Introduction

• Liviu Gr. Ixaru
• Guido Vanden Berghe
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 568)

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.

Keywords

Multistep Method Oscillatory Integral Oscillatory Function Ofthe Form Order ODEs
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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