fractional differentiation
The idea of Grunwald-Letnikov differentiation comes from the following formulas of backward (http://planetmath.org/BackwardDifference) and forward difference . Within this entry, will be used to denote the greatest integer function and will be used to denote the gamma function.
Backward difference
(1) |
(2) |
For derivatives of integer orders, we only requires to specifies one point . Fractional derivatives, like fractional definite integrals, require an interval to be specified for the function we are talking about.
Definition 1: Left-hand Grunwald-Letnikov derivative
(3) |
Forward difference
(4) |
(5) |
Definition 2: Right-hand Grunwald-Letnikov derivative
(6) |
Theorem 1: Properties of fractional derivatives
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Linearity: where are any real constants
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Theorem 2: Table of fractional derivatives
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where and
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for all
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Title | fractional differentiation |
Canonical name | FractionalDifferentiation |
Date of creation | 2013-03-22 16:18:46 |
Last modified on | 2013-03-22 16:18:46 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 21 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 26A06 |
Synonym | Grunwald-Letnikov differentiation |
Related topic | HigherOrderDerivativesOfSineAndCosine |
Defines | fractional derivative |
Defines | left-hand Grunwald-Letnikov derivative |
Defines | left hand Grundwald Letnikov derivative |
Defines | right-hand Grundwald-Letnikov derivative |
Defines | right hand Grundwald-Letnikov derivative |