Friday, July 23, 2021

An Introductory Look At Fractional Calculus

 In ordinary, single variable differential calculus one is used to seeing the symbolism:

d n f/ dxn  

For the nth derivative of a function f with respect to x when n is a non-negative integer. We know integration and differentiation are inverse processes so it is natural to associate the particular symbolism

d  -1 f/ [dx]-1    

with the indefinite integration of f with respect to x. However. one must stipulate a lower limit of integration in order that the indefinite integral be completely specified. For the purpose of this blog post I will associate the following:   

 -1 f/ [dx]-1       =  ò o   f(y)   dy

The preceding can be generalized to the case of multiple integration, whereby, for example:

 -2 f/ [dx]-2       =  ò o    dx 1  ò x1 o   f(x o)   dx o

 -n f/ [dx]-n    =  ò o    dx n-1  ò xn-1 o  dx n-2  .....ò x2    dx ò x1 o   f(x o)   dx o

And use is made of the identity:

ò a   f(y)   dy   =   ò x-a 0   f(y + a)   dy

To extend the formalism to lower limits, i.e. other than zero. Thus we may define:

 -1 f/ [d(x - a) ]-1     =   ò a   f(y)   dy

 -n f/ [d(x - a) ]-n     =  

 ò a    dx n-1  ò xn-1 o  dx n-2  ò x2    dx 1   ò x1 o   f(x o)   dx o

And caution obviously needs to be exercised when applying the contracted equivalent form, e.g.

f/ [d(x - a) ]n     =  d n/ dxn 

characteristic of a local operator to negative orders, or to fractional orders of either sign, given:

-n / [d(x - a) ]- n     -n/ [dx] -n 

The key point to bear in mind here is that the appearance of fractional orders is what defines fractional calculus. This use also marks the emergence of what we call differintegrals. From the simplistic basis provided here it is then possible to venture into the realm of differintegral operators and their application.  This in turn will introduce the differintegration of assorted functions which we will examine over time.  In the case of fractional calculus it is best not to try to imbibe too much at once. But ultimately - perhaps by sometime next year- we will want to look at the most powerful application of the theory.  That would be in the diffusive transport in a semi-infinite medium.   Before that, we will also be going into the role of the gamma function, and see how it applies to differintegration.  

In the meantime,  we can at least have a sampling of what awaits. In particular, the symbol  f (n) finds frequent use in fractional calculus as an abbreviation for  f/[dx] n .   This affords the benefit of brevity and in fractional calculus we look for such economy wherever we can find it.  Similarly, (-n)   will occasionally be used to represent the n-fold integration of f, with respect to x, the lower limits being unspecified.  Thus,  

(-n)   ò an    dx n-1  ò xn-1 an -1  dx n-2  ....      ò x2 a2   dx 1   ò x1 a1   f(x o)   dx o

where a 1,  a 2 ,  a n  are completely  arbitrary.  Note, however, that when we write a difference, e.g. 

(-n) (x)  -   (-n) (a)  we intend that the same lower limits (a 1,  a 2 ,  n) attach to each integral.

Finally, note that  -1 f/ [d(x - a) ]-1   has been used instead of   

 -1 f/ [d(x - a) ]-1 (x)  

since f and  -1 f/ [d(x - a) ]-1  are understood to be functions of the independent variable x.  If it is necessary to specify the value of x at which a function is to be evaluated, we simply write:   f(a)  or:

[ -1 f/ [d(x - a) ]-1]x= xo

Suggested problem:

Write the multiple integral form for   -3 f/ [dx]-3 

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