Fractional Derivatives as Binomial Limits Researc Question: Can te limit form of te iger-order derivative be extended to fractional orders? (atematics) Word Count: 669 words
Contents - IRODUCIO... Error! Boomar not defined. - FOR IEGER POWERS... - HE GEERALISAIO... Analogy wit ewton s binomial teorem... Proposing te fractional derivative... 4 - PROOF OF COVERGECE... 5 5 - PROPERIES... 6 6 - COPUIG SPECIAL CASES... 9 7 - COCLUSIO... 4 BIBLIOGRAPHY... 6
- IRODUCIO e iger-order derivative operator can be denoted as D n in Euler notation (or equivalently n d dx n in Leibniz notation), were d D is te first-order derivative operator and n is any integer (n dx can even be negative, resulting in an integral over a identity operator). n dimensional region, or, resulting in te e fractional derivative is a class of operators tat generalises D n to non-integer values of n, including rational, real and even complex values. e motivating property of taing rational q powers of te derivative operator, in particular, is tat ( D ) p q p D (colloquially, p q D is te q t root of D p operator) for integers p, q. Suc a construction is not necessarily unique, because tere are often multiple q t roots of suc an operator (for an analogy, given tat F sin(x) = sin(x) for an unnown operator F, ten F, tat is, te square root of F, could be te derivative operator, or multiplication by i, or multiplication by i, among oter tings). In tis paper, I deduce a special fractional derivative operator in limit form, similar to te standard definition of te derivative lim f ( x ) f ( x). y motivation for tis comes from studying te proof of te wave equation from pysics, were f ( x ) f ( x ) f ( x) te limit ( ) of appears directly from studying te force on a small element of te string, wic appens to be te same as D f, wic gives us te wave Stein, Elias, and Rami Saraci.. "Fourier Analysis: An Introduction." In Princeton lectures in Analysis, by Elias Stein and Rami Saraci, 6-7. Princeton: Princeton University Press.
equation. It is possible tat a similar limit-based expression for a fractional derivative may appear elsewere in matematics or pysics, but as not yet been recognized as suc. After finding te general form of a iger-order derivatives for integer orders, I propose a generalised form for non-integer orders (in fact, tere are two separate but similar generalisations tat depend on te direction tat te limit is taen in). e generalization is motivated by ewton s binomial teorem, wic generalizes te binomial teorem to non-integer exponents. e general reason wy tis generalization wors is te existence of te n binomial coefficients in te expansion, wic vanises for > n only for integer n. Finally, I demonstrate te computation of te fractional derivatives of a few simple functions, but not before proving tat our fractional derivative satisfies some basic properties one would expect of it, and defining a new, simpler notation wic maes te wole job of dealing wit suc infinite sums a lot cleaner. I SHOR, wile tere exist fractional derivative operators in te literature, tere are multiple ways to generalize an operator to a larger domain based on wic defining properties of te operator one wises to preserve. In tis paper, we will mae our generalization based on te limitbased definition of te iger-order derivative (eorem ). e primary novel contribution of tis paper is of course te generalization itself (made in Definitions and ) in fact, it turns out tat two distinct generalisations are possible, and we call tem te rigt-anded and left-anded fractional derivatives wic we will denote as D and D respectively. We note a similarity between our generalization and ewton s Binomial teorem, prompting te construction of a new notation to represent differentials, wic elps prove several properties of te generalization. We end by computing special cases/examples of our fractional derivative operator.
- FOR IEGER POWERS e n t derivative is recursively defined as D n f D n Df, were n is an integer, and I will use Euler s notation (of representing te derivative operator wit D ) and Lagrange s notation (f (x), f (n) (x), ) trougout tis paper until we develop a new notation in Section 5. egative values of n yield multiple indefinite (te indefiniteness of te integral can be understood to be a result of te Hilbert-space matrix for te derivative being non-invertible) integrals (single, double, triple and so on), and n = is te identity operator. So let s start wit te second derivative. D f ( x) lim f ( x ) f ( x) ow we replace te f (x), f (x + ) wit teir own limit forms (te limit form for f (x) evaluated at x and x respectively): D f x ( ) lim lim f ( x ) f ( x ) f ( x ) f ( x) f ( x ) f ( x ) f ( x) ow let s do te same ting wit D, starting wit te expression above. D f ( x) lim lim lim f ( x ) f ( x ) f '( x) f ( x ) f ( x ) f ( x ) f ( x ) f ( x ) f ( x) f ( x ) f ( x ) f ( x ) f ( x)
e coefficients of te terms in te numerator sow a suspicious resemblance to te rows of Pascal s triangle/to binomial coefficients anoter one in tis form. for, wit alternating signs. Let s expand 4 D f x f ( x ) f ( x ) f ( x ) f ( x) ( ) lim f ( x 4 ) f ( x ) f ( x ) f ( x ) lim f ( x ) f ( x ) f ( x ) f ( x) f ( x 4 ) 4 f ( x ) 6 f ( x ) 4 f ( x ) f ( x) lim 4 It is clear by now wy tis is te case at eac stage, te coefficient on eac term comes from te sum of te coefficients on two consecutive terms in te previous limit. Since te coefficients on te terms of te last line are from Pascal s triangle, te coefficients on te terms of tis line is a sequence of terms formed by adding consecutive elements of tat line of Pascal s triangle i.e. te next line of Pascal s triangle. is rule can be illustrated via Figure, a diagram of Pascal s triangle taen as te array of coefficients on te terms appearing in te limit forms of te iger-order derivatives. A divergence of arrows from an entry in Pascal s triangle indicates a minus sign between te labels on te diverging arrows and a division by (e.g. diverging arrows wit labels f ( x ) and f ( x ) indicates f ( x ) f ( x) ). A convergence of arrows towards an entry indicates te addition of te labels on te arrows to form te values of te entry itself (e.g. converging arrows wit labels f ( x ) and f ( x ) result in te entry taing a value of f ( x ) ). 4
Figure : Pascal's triangle as array of coefficients to terms of limit numerator We now formalize tis intuition into a proof te nature of te intuition strongly suggests an inductive proof. o formulate te statement of our teorem, we rewrite our results so far in te following notation: D f ( x) lim f x D f ( x) lim f x D f ( x) lim f x D f ( x) lim f x 4 4 4 D f ( x) lim 4 f x 4 Wic we may generalize to te following conjecture: 5
D f x f x HEORE : ( ) lim Proof First, observe tat wen =, te rigt-and-side reduces to lim f ( x ) f ( x) f ( x ) f ( x) lim, wic is equal to (by definition), Df ( x ). Hence te statement is true for te base case =. ow suppose tat te proposition is true for some =. en D f ( x) D Df ( x) f x ( ) ( ) f ( ) f x f x ( ) f x x Our desired end result contains all te f (x + b) terms grouped togeter (e.g. all te coefficients on say for instance, f x f x, so let s regroup te terms: terms, would be added up, rater tan ave multiple multiples of D f ( x) f x ( ) f x ( f x ( ) f x ( ) ( ) We furter split te first summation into te = term and te rest of te terms, and replace te index of te second summation to elp us group terms togeter. 6
7 ( ) ( ) ( ) ( ) ) ( ) ( D x x x x f x f f f f x x f f ( ) x x f f and are just te t and ( + ) t terms of te t row of Pascal s triangle. eir sum is equal to te element of Pascal s triangle in te row below tem, placed orizontally in between, i.e.. So we rewrite tis as: ( ) ( ) ( ) ( ) D f x f x f x f x f x f x Wic is our desired result for = +. Since our proposition is true for = and true for any + if it is true for, it must terefore be true for =,, 4 and by induction, for all.
It is interesting to observe tat te sum of te coefficients is equal to, wic is te binomial expansion for ( ) and terefore equal to zero. erefore directly setting = will always directly yield /, an undefined form, and evaluating te limit is necessary. Examples and properties We will verify eorem wit a few examples based on some iger-order derivatives we already now.. ird derivative of f(x) = x at x = 4 (value arbitrarily cosen to simplify calculation). ( x ) ( x ) ( x ) x e limit we re interested in is lim out, we get 6 x 6x 9 x x x x x lim. Expanding tis, and te numerator simplifies to regardless of x. us te limit is, including wen x = 4, wic confirms our standard nowledge about te tird derivative of a quadratic polynomial.. Second derivative of f(x) = sin(x) at x =.5π. e limit we re interested in is sin( x ) sin( x ) sin x lim. We may use L Hopital s rule to evaluate te limit (since we already now wat te derivatives of sin(x) are, and are just verifying tat our limit wors) we find tat after differentiating te numerator and denominator wit respect to, te limit becomes cos( x ) cos( x ) lim sin x, wic produces / at x =.5π. 8
9. Linearity of D, i.e. D [af(x)] = ad f(x) and D [f(x) + g(x)] = D f(x) + D g(x). (i) ( ) lim im ( l ) D af x af x a f x f x ad (ii) ( ) ( ) lim lim l ( ) ( ) im D f x g x f x g x f x g x D f x x D g
- HE GEERALISAIO Analogy wit ewton s binomial teorem Understanding of te motivation for tis generalization warrants a brief discussion of ewton s binomial teorem, wic made a similar generalization of te standard binomial teorem. e analogy is useful, because te sum in te numerator of te limit closely resembles te general binomial expansion of te expression (t ) except t is replaced wit f(x + ( )). ewton s binomial teorem generalizes te standard binomial teorem ( a b) a b to non-integer (rational, real or potentially complex) powers by replacing te summation wit an infinite sum: ( ) a b a b Were ( )...( )! Is te generalised binomial coefficient defined for potentially non-integer but integer. Replacing te summation wit an infinite summation does not mae a difference for integer as is zero for >. However, te resulting expression agrees wit te aylor expansion wen is not an integer. Weisstein, Eric.. Binomial eorem. ay. Accessed April, 7. ttp://matworld.wolfram.com/binomialeorem.tml.
Once again, for integer tis reduces to te standard binomial coefficient. It is important to note tat for to be equal to zero, at least one of te terms of te product in te numerator must be. However, for non-integer, all of,, are integers wile is not, terefore cannot be equal to any of tem, and tus none of te terms is zero. On te oter and for non-negative integer, would require tat is equal to one of,,, wic is satisfied if and only if and. Examples..5.5.5 4.75! 4.5 4.5.5.5.5.5 (.5) (.5) (.5) 475..4 8 8 8! 8!. erms to 5 for te.5 t row of Pascal s triangle.5.5.5.5.5.5 ;.5;.875;.5;.9;.7 4 5 Proposing te fractional derivative We mae te same generalization to generalize te limit expression to fractional powers of te derivative operator by replacing te summation in eorem (p. 7) wit an infinite sum. DEFIIIO : For any real number, we define te rigt-anded fractional derivative D f ( x) as follows: D f ( x) lim f x
e reason I call tis te rigt-anded derivative is tat it turns out tat tis is not te only possible generalization of te limit expression to fractional powers of te derivative operator. Observe tat tis expression starts wit te term f (x + ) and counts down as f (x + ( )), f (x + ( )), f (x + ( )) For values of tat are not positive integers, tis sequence does not include te term f (x), since tere is no non-negative integer suc tat =. However, tis is purely a result of te ordering we defined in eorem (p. 7) wile maing te generalization we started te series wit f (x + ) and counted down. In oter words, we ordered te integer-order derivative as: ( ) D f ( x) lim f x f x f x... ( ) f ( x ) f ( x ) f ( x) However, we could also ave re-ordered te terms so tat te expansion starts wit f(x) and continues to f(x + ). ( ) D f ( x) lim f ( x) f ( x )... f x f x f x Wic can be written in summation notation as: D f ( x) lim f x Or: D f ( x) lim f x
is is equivalent to te result in eorem (p. 7), as sown above (since te new summation was obtained simply by re-ordering te terms in eorem ). However, tis may not be te case wit our generalization in Definition (p. ). is is a result of te fact tat te identity does not generalize to non-integer in fact, if is not an integer and eiter one of te two is defined, te oter one must be undefined as lower number will not be an integer (it s only te upper number tat need not be an integer in te generalised binomial coefficient). For example if is.5 and is, ten is defined. However,.5 and is no longer defined. Since te identity does not generalize, te binomial coefficients on te terms will be different in te two alternative expansions (te expansion in Definition and te following expansion in Definition ). We mae te generalization in te same way as before, replacing te sum wit an infinite sum. DEFIIIO : For any real number, we define te left-anded fractional derivative D f ( x) as follows: ( ) D f ( x) lim f x You may notice tat te left-anded derivative seems to be quite similar to te rigt-anded one except for a few sign canges. Indeed, if we mae te substitution, we see tat: D f ( x) lim f x ote tat x x wen approaces zero, terefore tis is equivalent to
D f ( x) lim f x ( ) However, te two fractional derivatives are not necessarily equivalent, because te substitution means tat we re taing te limit from te opposite direction. erefore, it is fair to say tat te two derivatives are equivalent wen and only wen te limit exists. In tis circumstance, we will simply denote our fractional derivative as D f ( x ). Recall our earlier comment (beginning of p. 9) on te fact tat te limit in eorem (p. 7) needs to be computed because directly setting = results in an indeterminant form. It is important to verify tat tis is te case wit our generalization (te left and rigt anded fractional derivatives) as well if it is to be useful. In oter words, we need to prove tat lim converges to. f x 4
4 - PROOF OF COVERGECE HEORE : e limit lim f x setting = ), yields an indeterminate form wen f (x) is not te zero function., wen directly evaluated (i.e. by Proof e point of te teorem is tat te limit yields a / form, so it is not a constant value for all functions, and doesn t yield someting lie / wic never maes sense. f x equals In order to prove tis, it is sufficient to sow tat f x is for positive and divergent for negative. Since f(x) is not te zero function, we are in effect evaluating te sum: e easiest way to calculate tis sum is to go bac to ewton s binomial teorem, noting tat ( ) Indeed, is wenever > and undefined oterwise, as required. 5
5 - PROPERIES Some elementary results regarding rigt and left-anded fractional derivatives are listed below: After terms = (were is te greatest integer lower tan ), te rest of te terms all ave te same sign because wile te sign of eeps flipping, so does te sign of, because a new negative number is multiplied eac time. e fractional derivative is linear, i.e. te fractional derivative of a sum of two functions is te sum of te fractional derivatives of eac function, and D f ( x) D f ( x). is follows from te linearity of te summation operator itself. Before we sow a few more properties of tese operators, we first introduce te following notation: Consider some operator (specifically a functional) H x tat maps te function f to te scalar value f ( x ). We will write tis operator as, suppressing te x (te value of x sould be inputted separately), i.e. f f ( x ) for all functions f and elements x and in te domain of f. As an example, cos. x Here is a special matematical object (not a real or complex number or any suc standard ting) tat satisfies te properties a b a b a and m am, justifying te use of exponent notation (as te defining properties of exponents are satisfied). d We say tat te differential operator (not differentiation operator D ) dx d, were d is te same d as in d dx or d dx and is an infinitesimal (i.e. it approaces ). 6
Were te last definition of te differential operator is reasonable because df f ( x ) f ( x) (from te definition of te derivative df dx ), wic is te same as. x e motivation for using tis notation is tat it maes te entirety of eorem, as well as our generalisations in Definition (p. ) and Definition (p. 4) muc more natural, as direct results of te Binomial teorem by binomially expanding out te numerator of d dx after substituting d (yes, it causes Leibniz notation for repeated differentiation to mae a lot more sense, considering tat is simply alternative notation for dx). For example, d d dx Wic is te limit expansion for te tird derivative (a special case of eorem ) expressed in tis new notation. Indeed, using te definition of and te fact tat, tis can be written in standard notation as: d dx f ( x ) f ( x ) f ( x ) f ( x) lim is notation maes Definition (p. ) and Definition (p. 4) simply te ewtonian binomial expansion of d for non-integer, and eorem a case of te integer-power binomial teorem, i.e. simply te binomial expansion of d for positive integer. is maes it easier to find te properties we re looing for te properties follow: 7
R S. R D D / / D RS / S. D RS R S RS R RS. D / / D D D S 4. 5. Q P Q D D D D D I P were I is te identity operator. ese properties are muc more difficult to sow in te fully-expanded limit form. e fourt property, in particular, is te condition for a valid fractional derivative, and te fact tat our generalization satisfies it is proof tat it is a fractional derivative at all. Bologna, auro. 4. "Sort Introduction to Fractional Calculus." Universidad de arapaca. April 7. Accessed ay 5, 7. ttp://uta.cl/carlas/volumen9/indice/aurorevision.pdf. 8
6 - COPUIG SPECIAL CASES Extracting te indefinite integral from te fractional derivative As an example, let s try to simplify D f ( x) by setting in Definition (p. ). D f ( x) lim f x ( )( )...( ) lim f x! ( )! lim f x! lim f x lim f ( x ) f ( x )... x Observe tat tis is just te Riemann sum for f ( t ) dt, were a x a. Since a is unnown, tis is simply te indefinite integral f ( x) dx. Analogously, for te left-anded derivative: D f ( x) lim f x lim f x ( )( )...( ) lim f x! lim lim f ( x) f ( x ) f ( x )... a x a x f ( t) dt f ( t) dt f ( x) dx f x 9
Fractional derivative of a constant Here, we re calculating te limit lim C for some constant C. Computing te summation as we did in eorem, tis is equivalent to lim C. Wen >, i.e. wen we re computing a derivative, not an integral, te limit evaluates to zero. Half-derivative of a linear function. Consider te function f(x) = x and substitute it into te formula for te rigt-anded fractional derivative of order note tat te term x in te summand can be replaced wit x since x x wen approaces zero. D ( x) lim x lim x o evaluate te sum, we refer once again to ewton s Binomial eorem: erefore D ( x) lim
Wic we may furter simplify as follows: / D ( x) lim lim...!...!... ( ) ( )!...!...! Were te last step is acceptable because wen te index =, te summand clearly evaluates to.... We now try to convert tis summation into an index of j, ranging from to! infinity, were j =. D / ( x) j... j! j... ( j ) j j! j ( j)! j ( 4... j) j! j ( j)! j j (... j) j! j
j j j 4 j aing a few partial sums indicates tat tis sum is divergent, and Wolfram Alpa indicates tat te solution involves te usage of te comparison test. is divergence can be proven as follows: We try to compare te term j j wit j 4 j (j +, because j is undefined for j = wile our term is not). If we can prove tat j j j 4 j for all non-negative j, we will ave proved tat te series is divergent, since te armonic series (te sum of terms j over te index j) is itself divergent. e required condition is equivalent to j 4 j ( j ). On Pascal s triangle, j j j is te central term (i.e. te largest term) of te j t row, j + te number of terms, and 4 j te sum of terms in te row. en j ( j ) j is simply wat te sum would ave been if every term were as large as te central term. aturally, tis is greater tan te actual sum, and te inequality olds. us, te sum diverges, and te alf-derivative of f(x) = x does not exist. Fractional derivative of exponential and trigonometric functions Our failure to calculate te simple alf-derivative above and in fact discovering tat te function x does not ave a alf-derivative maes us worry tat te fractional derivative does not generally exist for most elementary functions. We terefore try once more wit anoter important function, te exponential function ax e. Here,
ax D e lim ( ) e ax e a lim e a( x) Once again, we may apply ewton s generalised binomial teorem as follows: ax e a lim e ax D e ax e lim e a ax e e lim a ax ae e lim ax a e a (L' Hospital's rule) Wic proves tat te corresponding result for integer-order derivatives generalizes exactly to fractional derivatives. One can now immediately calculate te fractional derivatives of cos(x) and sin(x) wit tis and te linearity of te fractional derivative we now from te section Properties : e e i e ( i) e ( ) e e D cos x D i e e i e ( i) e ( ) e e D sin x D i i i ix ix ix ix ix ix ix ix ix ix ix ix e expressions for bot functions is intriguingly similar, a result of te fact tat sin x can itself be written as sin x ix e i e ix e wile cos x ix e ix. Additionally, one may confirm tat setting to be an integer yields te standard sequence sin x, cos x, sin x, cos x..., confirming tat te fractional derivative agrees wit our existing nowledge.
7 - COCLUSIO We ave tus produced two distinct infinite-sum expansions for a fractional derivative operator wic generalizes te operator D to all real (or potentially even complex) values of, yielding two separate operators te left-anded fractional derivative and te rigt-anded fractional derivative. Among oter properties, we were able to sow tat bot te operators satisfy te necessary conditions for being classified as a fractional derivative operator. e paper leaves several avenues for future researc:. A more detailed analysis of convergence conditions for fractional derivatives several functions wose alf-derivative I tried to tae seemed to be divergent at all values of x, or divergent at all values of x except. Yet oters yielded complex solutions wose imaginary part alone converged wile te real part diverged. A study of te details of te convergence criteria is tus necessary.. Furter study is needed on te properties of te fractional derivative, suc as to see if tere are generalisations of te product rule, cain rule, etc. for te fractional derivative. On a related note, it migt be interesting to study ow te fractional derivative beaves wit complex orders.. It migt be wort studying ow fractional derivatives and integrals would beave wit te bacground of multiple variables, as I restricted my study to single-variable calculus in tis paper. Similarly, te approac could be extended to oter fields of analysis, suc as te calculus of variations. 4. It is possible tat similar to te aylor series, wic is a sum of weiged integer-order derivatives of a function, a function could also be written as a sum of weigted fractional 4
derivatives of itself. Suc a sum would be an integral, as te space of all real numbers is te continuum. e matematical content of tis paper as possible implications in solving fractional differential equations a class of differential equations tat are gaining prominence for modelling various penomena 4, suc as in pysics, especially pertaining to fluid mecanics. any believe tat unsolved problems relating to turbulence will eventually be expressed wit fractional differential equations 5. 4 West, Bruce J. 4. "Colloquium: Fractional calculus view of complexity: A tutorial." Reviews of odern Pysics 69-86. 5 Cen, Wen. 5. "Fractional and Fractal derivatives modeling of turbulence." ArXiv. ovember. Accessed ay 5, 7. ttps://arxiv.org/ftp/nlin/papers/5/566.pdf. 5
BIBLIOGRAPHY. Bologna, auro. 4. "Sort Introduction to Fractional Calculus." Universidad de arapaca. April 7. Accessed ay 5, 7. ttp://uta.cl/carlas/volumen9/indice/aurorevision.pdf.. Cen, Wen. 5. "Fractional and Fractal derivatives modeling of turbulence." ArXiv. ovember. Accessed ay 5, 7. ttps://arxiv.org/ftp/nlin/papers/5/566.pdf.. Stein, Elias, and Rami Saraci.. "Fourier Analysis: An Introduction." In Princeton lectures in Analysis, by Elias Stein and Rami Saraci, 6-7. Princeton: Princeton University Press. 4. Weisstein, Eric.. Binomial eorem. ay. Accessed April, 7. ttp://matworld.wolfram.com/binomialeorem.tml. 5. West, Bruce J. 4. "Colloquium: Fractional calculus view of complexity: A tutorial." Reviews of odern Pysics 69-86. 6