Looking to spice up your calculus classes? Try fractional order integrals and/or derivatives. These are great products, and they have been around since almost the beginning of calculus. In fact, these operators have quite an extensive history, so much so that I do not have space to go into it here.

Applications for fractional calculus are quite numerous. They are found mostly in the study of systems that exhibit memory: anomalous diffusion, fractional relaxation and the like (a good place to find information is the arXive, arxiv.org).

The most common formulation of fractional integration is the Riemann-Liouville fractional integral. There are other formulations of fractional integration but most of them can be shown to be special cases of Riemann-Liouville. It is relatively easy to obtain the Riemann-Liouville fractional integral in a manner accessible to students that have got at least one calculus course under their belts. Denote the integral of f(x) over the interval (a,x) as

.

Note that this operator, , has a left inverse, namely

.

Now consider the following identity

. (1)

Verifying this formula in class is a nice way to see if they get differentiating an integral. Once the students believe equation (1) it is easy to build up to higher orders:

. (2)

Now if you apply to equation (2) you get back your original function

.

Since is a left inverse for we can make a leap in logic and identify the n + 1 iteration of

, (3)

or,

. (4)

(4) is the Cauchy formula for iterated integration. At this point you might want to do something to verify the formula (maybe let n = 3 and apply it to a polynomial or trig function).

To obtain the Riemann-Liouville fractional integration formula from (4) first exchange the factorial for its gamma function equivalent

.

In this form there is nothing to stop us from letting n take on values that are not whole numbers. To make note of this new freedom, replace n with the Greek letter

. (5)

(5) is the Riemann-Liouville formulation of a fractional integral. There are more rigorous derivations of (5), but what is presented above should be accessible to undergraduate calculus students. The domain of can be extended in various directions. I was able to show [arXive:math-ph/0312051] that could even take on matrix values. Roughly speaking, an integral of almost any order of almost anything can be defined. We should note that (5) is well defined for negative values of (provided is not a negative integer) but it should not be interpreted as a fractional derivative.

For many functions the integral in (5) can be somewhat difficult to evaluate. Fortunately there are CAS, such as MAPLE, that can come to the rescue (doing a few fractional integrals is good practice for gaining proficiency with MAPLE).

Over the past few semesters I have been introducting fractional calculus when and if I get ahead of schedule. Of the 30 or so students in class about half do not appear interested. Though the look of relief on their faces is quite entertaining when they find out that the material will not be on the exam. The other half of the class finds the material interesting, a couple even find it very interesting. The presentation of this material usually starts a good discussion about what an integral really is and how to interpret it. This discussion benefits all the students, even the ones that were not interested. So, if you are looking for some spice for your calculus class, try a dash of fractional integration.

Mark Naber, Two-Year College Vice Chair

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