On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

Transcription

1 Cent. Eur. J. Phys DOI: /s Central European Journal of Physics On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling Research Article Guy Jumarie Department of Mathematics, University of Quebec at Montreal P.O. Box 8888, Downtown Station, Montreal Qc, H3C 3P8, Canada Received 2 April 213; accepted 22 May 213 Abstract: PACS 28: Keywords: It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments which might be clarifying to some readers on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features. 2.3.Gp; 2.3.Mv; 2.6.-x; 2.3.-f fractional calculus fractional Taylor s series fractional derivative systems modelling fractional derivative chain rule Versita sp. z o.o. 1. Introduction 1.1. Statement of the main problem Loosely speaking, there are two main trends in defining the framework of fractional calculus: on the one hand, the formal definition of fractional derivative as an anti- jumarie.guy@uqam.ca integral; and on the other hand, the approach via fractional difference which works exactly like the standard Leibniz classical differential calculus. In this model one can arrive easily at a fractional Taylor s series which refers to increment of fractional order, from where one can obtain various formulae for fractional derivative chain rules involving fractional derivative of functions of functions. It appears that these fractional derivative chain rules can provide different results for the same function, and at first glance, this could be thought of as a defect. 617

2 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling In the present short paper, we shall examine this question, and we shall show that, to some extent, the matter is quite right as it is, and is merely a result of the noncommutativity of fractional derivatives. The main conclusion is that the selection of the suitable fractional derivative chain rule for a given problem will depend upon the definition of the system which we are dealing with. The paper is organized as follows. To begin with, we shall bear in mind the essential of fractional calculus via fractional difference, and we shall take this opportunity to contribute new results on the fractional Taylor s series of multivariabe functions. Then we shall carefully derive some fractional derivative chain rules, which will be illustrated later with numerous examples, and quite in a natural way we shall arrive at the conclusion that care must be exercised in the fractionalization process of dynamical systems. But before we summarize some features which are hidden in fractional calculus. To summarize, the key lemma is as follows: a function which is continuous everywhere but is nowhere differentiable exhibits random-like properties and as a result should have several derivatives Background on some features hidden in fractional calculus Hurst exponent and fractional calculus The main, or at least, one of the main motives to investigate fractional differential manifolds is the fact that fractional calculus seems to be quite relevant to investigate some problems which occur in fractal space-time physics. Basically at least in a first elementary point of view the mathematical framework of fractal physics deals with functions fx of which the differential df satisfies the condition dfx = o dx H, H >, 1 where o. denotes the Landau s symbol and H is referred to as the Hurst exponent. This relation merely means that the classical equality dfx = f xdx no longer holds, fx is not differentiable, and that instead we should use a modelling in the form dfx = gxdx H, 2 which directly leads us to fractional derivative. Self-similarity, non-differentiability, fractional calculus Loosely speaking, qualitatively speaking, a self-similar function is a function which exhibits similar patterns when one changes the scale observation: the patterns generated by fx and fax, a >, look like the same. Formally, fx, x R, is similar of order H if one has the equality fax = a H fx, a >, H >, 3 which means that the landcapes in the vicinity of x and of ax look the same. Such a function satisfies the condition f =, and furthermore 3 provides the relation via the substitution a x, x 1 fx = x H f1. It turns that, in the special case when < H < 1, fx is not differentiable, but is differentiable of fractional order H. Coarse-grained phenomenon and fractional calculus In systems involving coarse-grained phenomena, everything happens as if the elemental point is not infinitely small, but on the contrary exhibits some thickness, what could be pictured by using dx H, < H < 1, instead of dx. In other words, we would be led to consider the rate of variation df/dx H, and once again we come across fractional derivative and fractional calculus. Continuity, non-differentiability, pseudo-randomness and fractional calculus We can examine a physical systems at three different levels of observation: microscopic, mesoscopic and macroscopic, and non-differentiability is quite relevant in the first one. It has been pointed out that a function which is continuous everywhere but is nowhere differentiable cannot be replicated, and on the contrary, exhibits randomlike features. Various samplings ofsuch a function on the same given interval will be different. Shortly, nondifferentiability infers pseudo-randomness, and in quite a direct way, we are suggested to use random white noise in the corresponding modeling framework. And of course, in a generalized approach, we shall introduce fractional white noises that is to say fractional derivative and fractional calculus. Purpose and organization of the article We have proposed recently a fractional calculus based on fractional difference [9 16] which is slightly different from the classical Riemann-Liouville framework [1 6, 2 28], and results in a useful fractional Taylor series [12] providing 2 as the first term. The fractional calculus so obtained is quite parallel to the classical calculus, and it involves non-commutative derivatives, what seems to be quite consistent with non commutative geometry. We used 618

3 Guy Jumarie it recently to outline an elementary theory of differential geometry of fractional order, and our purpose herein is to contribute some new results in this approach. For other points of view on fractional calculus, see for instance [7, 8, 17 19]. After a short background on the definition of the modified Riemann-Liouville derivative and the related fractional Taylor s series, we shall successively display some formulae involving fractional derivative of compounded functions, and some formulae involving integrals with respect to dx α, all prerequisite which we shall need for our purpose. We shall take this opportunity to introduce fractional derivative on the left via fractional difference on the left. Then, in this framework, we shall derive the proofs of some fractional derivative chain rules, and later, we shall examine some of the results of their applications in systems modelling. 2. Summary of fractional derivative via fractional difference 2.1. Fractional derivative on the right Definition 1. Let f : R R, x fx, denote a continuous but not necessarily differentiable function, and let h > denote a constant discretization span. Define the forward operator FW h by the equality the symbol := means that the left side is defined by the right one FW hfx := fx + h; 4 then the fractional difference on the right and of order α, < α < 1, of fx is defined by the expression +fx α := FW 1 α fx α = 1 k f [x + α kh], 5 k and its fractional derivative on the right is the limit f α + x := lim h α [fx f] h α = d α fx dx α Fractional derivative on the left Definition 2. Refer to the framework of the of the definition 1. The fractional difference on the left and of order α, < α < 1, of fx is defined by the expression fx α := 1 FW 1 α fx 7 α = 1 k fx kh, 8 k and its fractional derivative on the left is f α x := lim h α [fx f] h α. 9 Remark that one has the equality α fx = FW α h α +fx. These definitions, which are local definition as compared with the standard approach via integral, is close to the standard definition of derivative calculus for beginners, and as a direct result, the α-th derivative of a constant, < α < 1, is zero. In the following, we shall take interest mainly and only in the derivative on he right, and to shorten the writing in the following, we shall set f α + x f α x = d α fx/dx α, and here we are fully in Leibniz framework, that is to say both d α f and dx α denote finite increments. As a result, formally, most of the classical formulae will be able to be duplicated in a straightforward manner by making the substitution df d α f and dx dx α Application to the non-commutativity of modified Riemann-Liouville derivative Corollary 3 Riemann-Liouville definition revisited. As a direct result of the definition 1, the fractional derivative of order α, α <, can be written in the integral form f α x := 1 x x ξ α 1 fξdξ, α <. 1 Γ α For positive α one will set and = f α x := f α 1 x, < α < 1, 1 d Γ1 α dx x x ξ α fξ f dξ 11 f α x := f α n x n, n α < n + 1, n

4 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling Indication on the proof. It is sufficient to show that 6 and 11 have the same Laplace transform. Remark the difference between 1 and 11. The second one involves the constant f whilst the first one does not. We shall refer to this fractional derivative as to the modified Riemann Liouville derivative. With this definition, the Laplace s transform L{.} of the fractional derivative is L { f α x } = s α L {fx} s α 1 f, < α < The definition in equation 8 is different from Caputo s definition; and the two definitions are equivalent only when fx is differentiable. It follows from 13 that the fractional derivative, as so defined, is not commutative. Clearly one has D α D β fx D β D α fx, as a result of the Laplace s transforms L { D α D β fx } = s α+β L { fx s α+β 1 f } s α 1 f β and L { D β D α fx } = s α+β L { fx s α+β 1 f } s β 1 f α. The commutativity holds when and only when f β = f α. To some extent, this absence of commutativity is quite consistent with non-commutative geometry and quantum probability. Remark that the Schwarz equality holds, that is α x α yfx, y = α y α x fx, y, 14 and this is a direct result of the formal equality FW 1 α x FW 1β y = FW 1 β y FW 1α x which provides α x β yfx, y = β y α x fx, y. The point of importance should be understood as follows: the main motivation for introducing this new approach is that it provides a definition of fractional derivative for functions which are not necessarily differentiable, as it is the case with the useful white noise, for instance Fractional derivative and coarse-grained space One of the main motive and probably the main one for using fractional calculus in physics lies in the fact that, many often, the space which we are dealing with exhibits some coarse-grained phenomenon, in the sense that the physical point which generates this space is not infinitely small, but rather exhibits a thickness. In other words, if we denote by x the standard point of the space, then the differential dx cannot be infinitely small but rather is bounded by a minimum thickness which cannot be reduce at will. A way to take account of this phenomenon is to introduce the term h α, < α < 1, which, for small h, is larger than h: h α > h. For instance one can introduce the derivative fx h α = fx + h fx h α which can be considered and has been considered [17, 18] as a point of departure for fractional calculus. 3. Main results on Taylor s series of fractional order 3.1. Fractional Taylor s series for one-variable functions A generalized Taylor expansion of fractional order which applies to non-differentiable functions F-Taylor series in the following reads as follows [9] Proposition 4. Assume that the continuous function f : R R, x fx has fractional derivative of order kα, for any positive integer k and α, < α 1, then the following equality holds, fx + h = h αk αk! Dα k fx, < α 1, 15 where, and this is of paramount importance D α k fx = D α D α...d α fx, k times, is the derivative of order α +α α of fx, and with the notation Γ1 + αk =: αk!, with Γ. denoting the Euler gamma function. 62

5 Guy Jumarie For the proof, see the subsection 3.2. Remark that we write D α k instead of D kα as a result of the inequality D α D α D 2α. Formally, this series can be written in the form fx + h = E α h α D α x fx, 16 where D x is the derivative operator with respect to x, and E α u is the Mittag-Leffler function defined by the expression u k E α u = αk!. This fractional Taylor s series does not hold with the standard Riemann-Liouville derivative, and it applies to nondifferentiable functions only. In addition it is different from Osler s fractional series [25]. Indication on the proof. If we refer to the forward operator FW h defined by the equation4, then, on using 5, one can show that it satisfies the formal fractional differential equation D h α FW h = D x α FW h, of which the solution is FW h = E α h α Dx α h kα Dx α = k. kα! Another way to take this fractional Taylor s formula for granted is as follows. Firstly, it is a simple task to check that it applies to the Mittag-Leffler function. It is then sufficient to expand a theory of approximation of functions by sequences of Mittag-Leffler functions to get the result. Let us point out that the first two terms of this fractional Taylor s series, that is to say the corresponding Rolle s fractional theorem, has been already obtained by Kolwankar and Gangal [17, 18] who work with Cantor s sets. Corollary 5. Assume that m < α m + 1, m N {} and that fx has derivatives of order k integer, 1 k m. Assume further that f m x has a fractional Taylor s series of order α m =: β provided by the expression f m x + h = h kα m Γ[1+kα m] D α m k f m x, m < α m + 1. Then, integrating this series with respect to h provides fx + h = m h k f k x + h kβ+m f kβ+m x, k! Γkβ+m+1 k=1 β := α m The order of the derivation in f kβ+m x is of paramount importance and should be understood as D kβ f m x, since we start with the fractional Taylor s series of f m x Fractional Taylor s series for multivariable functions Approach via one-variable fractional Taylor series Expansion of α, α-order Assume that fx, y is non-differentiable with respect to x; then we can write the series fx + h, y = h kα kα! D α x k fx, y, 19 and then, on assuming now that the fractional derivatives with respect to x are themselves non-differentiable w.r.t. y, the fractional Taylor series w.r.t. y yields fx + h, y + l = = h kα kα! r= Expansion of α, β-order Here, one writes directly therefore l rα D α r rα! y Dx αk fx, y h kα l rα D α r kα!rα! y Dx αk fx, y. 2 fx + h, y + l = E α h α Dx α E α l β Dy β fx, y h kα l rβ = kα!rβ! Dkα x Dy rβ fx, y, 21 fx, y = r= r= x kα y rβ kα!rβ! Dkα x D rβ y fx, y x=,y= Approach via one-variable Mittag-Leffler function Proposition 6. Framework of Proposition 4, but extended to two independent variables. Under some mathematical conditions which are implicit in the result, the real-valued function fx, y of the scalar real-valued variables x and y can be expanded in the form of the two-variable fractional Taylor s series α fx + h, y + l = E α hdx + ld y fx, y α k hdx + ld y = fx, y. 23 kα! 621

6 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling Proof. The formal proof of this result is parallel to that of Proposition 4 and reads as follows. Step 1 Define the operator FW h,l fx, y := fx + h, y + l, from where we write the operator identity D t = hd x + ld y. 28 from where we set Step 5 Taking the αth-power of 28 yields the equality FW h,l tfx, y := fx + ht, y + lt, 24 where t is an auxiliary real-valued parameter such that t 1. Step 2 According to the standard Taylor s series for one variable applied to t, one can write the series D t α = hd x + ld y α, 29 which is associated with the equation FW h,l tfx, y = e hdx +ldyt fx, y. 25 D α t FWh,l t = hd x + ld y α FWh,l t, 3 Step 3 Taking the standard derivative of 25 with respect to t yields of which the solution is Dt FWh,l t fx, y = hd x + ld y e hd x +ld yt fx, y = hd x + ld y FWh,l tfx, y. 26 Step 4 Formally, we then obtain the operational differential equation FW h,l t = E α hdx + ld y α t α. 31 Step 6 Making t = 1 in 31 yields the result. D t FWh,l t = hd x + ld y FWh,l t. 27 For instance, close to,, 2 yields fx, y = f, + xα α! f x α, + yα α! f y α, + 2α! f xx 2α, + xα y α α! f 2α 2 xy, + y2α x 2α! f xy 2α, + o 2 + y 2 3α/ x2α Approach via multivariable Mittag-Leffler function Another formal approach based on operational calculus and which needs further investigation runs as follows. Proposition 7. Assume that the continuous function f : R 2 R, x, y fx, y has fractional derivative of order kα, for any positive integer k and α, < α 1, then the following equality holds, and fx + h, y + l = fx, y = 1 αk hdx + ld y fx, y, 33 αk! 1 αk xdx + yd y fx, yx=,y=. 34 αk! where f αk x is the derivative of order αk of fx, and with the notation Γ1 + αk =: αk!, 622

7 Guy Jumarie where Γ. denotes the Euler gamma function. Proof. The general solution of 41 is Proof. Step 1 On the definition of α fx, y We introduce the bi-variate forward operator F w h, l defined by the equality F w h, lfx, y := fx + h, y + l. 35 We then have successively hl fx, y = fx + h, y + l fx, y = fx + h, y + l fx, y + l +fx, y + l fx, y = x fx, y + l + y fx, y = x F wy f + y f = x F wy + y f = F wx 1 F wy + F wy 1 f = F wx F wy 1 f. 36 ux, y = E α ax + φy α, 44 where φy is to be determined. To this end, using the derivative rule 48, 44 yields u α x x, y = ux, y φ y α, and on equating to 42 we obtain the equation which yields ux, y φ y α = b α ux, y φ y = b We are then led to define hl α fx, y by the operator hlfx, α y = F wx F wy 1 α fx, y α = 1 k f x + α k h, y + α k l. 37 k that is to say therefore the result. φy = by, Step 2 Derivation of a bi-variate fractional Taylor s series Duplicating the rationale of fractional Taylor series, but by using now hl α fx, y, it is easy to obtain the two operational equations and D α h F wh, l = F w h, ld α x 38 D α l F w h, l = F w h, ld α y. 39 Further remarks and comments We can now get more insight on the kinds of mathematical assumptions which are required to support the above result. Shortly, we have to compare 2 and 21. At first glance, the difference between would be a matter of differentiability. Indeed, if fx, y is non-differentiable with respect to both x and y, one at a time, then 3.6 applies. Assume now that in 21 we write The solution of which is F w h, l = E α hdx + ld y α 4 hd x + ld y α = α hd x k ld y α k, k is provided by the following lemma. Lemma 8. Under suitable mathematical assumptions of regularity, the solution ux, y of the fractional system u α x x, y = a α ux, y, 41 u α y x, y = b α ux, y, 42 where a and b denote two constants, is ux, y = E α ax + by α. 43 and the like for the powers kα, then we arrive at the conclusion that if fx, y is non-differentiable w.r.t. y, but differentiable w.r.t. x, then it is the series 21 which applies. Likewise for the non-differentiability w.r.t. x and differentiability w.r.t. y. This remark remains to be deepened, but nevertheless it already points out that we have to be very careful when we deals with non-differentiable functions. And to the readers who wonder why we have such an obsession for non-differentiable functions, we shall merely flourish the two key-words fractal space-time and Gaussian white noise. 623

8 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling 4. Some useful formulae of fractional calculus 4.1. Fractional derivative of compounded functions The equation 15 provides the useful differential relation shortly, we can claim that α-th differentiability implies non-differentiability. Corollary 9. The following equalities hold, which are d α f = Γ1 + α df, < α < 1, 45 D α x γ = Γγ + 1Γ 1 γ + 1 αx γ α, γ >, 46 or in terms of fractional difference, α f = α! f, which holds for non-differentiable functions only. As a result, or, what amounts to the same we set α = n + θ D n+θ x γ = Γγ + 1Γ 1 γ + 1 n θx γ n θ, < θ < 1, uxvx α = u α xvx + uxv α x, 47 f [ux] α = f α u uu x α. 48 According to the αth-derivative D α u of uwith respect to u itself, one has the equality therefore the relation d α u du = 1! α 1 α! u1 α, 49 du α = 1 α!u α 1 d α u, 5 which provides on dividing both sides by dx α u x x α = 1 α!u α 1 u α x x. 51 therefore, on substituting into 48 f α x ux = 1 α!u α 1 f α u u uα x x. 52 The Leibniz rule 47 applies when and only when both ux and vx are non-differentiable at the considered point. 48 applies only when fu is non-differentiable whilst ux is differentiable Integration with respect to dx α The integral with respect to dx α is defined as the solution of the fractional differential equation dy = fxdx α, x, y =, < α < 1, 53 which is provided by the following result: Lemma 1. Let fx denote a continuous function, then the solution of the equation 53 is defined by the equality [9] y = = α x When x a, one has y = = α x x x fξdξ α x ξ α 1 fξdξ α, x, < α < fξdξ α x ξ α 1 fξ + adξ α, x a, < α < Useful example A special case of interest is the equation we set fx 1 in 53 in which case 54 yields dy = dx α 56 y = x α. 57 The fractional integration by part formula b a b u α xvxdx α = α! [uxvx] b a uxv α xdx α a

9 Guy Jumarie can be obtained easily by combining 45 with 47. Change of variable. Consider the variable transformation y = gx in which gx is a non-decreasing differential function then, according to 52, one has the equality f y dy α = f gx g x α dx α, < α < 1, and when gx has a positive fractional derivative of order β, < α, β < 1, one has f y dy α = Γ α 1 + β f gx g β x α dx αβ, < α, β < Fractional derivative of fractional integrals The relation between fractional integral and fractional derivative reads d α x fξdξ α dx α = Γ1 + αfx = α!fx, 59 d α dx α ux fξdξ α = α!f ux u x α. 6 The proof of 59 results from the combination of the equalities y α x = fx and d α y = α!dy which yields the useful formula x y α ξdξ α = α!y Fractional derivative of compounded functions 5.1. Main results Lemma 11 Leibniz rule for non-differentiable functions. Assume that ux and vx are two non-differentiable functions; then one has the fractional derivative chain rule Proof. uxvx α = u α xvx + uxv α x. 62 We start from the equality duv = vdu + udv which provides α!duv = vα!du + uα!dv. But since ux and vx both are non-differentiable, we can use the conversion formula 45 therefore the result. Lemma 12. Let us consider the compounded function fux. Assume that fu is αth-differentiable with respect to u, and that ux is differentiable with respect to x. Then one has the fractional derivative chain rule Proof. f [ux] α = f α u uu x α. 63 It is sufficient to write d α f ux du α = dα fu du α du α dx. α Lemma 13. i Assume that fu is differentiable with respect to u and that ux is αth-differentiable with respect to x. Then one has the fractional derivative chain rule f [ux] α = f/u 1 α f uu α u α x. 64 Proof. i One first remark that the αth-derivatives of with respect to yields the d α u, du α conversion formula d α u = 1 α! 1 u 1 α du α, 65 and like-wise for f, that is to say d α f = 1 α! 1 f 1 α df α. 66 ii This being the case, one has successively d α f dx α = dα f d α u 67 d α u dx α in which, according to 65 and 66 one has d α 1 α α f f df d α u =. u du Lemma 14. Assume that both fu and ux are αth-differentiable with respect to u and x respectively, then one has the equality f [ux] α = 1 α!u α 1 f α u uu α x. 68 Proof. We refer to the equation 67, but now we use only 65 to convert d α u into du α. 625

10 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling 5.2. Illustrative examples and tricks In this section, we shall apply directly the formulae above to various functions, to see the kind of results one so obtains and to check whether they look like what we would expect to obtain. For each example, we shall apply successively the formulae D α 1 f [ux] = f α u uu x α, 69 D α 2 f [ux] = f/u1 α f uu α u α x, 7 D α 3 f [ux] = 1 α! uα 1 f α u uu α x. 71 Example 15. For the function Comments. Here again, we have the same comments as above in the example 15. Indeed ux is differentiable in 77 and is not in 78 and 79. Example 17. We now refer to the Mittag-Leffler function f 3 u = E α λx α = E α λ 1/α x α = E α u α, where λ is a real-valued parameter and < α < 1. We successively obtain D α 1 f 3 ux = E α u α λ 1/α α, 8 f 1 ux = kux, 72 that is to say the well known famous formula one has D1 α f 1 ux = k 1 α! u1 α u α x, 73 D2 α f 1 ux = 1 α ku k α u α x u = ku α x, 74 D3 α f 1 ux = 1 α!u α 1 1 k 1 α! u1 α u α x = ku α x. 75 and D α 2 f 3 ux = = D α E α λx α = λe α λx α, Eα λx α 1 α [D λ 1/α u E α u α ] α λ 1/α x 1 α! x1 α λ 1 α! [E α λx α ] 1 α [D u E α u α ] α. 81 Comments. 74 and 75 are the expected solution. The discrepancy with 73 can be explained by the fact that uxis differentiable in 73 whilst it is not in 74 and 75. Example 16. We now consider the function But E α u α is not differentiable w.r.t. u in such a manner that 81 fails to apply. D3 α f 3 ux = 1 α! λ 1/α x α 1 Eα λx α λ 1/α 1 α! x1 α = λe α λx α. 82 to obtain f 2 ux = u n x, n N {}, 76 Comments. At first glance, everything is right and there is no inconsistency. D1 α f 2 ux = D2 α f 2 ux = n! n α! un α u α x, 77 u n 1 α nu n 1 α u α x u = n α u n 1 u α x, 78 u D3 α n 1 α f 2 ux = nu n 1 α u α x u = n α u n 1 u α x. 79 Example 18. Let f 4 u denote the function f 4 ux = fx + h, 83 where h denotes a given constant. We then have sucessively 626

11 Guy Jumarie D1 α f 4 x + h = f α 4 u, D α 2 f 4 ux = fx + h x + h 1 α f 4u α D α 3 f 4 ux = 1 α!x + h α 1 f α u x 1 α 1 α! = 1 x 1 α! x + h 1 x u 1 α! x1 α = x + h 84 1 α f4 u 1 α f 4u α, 85 1 α α f 4 u. 86 Comments. The result D α 1 f 4x+h in 84 is quite expected; but in contrast the derivatives D α 2 f 4x +h and D α 3 f 4x +h look like rather surprizing. Our claim is that these results which are derived from 7 and 71 do not hold here because they involve u α x, and then assume that ux := x + h would be non-differentiable. Example 19. Define f 5 ux = x a x b = u a 87 with < a, b < 1 and u := x 1+b/a. First of all, the direct Riemann-Liouville definition yields D α x x a x b = D α x x a+b = This being the case, we have successively a + b! a + b α! xa+b α. 88 D1 α a! f 5 ux = a α! ua α u D2 α a 1 α f 5 u = au a 1 α u [ 1 + b ] α x b/a = a 1 + b a! b 1 + b α! x1+ a a! 1 + b α x a+b α, 89 a α! a a α = u a 11 α+a 1α a α 1 + b a! 1 + b a α! x1+ b 1 + b = a α a! 1 + b α! xa+b α, 9 a D3 α f 5 u = 1 α!u α 1 a! 1 + b a α! ua α a! b 1 + b α! x1+ a α = 1 α!a! 1 + b a! a α! 1 + b α! xa+b α. 91 a a a α Remark that the Leibniz rule, which works here since one has < a, b < 1, yields a! DLeibnizx α a x b = a α! + b! x a+b α. 92 b α! Comments. We first remark that ux as defined in 87 is quite differentiable in such a manner that, at first glance, irrespective of any calculus, D α 2 f 5 and D α 3 f 5 should not apply here. All the other formulae yield fractional derivative of the form K x a+b α where K is a constant varying with the formula which is selected. This being the case, D α x x a+b Eq. 88 and D α x x a x b Eq. 82 deal respectively with x a+b and x a x b and we must conclude that, from the viewpoint of fractional derivative, they are not the same, in the sense that their fractional differentials or differential increments have not the same value. Example 2. Define with f 6 ux = x ab = x a b = u b, 93 < a, b < 1 and u := x a. According to the standard Riemann-Liouville definition of fractional derivative, a direct calculation uields the derivative Dx α x ab = ab! ab α! xab α 94 which so appears as the expected result to be obtained by other techniques. This being the case, the various fractional derivative chain rules above, we have successively 627

12 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling D1 α b! f 6 ux = b α! ub α ax a 1 α b! = b α! xab α a α x αa α = b!aα b α! xab α, 95 u D2 α b 1 α f 6 ux = bu b 1 α a! u a α! xa α = x ab a 1 α b α x ab 1 α a! a α! xa α = bα a! a α! xab α, 96 D3 α f 6 ux = 1 α!x aα 1 b! b α! xa b α a! 1 α!a!b! a α! xa α = a α!b α! xab α. 97 Comments. x a and u b both are not differentiable in such a manner that 95 and 96 are automatically disqualified. There remains 93 and 97 which involve two functions with different fractional increments More about the fractional derivative chain rules As a matter of fact, the differential of the function fux involves the differential dfu on the one hand, and the differential dux on the other hand, and they have various expressions depending upon the assumptions we make about. Indeed, we can write as well du = u xdx, 98 du = α! 1 u α xdx α, 99 df = f udu, 1 df = α! 1 f α udu α, 11 according to whether the functions so involved are differentiable or αth-differentiable. Combining these expressions, we obtain easily df = f udu = f uu xdx 12 df = f u α! 1 u α xdx α, 13 = α! 1 f α udu α = α! 1 f α u u x α dx α 14 [ α = α! 1 f α u α! 1 u x] α dx 2α. 15 Remark 21. It is tempting to re-write 13 in the form α!df = f uu α xdx α, and to use the conversion formula 27, that is to say d α f = α!df, to write d α f = f uu α xdx α, but on doing so we would be wrong because 29 applies to non-differentiable functions only, whilst here fu is differentiable. All we can say is that there exists a constant K such that f α x = K f uu α x, 16 and so, as a result of the equalities and which yields d u f = f udu d x f = α! 1 f α xdx α K f udu = α! 1 f α xdx α. Selection rule We are then led to work with the following selection rule regarding the suitable derivative chain among 69, 7 and 71. f non-differentiable and udifferentiable, select 69, f differentiable and unon differentiable, select 7 f non-differentiable and unon-differentiable, then select 71 f differentiable and u differentiable, then select none of 69, 7 and On the commutative property of fractional derivatives One of the specific properties of fractional derivatives, at least with the model defined via fractional difference, is that it is not commutative. Example 22. i Let us refer to the differential equation with the initial condition Dyx = yx, 17 y = 1,

13 Guy Jumarie the well known solution of which is yx = y o e t. ii This being the case, let us now consider the same equation, but with the point of view of fractional derivative. If we take for granted that fractional derivatives are commutative, then, at first glance, we would be entitled to re-write 17 in the form D 1/2 D 1/2 yx = yx 19 to which we have to add suitable initial conditions, and to this end, we shall select y = y 1/2 = 1, 11 in order to be as much consistent as possible with 17 and 18. Integrating 19 with respect to dx α yields D 1/2 yx y 1/2 = D 1/2 yx, 111 where the anti fractional derivative on the right side is selected to yield zero at x =. iii Let us now look for a solution in the form yx = E 1/2 λ x, then on substituting into 111 we obtain the equality λe 1/2 λ x 1 = λ 1 E 1/2 λ x 1, which provides λ = 1, therefore the sought solution yx = E 1/2 x. 112 Example 23. Let us compare the fractional derivatives D α+β E α x α with D α D β E α x α and D β D α E α x α. i As a result of the property one can write the equality D α E α x α = E α x α 113 D β D α E α x α = D β E α x α. 114 iithis being the case, if we take for granted that fractional derivatives are commutative, we should have the equality D β D α E α x α = D α D β E α x α. 115 Therefore, by equating the two right members of 114 and 115, D α D β E α x α = D β E α x α. 116 iii In other words, considered as a fractional differential equation of defined by D α, 116 would provide the equality D β E α x α = E α x α which clearly does not make sense. Example 24. Let us consider the function yx = x α. On the one hand one has with the modified Riemann- Liouville definition! α! D β D α x α = D β! xα α = whilst a simple calculation yields D α D β x α α! = D α α β! xα β α! α β! = α β! α β α! xα β α = α! β! x β. Our conclusion is that, on a general standpoint, fractional derivatives are not commutative, at least in our framework, and this is not surprising at all if we have in mind Laplace s transform of fractional derivative Fractional modeling and derivative chain rule Our claim is that the problems of fractional modeling and of derivative chain rules are mutually related and possibly come from the fact that the fractional calculus herein considered refers mainly to non-differential functions. Two ways are open for future research. In a first approach, one could assume that if a given function fx has several derivatives f α α 1 x,..., f n x, then one 629

14 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling can assume that it completely characterized by a mean derivative n ˆf α x = p k f α k x, k=1 where {p k } is a sequence of positive weighting coefficients. In a second approach, we can try to exhibit the very practical meaning of the various compounded derivatives in such a manner that we might be in a position to select that one which is the most suitable for a given physical problem. In the next section we shall outline the kinds of results one may so expect to obtain by using transformation of variables in fractional differential equations. then the equation 117 turns to be of which a solution is [ α x + ] 2 α y + α y φx, y = φx, y = E α 2x α E α y α. 119 Derivation. A direct calculation yields α x α x φx, y = α x α x + α x α y + α x α y + α y α y φx, y 6. Variable transformation in fractional differential equations and α yφx, y = α y φx, y, For illustrative purpose, let us consider the fractional partial differential equation in such a manner that the differential equation 117 now reads α x α x + α y φx, y =, 117 α x α x + 2 α x α y + α y α y + α y φx, y =. 12 i It is easy to check that a particular solution is φ x, y = E α x α E α y α. 118 Derivation. Indeed, on looking for a solution in the form Let us look for a solution in the form φx y = fx gy ; φx, y = fxgy, then on substituting into 12, we obtain the equation one obtains the condition f α+α x fx = gα y gy which provides the special solution = λ, λ R fx = E α λ x α, gy = E α λ y α and λ = 1 yields the result. ii Assume now that we make the transformation x = x, y = x + y, f α+α x fx + 2 f α x x fx g α y y gy of which a special solution is and + gα+α y y gy fx = E α 2x α gy = E α y α. + gα y y = gy It follows that, on the modeling standpoint, there is a deep relation between the variables which are selected to describe the considered system on the one hand, and the formal dynamics which are defined for them. 63

15 Guy Jumarie 7. Cautions about the fractionalization of dynamical systems 7.1. Formal substitution of fractional derivative for derivative In the following, we would like to illustrate how much we must be cautious when we construct the fractional model of standard dynamical systems, and to this end we assume that the starting system is the one-dimensional one defined by the nonlinear differential equation ẋ = dx dt = fx, t. 121 On the modeling standpoint, the key problem is to be sure that the fractional model so obtained is meaningful as compared to the initial dynamics. The most popular way to derive a fractional model of the dynamical equation 121 is to merely substitute the fractional derivative x α t for ẋt to write x α t = dα x = fx, t, < α < dtα In order to exhibit the practical meaning of this derivation on a physical standpoint, we shall proceed as follows. We re-write 122 in the form d α x dx α α dx = fx, t. 123 dt Using the conversion formula 49 which we bear in mind for convenience, Example 25. Assume that fx x; then 125 yields ẋ = [1 α!] 1/α x and the formal fractalization so appears as being merely equivalent to a scaling change in the gain coefficient of the system Fractal modeling with coarse-graining in space only We now assume that fractionalization is required because there is some coarse-graining with respect to x only whilst time is standard, and to take account of this feature, we make the substitution dx, dt dx α, dt into 121, to obtain the differential equation dx α dt = fx, t, < α < Therefore, on using the conversion formula α!x α 1 d α x = fx, tdt. 127 This being the case, we make the transformation dt = dτ α or refer to the lemma 1, equation 57 t = τ α, turns to be d α x dx α = 1 1 α! x1 α, 124 x 1 α 1 α! α dx = fx, t, dt and we eventually obtain the new model to write therefore or 1 α!x α 1 dα x dτ α = fx, τα, d α x dτ α = x1 α 1 α! fx, τα. 129 d α x dt = x1 α fx, t. 1 α! ẋt = [1 α!] 1/α x 1 1/α f 1/α x, t, 125 to be compared with 121. In other words, the substitution of fractional derivative for derivative in the dynamical eequation 121 is completely equivalent to a transformation of the non-linearity fx, t. Example 26. Assume once more thatfx x. Then 129 yields the fractinalequation d α x dτ α = x2 α 1 α!. 631

16 On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling Assume now that fx x α 1. Then the initial non-linear system ẋ = x α 1 is converted into the linear one x α τ = [1 α!] Fractal modeling with coarse-graining in both space and time Here we make the formal substitution dx, dt dx α, dt β to write dx α = fx, tdt β, < α, β < which provides xτ = 1 τ α. α Depending upon whether α > β or α < β we shall obtain the equation 7.3. Fractal modeling with coarse-graining in time only Here, we make the substitution dx, dt dx, dt α in 121 to obtain the equation dx = f 1/α x, tdt β/α, β > α 136 or dx α/β = f 1/β x, tdt, α > β 137 which both are related to the preceding ones. dx = fx, tdt α, < α < 1, 13 and the lemma 45 direct yields t xt = α t τ α 1 fx, τdτ. 131 A closer link with the equation 121, ẋ = f, can be obtained as follows. We make the change of variable dx d x α defined by the equation which provides see the lemma 1 dx = d x α, 132 x = x α. 133 Substituting this result into 13 provides the sought equation d x dt = f 1/α x α, t. 134 Example 27. When fx x, 134 yields d x dt = x1/α and as a result, the linear system is converted into a nonlinear one. But in contrast, when fx x α, one obtains the linear system d x dt = x. 8. Concluding remarks 8.1. General comments We believe that self-similarity, random noises, and fractal space-time, could be or rather should be encompassed in the framework of a suitable fractional differential calculus. Our claim is that the approach to fractional calculus via fractional difference combined with fractional Taylor s series provides exactly this sought fractional differential calculus in Leibniz sense, that is to say involving fractional increments, which should be quite useful for this kind of problems. Once the fractional differential calculus is soundly constructed, there remains to use it in systems modelling, and more especially to examine which kinds of results it provides when it is applied to their dynamical equations. At first glance, on the surface, it would be attractive or at least it is the simplest way to merely subtitute everywhere fractional derivative for derivative, something like a systematic fractionalization of the classical physical equations, but all the above derivations show that we must be very careful on doing so. Given the classical equation ẋt = fx, t, we could convert it in one of the following dynamics, which are d α x = fx, tdt α, dx = fx, tdt α, dx α = fx, tdt α, x 1 α dx α = 1 α!fx, tdt α, 632

17 Guy Jumarie and it is clear that the suitable model should be selected to comply with the physical significance of the problem under consideration, and there remains to be carefully investigated. In our dream of unified approach to micro physics and macro-physics, we shall have to consider combinations of differentiable macro dynamics with non-differentiable micro dynamics, and we so arrive, in quite a natural approach, to Nottale [23] view point. We shall finish with the following remarks. A fractional differential equation of special interest is y α x = λyx, < α < 1, y = 1, 138 where λ is a eal valued parameter. It is by now taken for granted that its solution is yx = E α λx α 139 and the most direct way to obtain this result is to seek a solution in the serial form yx = a k x kα. 14 The main problem is to select the suitable definition of fractional derivative in order that the Mittag-Leffler function be the solution of 138. i First of all, the fractional derivative of a constant should be equal to zero, what is satisfied by Caputo s derivative and modified Riemann-Liouville derivative, but is wrong with the classical Liouville definition. ii Next, the derivative αx α 1 of x α is not bounded when x =, and this feature could cause some defects in the use of the series 14 to find the solution of 138. iii This being the case, substituting 14 into 138 and using the fractional derivative formula 46, then a powerlike term identification direct yields a k = αk! 1. iv Remark that the Laplace s transform of 138 yields L {yx} = s α 1 s α λ with both Caputo s and modified definition. References [1] M. Al-Akaidi, Fractal Speech Processing Cambridge University Press, 24 [2] D. Baleanu, S. Vacaru, Fractional analogous models in mechanics and gravity theory, in Fractional Dynamics and Control Springer, New York, [3] D. Baleanu, S. Vacaru, Fractional exact solutions and solitons in Gravity, in Fractional Dynamics and Control Springer, New York, [4] L.M.C. Campos, IMA J. Appl Math 33, [5] L.M.C. Campos, Fractional calculus of analytic and branched functions, in R.N. Kalia Ed. Recent Advances in Fractional Calculus, Global Publishing Company, 1993 [6] M. Caputo, Geophys. J. R. Ast. Soc. 13, [7] M.M. Djrbashian, A.B. Nersesian, Fractional derivative and the Cauchy problem for differential equations of fractional order 3 Izv. Acad. Nauk Armjanskoi SSR, 1968 in Russian [8] C.F.L. Godinho, J. Weberszpil, J.A. Helayël-Nete, Chaos Solit. Fract., DOI: 1.116/j.chaos [9] G. Jumarie, Int. J. Syst. Sc. 24, [1] G. Jumarie, Appl. Math. Lett. 18, [11] G. Jumarie, Appl. Math. Lett. 18, [12] G. Jumarie, Comput. Math. Appl. 51, [13] G. Jumarie, Math. Comput. Model. 44, [14] G. Jumarie, Chaos Solit. Fract. 32, [15] G. Jumarie, Acta Math. Sinica, DOI: 1.17/s [16] G. Jumarie, Inf. Sci., DOI:1.116/j.ins [17] K.M. Kolwankar, A.D. Gangal, Pramana J. Phys. 48, [18] K.M. Kolwankar, A.D. Gangal, Phys. Rev. Lett. 8, [19] A.V. Letnikov, Math. Sb. 3, [2] J. Liouville, J. Ecole Polytechnique 13, [21] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional, Differential Equations Wiley, New York, 1933 [22] K. Nishimoto, Fractional Calculus Descartes Press Co., Koroyama, 1989 [23] L. Nottale, Fractal Space Time in Microphyssics World Scientific, Singapore, 1993 [24] K.B. Oldham, J. Spanier, The Fractional Calculus, Theory and Application of Differentiation and Integration to Arbitrary Order Acadenic Press, New York, 1974 [25] T.J. Osler, SIAM. J. Math. Anal. 2, [26] I. Podlubny, Fractional Differential Equations Academic Press, San Diego, 1999 [27] B. Ross, Fractional Calculus and its Applications, Lectures Notes in Mathematics 457 Springer, Berlin, 1974 [28] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives, Theory and Applications Gordon and Breach Science Publishers, London,