fracdiff  Fractional order differentiation

Calling Sequence


fracdiff(f, x, , method = mth, methodoptions = mthopts)


Parameters


f



algebraic expression

x



name




real number, or a name representing a real number, not an integer

mth



(optional) method of calculation, can be direct (default), laplace or series

mthopts



(optional) list of options for the specified method of calculation





Description


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Given an algebraic expression, f, fracdiff computes the th derivative of f with respect to x, where is not an integer. For integer (_nu_) order differentiation use diff  see also symbolic integer order differentiation.

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By default, the direct method is used to calculate the fractional derivative using the DavisonEssex (DE) definition, that is, first differentiate n times, then integrate n times, where n = ceil(), using the standard formula for iterated integrals. The resulting DE definition of fractional derivative is

>

diff(f(x),[x$nu]) = 1/GAMMA(nnu)*Int((xt)^(nnu1)*diff(f(t),[t$n]),t = 0 .. x);

 (1) 
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In the literature, the term fractional derivative is sometimes reserved for the RiemannLiouville (RL) fractional derivative, defined by

>

diff(f(x),[x$nu]) = 1/GAMMA(nnu)*Diff(Int((xt)^(nnu1)*f(t),t = 0..x),[x$n]);

 (2) 

The DavisonEssex and the RiemannLiouville definitions are different in the following aspect: in the DE formula, differentiation is performed first, then integration; in the RL formula it is the other way around. The DE definition implemented, thus, maps constants to zero, imitating integer order differentiation, while the RL definition does not. This property of the DE definition makes it suitable to work with initial value problems for fractional differential equations.



Examples


As is the case of the integer order derivative of a constant, the fractional order derivative of a constant is zero unless the differentiation order is zero.
For example, take in the DavisonEssex definition implemented in Maple, and compute the value at =0
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 (3) 
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 (4) 
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 (5) 
Note that in the lefthand side the computation was performed assuming that is an integer (see diff,symbolicorder). Recalling that n = ceil(nu), at =0 ( an integer) the DE definition is valid in a limiting sense
>


 (6) 
For > 1 and not an integer, the DE fractional derivative of a constant is equal to zero. For example, for = 1/2, the righthand side of val above becomes
>


 (7) 
The fractional derivative of order 1/2 of the cosine function
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 (8) 
For some cases, as the first in this block, the result can only be approximated (series method)
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>


 (9) 
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 (10) 
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 (11) 
The routines have limited ability to deal with symbolic fractional order; mainly they need to know the ceiling (least integer upper bound) for the order.
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>


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 (12) 
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 (13) 
This example also shows the simplified case of the DavisonEssex formula used to calculate the derivatives of the monomials in the series method.


References



Benghorbal, M. Power Series Solution of Fractional Differential Equations and Symbolic Derivatives. PhD Thesis, University of Western Ontario, Canada, 2004.


Davison, M., and Essex, G. C. "Fractional Differential Equations and Initial Value Problems." The Mathematical Scientist, (December 1998): 108116.


Liouville, J. Collected Works.


