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Volume 1, Issue 2
A Paradoxical Consistency Between Dynamic and Conventional Derivatives on Hybrid Grids

Qin Sheng

Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 198-213.

Published online: 2008-01

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  • Abstract

It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important rôle in hybrid modeling and computations. Being constructed on various kinds of hybrid grids, that is, time scales, dynamic derivatives offer superior accuracy and flexibility in approximating mathematically important natural processes with hard-to-predict singularities, such as the epidemic growth with unpredictable jump sizes and option market changes with high uncertainties, as compared with conventional derivatives. In this article, we shall review the novel new concepts, explore delicate relations between the most frequently used second-order dynamic derivatives and conventional derivatives. We shall investigate necessary conditions for guaranteeing the consistency between the two derivatives. We will show that such a consistency may never exist in general. This implies that the dynamic derivatives provide entirely different new tools for sensitive modeling and approximations on hybrid grids. Rigorous error analysis will be given via asymptotic expansions for further modeling and computational applications. Numerical experiments will also be given.

  • Keywords

Dynamic derivatives, conventional derivatives, time scale theory, approximations, error estimates, hybrid grids, uniform and nonuniform grids, asymptotic expansions.

  • AMS Subject Headings

34A45, 39A13, 74H15, 74S20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-1-198, author = {}, title = {A Paradoxical Consistency Between Dynamic and Conventional Derivatives on Hybrid Grids}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2008}, volume = {1}, number = {2}, pages = {198--213}, abstract = {

It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important rôle in hybrid modeling and computations. Being constructed on various kinds of hybrid grids, that is, time scales, dynamic derivatives offer superior accuracy and flexibility in approximating mathematically important natural processes with hard-to-predict singularities, such as the epidemic growth with unpredictable jump sizes and option market changes with high uncertainties, as compared with conventional derivatives. In this article, we shall review the novel new concepts, explore delicate relations between the most frequently used second-order dynamic derivatives and conventional derivatives. We shall investigate necessary conditions for guaranteeing the consistency between the two derivatives. We will show that such a consistency may never exist in general. This implies that the dynamic derivatives provide entirely different new tools for sensitive modeling and approximations on hybrid grids. Rigorous error analysis will be given via asymptotic expansions for further modeling and computational applications. Numerical experiments will also be given.

}, issn = {2079-7338}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nmtma/6048.html} }
TY - JOUR T1 - A Paradoxical Consistency Between Dynamic and Conventional Derivatives on Hybrid Grids JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 198 EP - 213 PY - 2008 DA - 2008/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/nmtma/6048.html KW - Dynamic derivatives, conventional derivatives, time scale theory, approximations, error estimates, hybrid grids, uniform and nonuniform grids, asymptotic expansions. AB -

It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important rôle in hybrid modeling and computations. Being constructed on various kinds of hybrid grids, that is, time scales, dynamic derivatives offer superior accuracy and flexibility in approximating mathematically important natural processes with hard-to-predict singularities, such as the epidemic growth with unpredictable jump sizes and option market changes with high uncertainties, as compared with conventional derivatives. In this article, we shall review the novel new concepts, explore delicate relations between the most frequently used second-order dynamic derivatives and conventional derivatives. We shall investigate necessary conditions for guaranteeing the consistency between the two derivatives. We will show that such a consistency may never exist in general. This implies that the dynamic derivatives provide entirely different new tools for sensitive modeling and approximations on hybrid grids. Rigorous error analysis will be given via asymptotic expansions for further modeling and computational applications. Numerical experiments will also be given.

Qin Sheng. (2020). A Paradoxical Consistency Between Dynamic and Conventional Derivatives on Hybrid Grids. Numerical Mathematics: Theory, Methods and Applications. 1 (2). 198-213. doi:
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