Looking Again At Fractional Calculus

 Recall that in ordinary, single variable differential calculus one is used to seeing the symbolism:

d ⁿ f/ dx ⁿ  

Or the differential operator equivalent:  (D _x y) ⁿ

For the nth derivative of a function f with respect to x when n is a non-negative integer. We also 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 on fractional calculus I will associate the following:   

d  ^-1 f/ [dx]^-1       =     ò ^x _o   f(y) dy

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

d  ^-2 f/ [dx]^-2       =  ò ^x _o    dx ₁  ò ^x1 _o   f(x _o)   dx _o

d  ^-n f/ [dx]^-n    =  ò ^x _o    dx _n-1  ò ^xn-1 _o  dx _n-2  .....ò ^x2 _o    dx ₁ ò ^x1 _o   f(x _o)   dx _o

And use is made of the identity:

ò ^x _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:

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

d  ^-n f/ [d(x - a) ]^-n     =  

 ò ^x _a    dx _n-1  ò ^xn-1 _o  dx _n-2  ò ^x2 _o    dx ₁   ò ^x1 _o   f(x _o)   dx _o

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

d ⁿ f/ [d(x - a) ]ⁿ     =  d ⁿ/ dxⁿ 

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

d ^-n / [d(x - a) ]^{- n}   ≠  d ^-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. 

We now look at distinct examples of fractional calculus applications:

1.  Composition Rule for Mixed Integer Orders:

The identities:

d ⁿ/ dxⁿ   [d ^N f/ [dx] ^N] =   d ^n+N f/ [dx] ^N+n]  

= d ^N/ dx^N   [d ^N f/ [dx] ^N]

And:

d  ^-n f/ [d(x - a) ]^-n   [d  ^-N f/ [d(x - a) ]^-N   = d ^-n-N f/ [d(x - a) ]^-n-N

=  d  ^-N f/ [d(x - a) ]^-N  [d  ^-n f/ [d(x - a) ]^-n

Are obeyed when n and N are non-negative integers.  Indeed, these identities are basic to the concept of multiple differentiation and integration.

2. The Chain Rule for Differentiation:

The chain rule:

d g(f(x)  / dx = d(g(u)/ du  [ d/dx (f(x) ] = g  ⁽¹⁾  f  ⁽¹⁾

Is that which enables g(u) to be differentiated with respect to x if the derivation of g(u) with respect to u and of u with respect to x are known, is one of the most useful in differential calculus.   The rule can be extended to higher orders of differentiation as well as fractional orders.  For economy we can use:   

 g ⁽ⁿ⁾   and     f  ⁽ⁿ⁾

To denote:  d ⁿ/ duⁿ   g(u)   and:   d ⁿ/ dxⁿ  f(x)  respectively.  

 By approximation of the basic chain rule and Leibniz' rule to differentiate a product we have:

d ²/ dx²  g (f(x)) = d/dx [d/ dx g (f(x)) ]  =  

d/dx [d/ du g(u)] [d/dx f(x)]

= [ d/dx d/du g(u)] [d/dx f(x)] +  [d/du g(u)] [d ²/ dx² f(x)]

= [du/dx  d ²/ du ²  g (u)  ]  [d/ dx f(x)] + [d/du g(u)][d ²/ dx² f(x)]

=  g  ⁽¹⁾  f  ⁽¹⁾    +    g  ⁽¹⁾  [ f  ⁽¹⁾]²

The expression of a similar procedure yields successively:

d ³/ dx³  g (f(x)) = g  ⁽¹⁾  f  ⁽³⁾    +   3 g  ⁽²⁾  f  ⁽¹⁾  f  ⁽²⁾   +    g  ⁽¹⁾  [ f  ⁽¹⁾]²

d⁴/ dx⁴  g (f(x)) = g  ⁽¹⁾  f  ⁽⁴⁾    +   4 g  ⁽²⁾  f  ⁽¹⁾  f  ⁽³⁾   +    6g  ⁽²⁾  [ f  ⁽²⁾]²   +    

6g  ⁽³⁾  [ f  ⁽¹⁾]² [ f  ⁽²⁾]  +  g  ⁽⁴⁾  [ f  ⁽¹⁾]⁴

d⁵/ dx⁵  g (f(x)) = g  ⁽¹⁾  f  ⁽⁵⁾    +   5 g  ⁽²⁾  f  ⁽¹⁾  f  ⁽⁴⁾   +    

10 g  ⁽³⁾  [ f  ⁽¹⁾]²   f  ⁽³⁾   +    

30 g  ⁽³⁾ f  ⁽¹⁾ [ f  ⁽²⁾]²   + 10  g  ⁽⁴⁾  [ f  ⁽¹⁾]³  f  ⁽²⁾  +  g  ⁽³⁾  [ f  ⁽¹⁾]⁵

^{The generalization to large n is accomplished via Faa di Bruno's formula, i.e.}

Faà di Bruno's formula - Wikipedia

^{But that is weighted by complexity and hence of little general utility in its applications.}    

        ³. Doing Fractional Derivatives:         

The key thing here is to reckon in the Gamma function:

G (x) = (x - 1)!

 along with the Riemann-Liouville fractional integral.

(But we cannot use n < 0 because the Gamma function which we will need, is not defined for numbers less than zero).  This means in taking any fractional derivative we will be trying to preserve:

d ^N f/ dx ^N =  I ^N f(t)  =   f(t)

Where I denotes an integral containing the Gamma function, i.e. the Riemann-Liouville fractional integral where we can spot the Gamma function present in the denominator preceding the integral from a to x.  Note the condition Re(u) > 0.

The fractional derivative then will have the form:

Which we can use to do simple-   not too complex! - fractional derivatives.  Then the full form for a fractional differentiation can now be expressed.

Where we have used the Cauchy form for repeated integration and regular positive integer differentiation.  

 We now consider discrete examples of how to take fractional derivatives.  For the unit function (f= 1) we have:

d^q (1)/ [d(x - a) ]^{q    =}  lim  _N ® oo  {N/ (x – a)} ^q   G(N - q)/ G(1 - q) G(N) 

 Where the symbol  denotes the Gamma function, see e.g.

Looking Again At The Beta - And Gamma - Functions (brane-space.blogspot.com)

Meanwhile, the fractional derivative of a series is:

d  ^-1 f/ [d(x - a) ]^-1   (å ^¥ _j=0   f _j)  =  å ^¥ _j=0  d  ^-1 f _j / [d(x - a)]^-1   

                  a <   x   <  b                         

provided the series converges uniformly in the interval a <  x  <  b        

Much easier examples include:

The zero function (or any constant) for which:

       d^q (C)/ [d(x - a) ]^q   =   C  d^q (1)/ [d(x - a) ]^q    

= C [x - a) ]^q  / G(1 - q)         

Specifically:  If q = 1/2,    G(1 - q)   =  G(1 - ½) =  1.772

d^q (C)/ [d(x - a) ]^q   =  d^q (0)/1.772   =   0

Other simple examples:   

a)     D ^a  (x ⁿ) = G(n + 1)/ G(n + 1 -  a)  

  b)  D ^1/2 (1)  =  1/ Ö p x                                  

c)  D ^a  (sin (x)) =   sin  (x +   a p / 2)

See Also:

• Fractional Gamma Functions
• 
  

Suggested Problems for the Calculus Whiz:

1) Let f(x) =  y =   x ³   - 3 x ²  +   5x  - 4

And: g(u) =  x =   u ²  +  u

Use the chain rule to show:  d y/ du =  dy/dx  (dx/du)

2) Find the fractional derivatives: D ^1/2  (x ⁰)  and D ^1/2  (x ²)

3)  Find the fractional derivative: D ^1/2  (sin (x))

4) Is D ^1/2 (1)   the same as: d ^q (1)/ [d(x - a) ]^q 

^{For q = 1 in the 2nd?  Explain.}

Two Week Alberta (Canada) Getaway Just The Ticket To Escape Bonkers 24/7 U.S. Media Noise Chamber

 

                     Interior of main lodge at Cathedral Mountain Lodge, Banff, Alberta                                       

                      About to dig in to evening meal of Chicken Duxelle at Lodge

Lake Minnewanka in Alberta - restful nature surrounds you.

Just after a morning swim in Canmore, Alberta- 3 Sisters Mtns. in background

Janice at 2- Jack Lake in Alberta

Alberta, Canada, became the best two-week rejuvenation and restoration venue or us in some time.  Just to be out of the 24/7 U.S news cycle meant the promise of a newfound mental balance for Janice and myself. And a welcome refuge of sanity and sensibility.   The 2 week getaway was actually organized some months earlier, around the time we got back from Boulder. True, no blog posts would appear on Brane Space for 2 weeks but in truth there hadn’t been that many daily reads of existing posts (maybe 20-24) the past few months with only a few exceptions but none reaching 100.

Weighing the cost-benefits (i.e. in mental health) of more blogging (to get more clicks) vs. just kicking back in Alberta and enjoying its great outdoors – the latter won.  The United flight out of Denver International on the 18^th was short (1h 55 m) and delightful as we flew over the heart of the Rockies into western Canada.  Janice – as per  our earlier trips (2017, 2018) used a wheel chair because of her spinal stenosis – which expedited our transit through TSA pre-check and then Canadian customs on landing at Calgary.

Oddly, the longest part of the trip was the 90 minute wait to get a rental car at the Calgary airport.  Apparently, according to the Budget Rental car agents, too many people opted to keep their rentals a bit longer thereby creating a supply problem at the issuing end. So nearly 10 of us had to patiently wait until the cars came in, were cleaned and readied for the next driver.

But by 5:15 we had a terrific car (2023 Nissan Murano) with an AI –directed navigation system and all we needed to head for Canmore.  Less than an hour and a half later we drove into the underground parking of The Malcolm hotel- named after the Scottish king who ruled 35 years from 1053.   The place was magnificent to say the least with all the creature comforts and  amenities one could want. Set in the middle of some of the most surreal mountain scenery The Malcolm offered us the ideal place to relax and recoup for 3 days, including use of its hot tub and pool.  

By Thursday, Sept. 21 we set out at 11:05 for the Cathedral Mountain Lodge, our next holiday stop.  This trip (104 km) lasted just under 80 minutes and took us right into some of Alberta’s most remote mountainous regions – ending up at a totally rustic cabin. The cabin was small (about 12’ by 18’) but comfortable and featured a wood-burning stove with logs all pre-cut. One only had to venture outside to get a whiff of the smoke from multiple such stoves.  As far as food, we only needed to travel as far as the main lodge to get dinner every night at 6:30 p.m. and breakfast at 8:00 a.m.  Nothing fancy here, just good, solid stick to the ribs vittles.

While at the Cathedral Lodge we drove 5km out to see the enchanting Emerald Lake where we ambled amongst the bucolic setting and along the jutting wood bridge taking lots of photos. This was before finding a picnic bench to eat our light lunch (apples, and BBQ chips).  Later on we drove to Takakkaw Falls, some 15 km further on the Yoho Valley Road in Banff National Park. (For which we had to purchase two senior passes before we could enter the National Park- that we did in Canmore)  

Of course, merely because one arrives in a fantastic natural setting doesn't mean one can let one's guard down - say from grizzly bears where one killed two people in the same Banff National Park, e.g.

Grizzly killed after 2 people found dead in bear attack at Canadian park, officials say

The peaceful lodge itself offered a quiet, contemplative environment free of the political cacophony in the U.S. and latest GOP House crazies’ antics, with the shutdown.  To that point there was no television-cable access, period. It may surprise many to grasp this is a healthy aspect.  Consider: According to a Global Information Industry report (as long ago as 2008), Americans consumed information for about 1.3 trillion hours, an average o almost 12 hours per day.  Consumption then totaled 3.6 zettabytes and 10,845 trillion words, corresponding to 100,500 words and 34 GB for an average person on an average day. 

Now, 15 years later, one can infer all those stats are significantly higher. Is this positive? Is it constructive?  No, absolutely not. This is given all the fragmentation and accompanying polarization which is also fragmenting minds.  Especially since Donald Trump ascended to the political and media stage in 2015.  His emergence alone balkanized the media field even more, including social media – spawning a host of authoritarian and regressive websites where covert planning could lead to such outrages as the 2021 insurrection.

Forget also a proper processing of events – far less any critical thinking – given the situation bears more similarity to trying to ingest a mass data dump than anything else. Hence, little to no empowerment, insight or thought mastery but instead endless distraction, consumption of lies and propaganda – leading to numerous outbursts o extremism such as the threats to election officials by the MAGA maggots.

One wonders what opposite effect “pulling the plug” might have had on the millions of denizens that fell for Trump’s lies. Indeed, one can imagine it would not have led author David Ulin to have made such a statement – as recently as February, 2018, to the effect: 

“We are more divided and divisive. We live inside the psychodrama of  disruptive would-be king.  I am writing this in February, 2018, in a nation that feels to me as if it is coming apart at the seams.”

And that was basically, 3 years before the Trump-incited insurrection, and more than 5 ½ years before Trump called for the execution of Gen.Mark Milley for “treason”.  This was because in December, 2020 with Trump off the rails and still with access to the nuclear codes, Milley sought to calm Chinese fears that missiles might be launched.  So Gen. Milley rightly protected this nation and the world from a confirmed madman.

But despite Trump’s execrable threat, not a peep about it in the mainstream press. This is what author Ulin means by the “commonwealth of information unmoored from reality.” And if that is so, how can citizens – who depend on the media’s consistent and reliable information – not become unmoored from reality?  Fortunately, none of these concerns intruded on us while on our two week vacay in Alberta.  None of it intruded until the GOP House scheduled its shutdown to hold the nation hostage. Showing the extent to which they are allied with the orange Traitor who Noam Chomsky has called “the worst criminal in human history.”

No, Trump hasn’t butchered 6 million….yet.  But he’s set the stage for mass butchery by infecting the minds of the most susceptible to his lies – the MAGA extremists- with his sullen seeds of violence.  Former Bush advisor Matthew Dowd has estimated at least one third of the country may be infected by his anti-democracy mind virus and the need now is to halt the spread to the ever  -distractable swing voters.

  Thankfully, Trump and his debasing conduct were only a distant blur while hiking along the trails near Canmore’s river and in the woods near the Cathedral Lodge. In the former case the majestic views of the ‘Three Sisters’ mountains wiped out any pestering thoughts of a pestilence like Donald Trump. It was a blessing and respite we wished for all our fellow citizens.  

At least, when we returned yesterday evening we were pleased to see the positive news that the insane effort at a government shutdown was halted.  Now, we need to see if sanity is preserved through Nov. 17 - the next crisis data for the Chaos Caucus.

See Also:

• Who Shot And Killed Bear Number 148?