From: Dave Rusin <rusin@math.niu.edu>
Date: Tue, 24 Nov 1998 17:36:41 -0600 (CST)
To: david_alessio@my-dejanews.com
Subject: Re: Fractional Calculus Question
Newsgroups: sci.math.symbolic

In article <73455a$qv9$1@nnrp1.dejanews.com> you write:
>Hello,
>
>I recall seeing a clever derivation of
>
>d^(1/2)/dx x    (the 1/2 derivative of x)
>and
>d^(1/2)/dx x^n or was it d^v/dx x where real v > 0
>
>using double integrals.  Does any one recall this?

The biggest problem is deciding what  d^(1/2)/dx  is supposed to mean,
and in particular, making a definition with  d^(1/2)/dx  o d^(1/2)/dx  = d/dx .

One possible candidate is to use Laplace transforms: if  f(x)  is a function
and  F(s)  its Laplace transform, then the transform of  d^nf/dx^n  is
(under suitable conditions)  s^n F(s). So you can define  d^(1/2)f / dx^(1/2)
to be that function whose Laplace transform is  s^(1/2) F(s), then 
invert back. 

If  f(x) = x, F(s) = 1/s^2, so s^(1/2) F(s) = s^(-3/2), which is the
transform of  sqrt(x)/Gamma(3/2) = 2/sqrt(pi) sqrt(x).

Is this "right"? Well, you have to decide why you wanted to consider
fractional derivatives in the first place, and decide what the "right"
definition is. 

dave
==============================================================================

To: "Dave Rusin" <rusin@math.niu.edu>
Date: Mon, 07 Dec 1998 12:41:50 -0700
From: "David Alessio" <david_alessio@my-dejanews.com>
Subject: Re: Fractional Calculus Question

On Tue, 24 Nov 1998 17:36:41   Dave Rusin wrote:
>In article <73455a$qv9$1@nnrp1.dejanews.com> you write:
>>Hello,
>>
>>I recall seeing a clever derivation of
>>
>>d^(1/2)/dx x    (the 1/2 derivative of x)
>>and
>>d^(1/2)/dx x^n or was it d^v/dx x where real v > 0
>>
>>using double integrals.  Does any one recall this?
>
>The biggest problem is deciding what  d^(1/2)/dx  is supposed to mean,
>and in particular, making a definition with  d^(1/2)/dx  o d^(1/2)/dx  = d/dx .
>
>One possible candidate is to use Laplace transforms: if  f(x)  is a function
>and  F(s)  its Laplace transform, then the transform of  d^nf/dx^n  is
>(under suitable conditions)  s^n F(s). So you can define  d^(1/2)f / dx^(1/2)
>to be that function whose Laplace transform is  s^(1/2) F(s), then 
>invert back. 
>
>If  f(x) = x, F(s) = 1/s^2, so s^(1/2) F(s) = s^(-3/2), which is the
>transform of  sqrt(x)/Gamma(3/2) = 2/sqrt(pi) sqrt(x).
>
>Is this "right"? Well, you have to decide why you wanted to consider
>fractional derivatives in the first place, and decide what the "right"
>definition is. 
>
>dave
>

Hi David,


I have used fractional calculus to solve equations
involving a Riemann-Liouville integral (similar to
the classic Tautochrome problem).

I've seen the Laplace transform used to solve this
type of problem but, have not seen it applied to 
fractional derivatives.

I enjoyed your reply--thank you for taking the time.

Thanks,
-david


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