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Introduction

  • Liviu Gr. Ixaru
  • Guido Vanden Berghe
Chapter
  • 378 Downloads
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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Copyright information

© Springer Science+Business Media Dordrecht 2004

Authors and Affiliations

  • Liviu Gr. Ixaru
    • 1
  • Guido Vanden Berghe
    • 2
  1. 1.“Horia Hulubei”, Department of Theoretical PhysicsNational Institute for Research and Development for Physics and Nuclear EngineeringBucharestRomania
  2. 2.Department of Applied Mathematics and Computer ScienceUniversity of GentGentBelgium

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