the first and the second derivative of y(x), for a set of quadrature rules, and for some simple interpolation formulae.

Liviu Gr. Ixaru, Guido Vanden Berghe

Chapter 5. Linear Multistep Solvers for Ordinary Differential Equations

Abstract

The solution of the initial value problem for ordinary differential equations is one of the main topics in numerical analysis. The linear multistep methods (algorithms) form a class of methods which benefitted from much attention over the years. The theory of these methods is basically due to Dahlquist, [14], and a series of well-known books which cover both theoretical and practical aspects are available, to mention only the books of Henrici, [22], Lambert, [40], Hairer, Nørsett and Wanner, [20], Shampine, [52], Hairer and Wanner, [21], and Butcher, [4].

Liviu Gr. Ixaru, Guido Vanden Berghe

Chapter 6. Runge-Kutta Solvers for Ordinary Differential Equations

Abstract

Since the original papers of Runge [24] and Kutta [17] a great number of papers and books have been devoted to the properties of Runge-Kutta methods. Reviews of this material can be found in [4], [5], [12], [18]. Kutta [17] formulated the general scheme of what is now called a Runge-Kutta method.

Liviu Gr. Ixaru, Guido Vanden Berghe

Backmatter

Titel: Exponential Fitting
verfasst von: Liviu Gr. Ixaru
Guido Vanden Berghe
Copyright-Jahr: 2004
Verlag: Springer Netherlands
Electronic ISBN: 978-1-4020-2100-8
Print ISBN: 978-90-481-6590-2
DOI: https://doi.org/10.1007/978-1-4020-2100-8