Characterization of non-differentiable points in a function by Fractional derivative of Jumarrie type

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1 Chrcteriztion of non-differentible points in function by Frctionl derivtive of Jumrrie type Uttm Ghosh (), Srijn Sengupt(), Susmit Srkr (), Shntnu Ds (3) (): Deprtment of Mthemtics, Nbdwip Vidysgr College, Nbdwip, Ndi, West Bengl, Indi; Emil: ():Deprtment of Applied Mthemtics, Clcutt University, Kolkt, Indi Emil: (3)Scientist H+, RCSDS, BARC Mumbi Indi Senior Reserch Professor, Dept. of Physics, Jdvpur University Kolkt Adjunct Professor. DIAT-Pune E-UGC Visiting Fellow Dept. of Applied Mthemtics, Clcutt University, Kolkt Indi Emil (3): The Birth of frctionl clculus from the question rised in the yer 695 by Mrquis de L'Hopitl to Gottfried Wilhelm Leibniz, which sought the mening of Leibniz's nottion for the derivtive of order N when N = /. Leibnitz responses it is n pprent prdo from which one dy useful consequences will be drown. Abstrct There re mny functions which re continuous everywhere but not differentible t some points, like in physicl systems of ECG, EEG plots, nd crcks pttern nd for severl other phenomen. Using clssicl clculus those functions cnnot be chrcterized-especilly t the nondifferentible points. To chrcterize those functions the concept of Frctionl Derivtive is used. From the nlysis it is estblished tht though those functions re unrechble t the nondifferentible points, in clssicl sense but cn be chrcterized using Frctionl derivtive. In this pper we demonstrte use of modified Riemnn-Liouvelli derivtive by Jumrrie to clculte the frctionl derivtives of the non-differentible points of function, which my be one step to chrcterize nd distinguish nd compre severl non-differentible points in system or cross the systems. This method we re etending to differentite vrious ECG grphs by quntifiction of non-differentible points; is useful method in differentil dignostic. Ech steps of clculting these frctionl derivtives is elborted.. Introduction The concept of clssicl clculus in modern form ws developed in end seventeenth by Newton n d y nd Leibnitz []. Leibnitz used the symbol n d to denote the n-th order derivtive of f ( ) []. n d y From the bove developed nottion de L Hospitl sked Leibnitz wht is the mening of n d,

2 for n =/gives the birth of frctionl derivtive. Recently uthors [3-8] re trying to generlize the concept of derivtive for ll rel nd comple vlues of n. Agin in generlized nottion when n is positive it will be derivtive nd for negtive n it will be the notion of integrtion. In n d y these erly methods the derivtive of constnt is found non-zero. So n n denotes the d generlized order derivtive nd integrtion, or generlized differ-integrtion. The bsic definition of generlized derivtive re the formuls from Grunwld-Letinikov(G-L) definition, Riemnn-Liouville (R-L) nd Cputo definition [6,8].The Riemnn-Liouvelli definition retuns non-zero for frctionl derivtive of constnt. This differs from the bsic definition of clssicl derivtive. To overcome this gp Jumrie [] modified the definition of the frctionl order derivtive of left Riemnn-Liouville. In this pper we hve modified the right Riemnn-Liouville frctionl derivtive nd used both the modified definition (left nd right) of derivtive to find the derivtive of the non-differentible functions nd the result is interpreted grphiclly. The clculus on rough unrechble functions is developed vi Kolwnkr-Gngl (KG) derivtive Locl Frctionl Derivtive (LFD) [6]-[], where limit is tken t unrechble point to get LFD. In this pper here there is no limit concept; insted the modifiction is done on clssicl Riemn-Liouvelli frctionl derivtive by constructing n offset function nd doing integrtion for the defined intervl; for the left nd right frctionl derivtive. The orgniztion of the pper is s follows in section. some definition with emples is given, in section. frctionl derivtive of some non-differentible is clculted with grphicl results presented.. Some definitions. Grunwld-Letinikov definition Let f ( t) be ny function then the -th order derivtive of f ( t ) is defined by n Dt f() t = lim h f( t rh) r= r h nh t n! = lim h f( t rh) r!( r)! h nh t r= t ( ) ( ) = t τ f τ dτ ( )! f ( τ ) = Γ ( ) ( t τ) t + dτ () Where is ny rbitrry number rel or comple nd! Γ ( + ) = = r r!( r)! Γ ( r+ ) Γ( r+ ) The bove formul becomes frctionl order integrtion if we replce by which is

3 t () ( ) ( ) Dt f t = t τ f τ dτ Γ( ) () Using the bove formul we get for f () t = ( t ) γ, t γ ( + ) γ t ( ) = ( ) ( ) Γ ( ) D t t τ τ dτ Using the substitution τ = + ξ( t ) we hve for τ =, ξ = nd for τ = t, ξ = ; dτ = ( t ), ( t τ ) = t ξ( t ) = ( t )( ξ); ( τ ) = ξ ( t ), we get the following t γ ( + ) γ t ( ) = ( ) ( ) Γ ( ) D t t τ τ dτ γ γ ( ) ( ) ( ) ( ) = t ξ ξ t t Γ ( ) γ ( t ) = ( ) Γ ( ) γ ( + ) ξ ξ ξ γ ( t ) = B( γ, + ) Γ ( ) Γ ( γ + ) γ = ( t ),( <, γ > ) Γ ( γ + ) d We used Bet-function ( + ) γ ξ ξ γ Γ( ) Γ ( γ + ) B(, + ) = ( ) = defined s Γ ( + γ + ) def pq u u du p q Γ( p) Γ( q) B(, ) = ( ) = Γ ( p + q) k If the function f () t be such tht f ( t), k =,,3..., m+ is continuous in the closed intervl[ t, ] nd m < m+ then m ( k) + k t f ( )( t ) m ( m+ ) () = + ( ) ( ) Dt f t t τ f τ dτ Γ ( + k+ ) Γ ( + m+ ) k = (3). Riemnn-Liouville (R-L) definition of frctionl derivtive Let the function f () t is one time integrble then the integro-differentil epression 3

4 m+ t d m Dt f() t = ( t τ ) f( τ) dτ Γ ( + m+ ) dt (4) is known s the Riemnn-Liouville definition of frctionl derivtive [6] with m < m+. In Riemnn-Liouville definition the function f () t is getting frctionlly integrted nd then differentited m + whole-times but in obtining the formul (3) f () t must be m + time differentible. If the function f () t is m + whole times differentible then the definition (), (3) nd (4) re equivlent. Using integrtion by prts formul in (), tht is t () ( ) ( ) Dt f t = t τ f τ dτ Γ( ) we get t f( )( t ) Dt f() t = + ( t τ ) f ( τ) dτ Γ ( + ) Γ ( + ) (5) The left R-L frctionl derivtive is defined by k t d k () = ( ) ( ) Dt f t t τ f τ dτ k k Γ( k ) dt < (6) And the right R-L derivtive is t k b d k () ( τ ) ( τ) τ Db f t = t f d k k Γ( k ) dt < t (7) In bove definitions k tht is integer just greter thn lph nd > From the bove definition it is cler tht if t time t the function f () t describes certin dynmicl system developing with time then forτ < t, where t is the present time then stte f () t represent the pst time nd similrly if τ > t then f () t represent the future time. Therefore the left derivtive represents the pst stte of the process nd the right hnd derivtive represents the future stge..3 Cputo definition of frctionl derivtive In the R-L type definition the initil conditions contins the limit of R-L frctionl derivtive such s lim D t = b etc tht is frctionl initil stes. But if the initil conditions re f ( ) = b, f ( ) = b... type then R-L definition fils nd to overcome these problems M. Cputo [5] proposed new definition of frctionl derivtive in the following form 4

5 t ( n) C f ( τ ) t + n Γ( n) ( t τ) D f() t = dτ, n < < n Under nturl condition on the function f () t nd s n the Cputo derivtive becomes conventionl n-th order derivtive of the function. The min dvntge of the Cputo derivtive is the initil conditions of the frctionl order derivtives re conventionl derivtive type-i.e requiring integer order sttes. In R-L derivtive the derivtive of Constnt (C) is non-zero. Since t Dt C = ( t τ) Cd Γ ( ) Ct = ( ) d, Where τ t Γ ( ) = Ct ( ) = Γ ( ) Ct for < = Γ( ) for >.4 Jumrie definition of frctionl derivtive On the other hnd to overcome the misconception derivtive of constnt is zero in the conventionl integer order derivtive Jumrie [] revised the R-L derivtive in the following form ( ) = ( ξ) ( ξ) ξ, for < D f f d Γ ( ) ( n) ( ) ( n) τ d = ( ξ) [ f( ξ) f()], for Γ( ) d < < = f ( ) for n < n+, n. The bove definition [] is developed using left R-L derivtive. Similrly using the right R-L derivtive other type cn be develop. Note in the bove definition for negtive frctionl orders the epression is just Riemnn-Liouvelli frctionl integrtion. The modifiction is crried out in R-L the derivtive formul, for the positive frctionl orders lph. The ide is to remove the offset vlue of function t strt point of the frctionl derivtive from the function, nd crry out R-L derivtive usully done for the function. First we wnt to find the derivtive of constnt (C) using right R-L derivtive, (8) 5

6 d Dt ( C) = ( ξ ) C Γ( ) d b C ( b ), for < = Γ ( ), other wise for ( > ) Since for ny function f ( ) in the intervl[ b, ] which stisfies the conditions of modified frctionl derivtive [6] cn be written s f ( ) = f ( b) [ f ( b) f ( )] D f ( ) = D f ( b) D [ f ( b) f ( )] For <, b [ ( ) ( )] = ( ) ( ) Γ( ) D f b f ξ f ξ For < <, ( ) D f b f f f ( ) ( ) [ ( ) ( )] = ( ) = ( ) b ( ) d f f b ( ) = ( ξ ) [ ( ) f ( ξ )] Γ( ) d nd for n < n +, ( ) ( n) ( n) f ( ) = [ f ( )] Thus finlly we cn define in the following form b D f() = ( ξ ) f() ξ, Γ ( ) for < d = ( ξ ) [ f( b) f( ξ)], for < < Γ( ) d ( n) ( ) ( n) b = f () for n < n+, n. Emple: Use the bove definition to continuous nd differentible function f ( ) = c, b. By using the Jumrie definition (s bove) we obtin for left frctionl derivtive 6

7 ( ) d fl ( ) = ( ξ) [ f ( ξ) f ( )], < <. Γ( ) d d d = ( ξ ) [ ξ c ( c)] = ( ξ) [ ξ ] d Γ( ) d Γ( ) d ξ d = ( ξ) [ ( ξ) + ( )] Γ( ) d [ ( ) ( )( ) ] d = ξ + ξ Γ( ) d d ( ξ) ( ξ) = + ( ) Γ( ) d d ( ) ( ) = Γ( ) d d ( ) ( ) = = Γ( ) d ( )( ) Γ( ) we used here nγ ( n) =Γ ( n+ ) Therefore b ( ) b b + + fl = = Γ( ) Γ( ) Agin using our right modified definition we obtin b ( ) d fr ( ) = ( ξ ) [ f ( b) f ( ξ)], <. Γ( ) d < b d = ( ξ ) [ ξ c ( b c)] Γ( ) d b d = ( ξ ) [( ξ ) + ( b)] Γ( ) d b [( ) ( )( ) ] d = ξ + b ξ Γ( ) d 7

8 d ( ξ ) ( ξ ) = + ( b) Γ( ) d d ( b ) ( b ) = Γ( ) d ( b ) = Γ ( ) b Therefore b ( ) b b + + fr b = = Γ( ) Γ() Thus in both the cses (for L nd R) vlue of ( ) + f differentible functions both the vlues re equl, nd is equl to.5 Unrechble function b is equl. Thus for continuous nd b Γ( ) There re mny functions which re continuous for ll but not-differentible t some points or t ll points. These functions re nmed s unrechble functions. The function f ( ) = / is unrechble t the point = /. The function f ( ) = tn( ) is unrechble t the point = π /. These functions re non-differentible i.e. unrechble functions, t some points in the intervl. To study those functions we re considering the following emples nd the frction derivtive will be found using Jumrie modified definition.. Frctionl derivtive of Some Unrechble Functions Emple : f( ) = (/) defined on [, ].This function is continuous for ll in the given intervl but not differentible t = /i.e. we cnnot drw tngent t this point to the curve. The curve is symmetric bout the non-differentible point which is cler from the figure. To study behvior of the function t we wnt to find out frctionl derivtive t = / using the modified frctionl derivtive of Jumrie. 8

9 y Fig.. Grph of the function -/ () The frctionl order derivtive using Jumrie modified definition is When ( ) d fl ( ) = ( ξ) [ f ( ξ) f ()],. Γ( ) d < < / f ( ) = + (/) ( ) d d L Γ( ) d Γ( ) d f ( ) = ( ξ ) [ f ( ξ) f ()] = ( ξ) [ ξ + (/ ) (/ )] d ( ) [( ) ] [( ) ( ) ] d = ξ ξ = ξ ξ Γ( ) d Γ( ) d d ( ξ) ( ξ) = + Γ( ) d d = = Γ ( ) d ( )( ) Γ( ) Agin when / f ( ) = (/) the frctionl derivtive from zero to hlf nd beyond is done in two seprte segments s below 9

10 / ( ) d fl ( ) = + ( ξ) [ f ( ξ) f ()] ( ) d Γ / = + + / d ( ξ) [ ξ (/) (/)] ( ξ) [ ξ (/) (/)] Γ( ) d / / d ( ξ) ( ξ) = + + ( ξ) [ ξ ] Γ( ) d / / d ( ξ) ( ξ) = + + ( ξ) [( ) ( ξ)] Γ( ) d / / d ( ξ) ( ξ) ( ξ) ( ξ) = + + ( ) Γ( ) d d ( /) ( /) ( /) ( /) = + + ( ) Γ( ) d ( / ) = Γ( ) Therefore f ( ) L /, / Γ ( ) ( ) = ( / ), / Γ( ) frctionl derivtive of mod(-.5) frctionl derivtive of mod(-.5) f dot lph f dot lph Fig. Grph of the function fl( ) ( ) for different vlues of lph.

11 From the bove epression it is cler tht though f (/ ) derivtive vlue s = / ( ) f L eists t =/ nd equls to 3/ (/ ) f L(.5) (/ ) = Γ(3/ ) s f (/ ) = (.5) L π with f L does not eists but the frctionl ( ) (/ ) (/ ) =. For =.5 Γ ( ) we get the Γ (3/ ) = π / we get vlue of left hlf derivtive t (b) The frctionl order derivtive using right R-L definition nd modifying the sme s Jumrrie, on sme function we get When ( ) d fr ( ) = ( ξ ) [ f () f ( ξ)], < <. Γ( ) d / f ( ) = + (/) / ( ) d fr ( ) = + ( ξ ) [ f () f ( ξ)] ( ) d Γ / = + + / d ( ξ ) [ ( ξ )] [( ξ ) ( )( ξ ) ] d Γ( ) d ξ / When / d + = Γ ( ) d ( ) (/ ) + ( ) ( ) = Γ ( ) f ( ) = (/) (/ ) (/ ) ( ) (/ )

12 ( ) d f R ( ) = ( ξ ) ( ξ (/ ) (/ ))] Γ( ) d d = ( ξ ) [( ξ ) + ( )] Γ( ) d d ( ξ ) ( ξ ) = + ( ) Γ( ) d d ( ) ( ) = + ( ) Γ( ) d ( ) = Γ ( ) Therefore f ( ) R { ( ) (/ ) }, for / Γ( ) ( ) = ( ), for / Γ ( ) frctionl derivtive of mod(-.5) f dot lph Fig.3 Grph of the function fr( ) ( ) for different vlues of lph. From figure 3 it is cler tht the right modified derivtive eist for this non-differentible function. Thus both the cses we noticed tht the function is not differentible t =/ but its frctionl order derivtive eists. The vlue of hlf right derivtive (.5) f (/ ) R = From the bove two emples it is cler tht for differentible functions the modified definition (both left nd right) of the frctionl derivtive gives the sme vlue t ny prticulr point but for those functions hving non-differentibility t some point gives different vlue for the LFFT π

13 nd RIGHT MODIFIED DERIVATIVE. The difference in vlues of the frctionl derivtive t the non-differentible points indictes the PHASE trnsition t the non-differentible points. The difference of the LFFT nd RIGHT MODIFIED DERIVATIVE is here defining s the indictor of level of phse trnsition. In emple- we consider function which is symmetric bout the non-differentible point nd the function is liner in both sides of the non-differentible points. Now we re considering function which is non-symmetric with respect to the non-differentible point nd liner in both side of the non-differentible point. Emple : Let 6,.5 f( ) = 49 6,.5 3 which rises in pproimtion of led in ECG grph of V5 pek of ptient 9 y Fig-4: ECG pek of V5 From the figure it is cler tht this function is continuous for ll vlues of in the given intervl nd non-symmetric bout the point =.5 (the non-differentible point). We give trnsltion z = nd rewrite the functionl form by sme nottion nd the intervl[,3 ] to[, ] nd the trnslted function is + 4,.5 f( ) =. 7 6,.5 This function is continuous t for ll but not-differentible t =/ but nture of discontinuity is different from the function f( ) = in [, ]. () The frctionl order derivtive using Jumrie modified definition is 3

14 When ( ) d fl ( ) = ( ξ) [ f ( ξ) f ()], Γ( ) d < < d = ( ξ) [ f ( ξ) 4] Γ( ) d / f ( ) = + 4 f () = 4 ( ) d d L = ξ ξ + ξ = ξ ξ Γ( ) d Γ( ) d ξ f ( ) ( ) [ 4 4] d [ ( ) ( ) ] d d ( ξ) ( ξ) d = = Γ( ) d Γ( ) d d = = Γ( ) d ( )( ) Γ( ) When / f ( ) = 7 6here we require the vlue t f () = 4nd lso intervl [ ],.5 where the function is f( ) = + 4, nd do the integrtion in two segments / d f ( ) = + ( ξ) [ f( ξ) f()] ( ) L Γ( ) d / = + / d ξ( ξ) ( ξ) {7 6ξ 4} Γ( ) d / = + + / d ( ( ξ))( ξ) ( ξ) {3 6 6( ξ)} Γ( ) d / = Γ ( ) d d / ( ξ) ( ξ) ( ξ) ( ξ) 6 + (3 6 ) ( /) ( /) ( /) 5 + d ( /) ( /) ( /) = ( ) Γ( ) d ( /) + (3 6 ) 6 + ( /) Γ( ) = + = { 6( / ) + } Γ( ) ( /) (6 8) / 4

15 Therefore f ( ) L Γ ( ) ( ) = + Γ( ) { 6( / ) } for / for / Therefore though the function is not differentible t = / but the -order derivtive ( ) (/ ) t =/ eists nd equls to f L (/ ) =. The grphicl presenttion of fl( ) ( ) for Γ ( ) different vlues of lph is shown in the figure-5, from the figure it cler tht the non-differentible point =/. f ( ) L ( ) eists t frctionl derivtive of non-symmetric function 5 f dot lph Fig.5 Grph of the function fl( ) ( ) for different vlues of lph. The frctionl order derivtive using right R-L definition on sme function we get When / ( ) d fr ( ) = ( ξ ) [ f () f ( ξ)], < <. Γ( ) d 5

16 / d f ( ) = + ( ξ ) [ f () f ( ξ)] ( ) R Γ( ) d / = / d ( ξ ) [( ξ ) ( 3)] 6 [ ( ξ ) ( )( ξ ) ] d Γ( ) d ξ / / d ( ξ ) ( ξ ) ( ξ ) ( ξ ) = + (+ 3) + 6 ( ) Γ( ) d / (/ ) (/ ) + ( + 3) + d = Γ( ) d ( ) (/ ) ( ) (/ ) 6 ( ) (/ ) (/ ) (/ ) + + = Γ( ) 6 6 8(/ ) + 8(/ ) + (/ ) ( ) 6( ) 6( ) = Γ( ) When / ( ) d 6 d R Γ( ) d Γ( ) d ξ f ( ) = ( ξ ) (7 6ξ )] = ( ξ ) [( ) ( ξ )] d 6 d ( ξ ) ( ξ ) = + ( ) Γ( ) d 6 d ( ) ( ) ( ) = + ( ) = 6 Γ( ) d Γ( ) 6

17 Right frctionl derivtive of non-symmetric nd non-liner function 5 f dot lph Fig-6 Grph of the function fr( ) ( ) for different vlues of lph. Therefore f ( ) R 6( ) 6( ), for / Γ( ) ( ) = ( ) 6, for / Γ( ) Thus; though the considered function is not differentible t = /but its right modified ( ) (/ ) frctionl derivtive eists nd its vlue is f R (/ ) = 6 which differ from the vlue Γ ( ) ( ) (/ ) f L (/ ) = Γ ( ) Jumrie modifiction. of the derivtive t = /obtined by left modified R-L derivtive by Here the difference indictes there is phse trnsition from the left hnd to the right hnd side (/ ) bout the point =/ nd the level or degree of phse trnsition is 6. Γ( ) Emple 3: Let 3+ 4,.5 f( ) = 34 3,.5 which rises in mpping of led in ECG grph. This function is continuous for ll vlues of in[, ] but not differentible t =/. 7

18 y Fig- 7 Grph of the function 3+ 4,.5 f( ) = 34 3,.5 () The frctionl order derivtive using Jumrie modified definition is When / ( ) d fl ( ) = ( ξ) [ f ( ξ) f ()], Γ( ) d < < ( ) 3 d fl ( ) = ( ξ) ξ Γ( ) d d = ( ξ) [ f ( ξ) 4] Γ( ) d 3 d = ( ξ) { ( ξ)} Γ( ) d 3 ( ) ( ) 3 3 d ξ ξ d = + = = Γ( ) d Γ( ) d Γ( ) 8

19 When/. / ( ) 3 d fl () = ( ξ) () ξ + ( ξ) ( ξ) Γ( ) d / = = + + / 3 d ( ξ) { ( ξ)} ( ξ) {( ξ) ( )} Γ( ) d / / 3 d ( ξ) ( ξ) ( ξ) ( ξ) = + + ( ) Γ( ) d 3 d ( ξ) ( / ) ( / ) ( / ) = ( ) Γ( ) d Γ( ) 3{ ( / ) } / / / 3 frctionl derivtive of symmetric liner function f dot lph Fig-8 Grph of the function fl( ) ( ) for different vlues of lph. From the figure 7 nd 8 it is cler tht though this function is not differentible t frctionl derivtive of order with < < eists t = /. Therefore =/ but its f ( ) L ( ) 3 for / Γ ( ) = 3{ ( / ) } for / Γ( ) Here 9

20 f 3 ( =.5) = (/). Γ ( ) ( ) L In previous ll the problems we consider the functions which re liner in both side of the nondifferentible point. In the net emple we consider function which is liner in one side nd non-liner in other side of the non- differentible point. Emple 4: Let 4 + +,. 5 f( ) = 5,.5 This function is continuous for ll vlues of in[, ] but not differentible t =/ which is cler from figure-9. y Fig-9 grph of the function defined bove. () The frctionl order derivtive using Jumrie modified definition is ( ) d fl ( ) = ( ξ) [ f ( ξ) f ()], < Γ( ) d < For / f( ) = 4 + +

21 ( ) d L Γ( ) d f ( ) = ( ξ) [4ξ + ξ] ( ) [ ] d = ξ ξ + ξ + ξ + ξ + + Γ( ) d ( ) [4( ) (8 )( ) 4 ] d = ξ ξ + ξ + + Γ( ) d d = ξ + ξ + + ξ Γ( ) d 4( ) (8 )( ) ( )( ) d 4( ξ) ( ξ) ( ξ) = + (8+ ) (+ 4 ) Γ( ) d d 4 = (8+ ) + (+ 4 ) Γ( ) d 3 = 8 8 ( ) + = + Γ Γ( ) 8 Γ( ) = + For / f( ) = 5 f L ( ) (4 ) (/ ) = Γ(3 ) For clcultions for the frctionl derivtive for / we need to clculte, the frctionl derivtive from strt point f (), nd thus tke the function in the region /which is f( ) = nd strt point vlue is f () = ; nd do the integrtion in two segments first,.5 nd then[.5, ], s demonstrted in the following steps. in [ ]

22 ( ) d fl ( ) = ( ξ) [ f ( ξ) f()],< < f () = Γ( ) d = / d ( ξ ) [4( ξ) (8 )( ξ) 4 ] ( ξ) [5 ξ ] d Γ( ) d ξ / / ( ξ) [4( ξ) (8+ )( ξ) ] + d = Γ ( ) d ( ξ ) [(3 ) + ( ξ)] / 3 / 4( ξ) ( ξ) ( ξ) + (8+ ) (+ 4 ) d 3 = ( ) Γ d ( ξ) ( ξ) (3 ) + / 3 3 ( /) ( /) 4 (8 + ) d + 3 = Γ ( ) d ( /) ( /) ( /) ( /) (+ 4 ) + + ( /) ( /) = Γ(3 ) Γ( ) Γ(3 ) Γ( ) Here lso f ( ) L f L 8 ( ) (4 ) (/ ) = Γ(3 ). + for.5 Γ( ) ( ) = ( /) ( /) for.5 Γ(3 ) Γ( ) Γ(3 ) Γ( ) Thus though this function ws non-differentible t = / but the frctionl derivtive eists t = /. The grph of the frctionl derivtive for different vlues, < < is shown in figure-.

23 6 frctionl derivtive of non-symmetric nd non-liner function f dot lph Fig- Grph of the function fl( ) ( ) for different vlues of lph. (b) The frctionl order derivtive using right R-L definition on sme function we get ( ) d fr ( ) = ( ξ ) [ f () f ( ξ)], < < Γ( ) d When / f ( ) = This clcultions requires f () = 3 the end point of the function nd the function in the intervl.5, which is f ( ) = 5. We need to crry this integrtion in two segments s demonstrted [ ] below. / ( ) d fr ( ) = + ( ξ ) [ f () f ( ξ)] f () = 3 ( ) d = Γ( ) = Γ ( ) Γ / d d d d / ( ξ ) [4( ξ ) + (8+ )( ξ ) + (4 + )] + ( ξ ) [( ) ( ξ )] / 3 / ( ξ ) ( ξ ) ( ξ ) 4 + (8 + ) + (4 + ) + 3 ( ξ ) ( ξ ) ( ) / 3

24 When / = 4 + (8+ ) + (4 + ) d 3 ( ) d ( ) (/ ) ( ) (/ ) ( ) 3 (/ ) (/ ) (/ ) Γ ( ) 8(/ ) 8( ) = + Γ( ) Γ( ) Γ(3 ) f ( ) = 5 f () = 3 ( ) d fr ( ) = ( ξ ) (5 ξ 3)] Γ( ) d d = ( ξ ) [( ) ( ξ )] Γ( ) d d ( ξ ) ( ξ ) = ( ) Γ( ) d = = Γ( ) d Γ( ) 6 d ( ) ( ) ( ) + 6 frctionl derivtive of non-symmetric nd both side non-liner function f dot lph Fig-: Grph of the function fr( ) ( ) for different vlues of lph. Therefore 4

25 f ( ) R ( ) 8(/ ) 8( ), for / ( ) + ( ) (3 ) Γ Γ Γ ( ) = ( ), for / Γ( ) The frctionl right modified derivtive of the function is shown in the figure-. The vlue of the derivtive t =/ is f R ( ) (/ ) = Γ(3 ) This vlue of right derivtive differs from the vlue of the Jumrie left derivtive to this function which indictes there is phse trnsition t = /. The previous function ws liner in one side of the differentible function nd non-liner in other side, of the trnsition point. Now we consider function which is non-liner in both sides with respect to the non-differentible point, but continuous t tht trnsition point. Emple 5: Now consider the function 4 + 3,. 5 f( ) =. 5 4,.5 This function is not symmetric with respect to the non-differentible point =/, refer figure-. To chrcterize this function the frctionl derivtive of this function is clculted using both side modified definition of derivtive. y Fig- grph of the function defined bove. () The frctionl order derivtive using Jumrie modified definition is ( ) d fl ( ) = ( ξ) [ f ( ξ) f ()], < Γ( ) d < 5

26 For.5 f( ) = f() =3 ( ) d fl ( ) = ( ξ) [ f( ξ) f()] Γ( ) d ( ) [4( ) 8 ( ) 4 ] d = ξ ξ ξ + Γ( ) d 3 d 4( ) 8 ( ) ( ) ξ ξ ξ = + 4 Γ( ) d d = + 4 = Γ( ) d 3 Γ(3 ) For.5 f( ) = 5 4 In this clcultions we require the vlue t the strt point which is f () = 3 lso requiring the function in the intervl[,.5 which is f( ) = 4 + 3, nd integrtion is done in two segments [..5] nd [.5,] ] ( ) d fl ( ) = ( ξ) [ f( ξ) f()] Γ( ) d = + + / d ( ξ ) 4ξ ξ 3d Γ( ) d ξ / / ( ξ) [4( ξ) 8 ( ξ) + 4 ] + d = Γ( ) d ( ξ) [( 4 ) 4( ξ) + 8 ( ξ)] / 3 / 4( ξ) 8 ( ξ) ( ξ) d 3 = ( ) 3 Γ d 4( ξ) 8 ( ξ) ( ξ) ( 4 ) 3 / 3 3 ( / ) 8 {( / ) } ( / ) 4 4 d + 3 = Γ ( 3 ) d 4( / ) 8 ( / ) ( / ) + + ( 4 ) 3 6

27 8 8( / ) 6( / ) ( 3 + 3) = Γ(3 ) Γ( ) Γ(3 ) Therefore f ( ) L 8,.5 Γ (3 ) ( ) = 8 8( / ) 6( / ) Γ (3 ) Γ( ) Γ(3 ),.5 4 frctionl derivtive of non-symmetric nd both side non-liner function f dot lph Fig-3 Grph of the function fl( ) ( ) for different vlues of lph. (b) The frctionl order derivtive using right R-L definition on sme function we get 4 + 3,. 5 f( ) = 5 4,.5 ( ) d fr ( ) = ( ξ ) [ f () f ( ξ)],< <. Γ( ) d When / f ( ) = f () = We need here the integrtion process in two intervls [,.5] nd [.5,] s demonstrted below 7

28 / ( ) d fr ( ) = + ( ξ ) [ f () f ( ξ)] ( ) d Γ / = + + / d ( ξ ) [ 4ξ 3] ( ξ ) [ 5 4 ξ ] d Γ( ) d ξ / / ( ξ ) [4( ξ ) + 8 ( ξ ) + (4 + )] + d 4 ( ξ ) [( ) ( ξ ) ( ξ )] / = Γ ( ) d 3 / ( ξ ) ( ξ ) ( ξ ) (4 + ) + d 3 = 3 Γ( ) d ( ξ ) ( ξ ) ( ξ ) 4 ( ) 3 / 3 (/ ) (/ ) (/ ) (4 + ) + d 3 = Γ( ) d 3 3 ( ) (/ ) ( ) (/ ) ( ) (/ ) 4 ( ) 8 3 ( ) ( ) (/ ) ( ) = Γ(3 ) Γ( ) Γ( ) Γ(3 ) When / f ( ) = 5 4 f () = ( ) d R Γ( ) d f ( ) = ( ξ ) (5 4ξ )] Therefore ( ) [( ) ( ) ( )] 4 d = ξ ξ ξ Γ( ) d 3 ξ ξ ξ ( ) 6 d ( ) ( ) ( ) = Γ( ) d 3 6 d ( ) ( ) ( ) ( ) ( ) = 8 8 Γ( ) d 3 = (3 ) Γ Γ() 3 ( ) 8

29 f ( ) R ( ) ( ) (/ ) ( ) , for / Γ (3 ) Γ ( ) Γ ( ) Γ (3 ) ( ) = ( ) ( ) 8 8, for / Γ(3 ) Γ( ) 4 frctionl derivtive of non-symmetric nd both side non-liner function f dot lph Fig-4 Grph of the function fr( ) ( ) for different vlues of lph. 3. Conclusion From the bove emples it is cler tht some functions which re unrechble t nondifferentible points in the defined intervl in clssicl derivtive sense but they re differentible in frctionl sense. But the modified frctionl derivtive in both left nd right sense gives sme vlue for differentible functions but gives different vlue for nondifferentible cse. For non-differentible the difference in vlues of the frctionl derivtive in Jumrie modified nd right R-L modified sense indictes there is phse trnsition bout the nondifferentible points. The difference in vlues indictes the level of the phse trnsition. These re useful indictors to quntify nd compre the non-differentible but continuous points in system. This method we re etending to differentite vrious ECG grphs by quntifiction of non-differentible points; is useful method in differentil dignostic. 4. Reference [] Ross, B., The development of frctionl clculus Histori Mthemtic, (), pp [] Diethelm. K. The nlysis of Frctionl Differentil equtions. Springer-Verlg.. 9

30 [3]Kilbs A, Srivstv HM, Trujillo JJ. Theory nd Applictions of Frctionl Differentil Equtions. North-Hollnd Mthemtics Studies, Elsevier Science, Amsterdm, the Netherlnds. 6;4:-53. [4] Miller KS, Ross B. An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions.John Wiley & Sons, New York, NY, USA; 993. [5] Smko SG, Kilbs AA, Mrichev OI. Frctionl Integrls nd Derivtives. Gordon nd Brech Science, Yverdon, Switzerlnd; 993. [6] Podlubny I. Frctionl Differentil Equtions, Mthemtics in Science nd Engineering, Acdemic Press, Sn Diego, Clif, USA. 999;98. [7] Oldhm KB, Spnier J. The Frctionl Clculus, Acdemic Press, New York, NY, USA; 974. [8 ]Ds. S. Functionl Frctionl Clculus nd Edition, Springer-Verlg. [9] Gemnt, A., On frctionl differentils. Phil. Mg. (Ser. 7), 94. 5(4), pp [] G. Jumrie, Frctionl prtil differentil equtions nd modified Riemnn-Liouville derivtives. Method for solution. J. Appl. Mth. nd Computing, 7 (4), Nos -, pp [] G. Jumrie, Modified Riemnn-Liouville derivtive nd frctionl Tylor series of nondifferentible functions Further results, Computers nd Mthemtics with Applictions, 6. (5), [] H. Kober, On frctionl integrls nd derivtives, Qurt. J. Mth. Oford, 94 (), pp [3] A.V. Letnivov, Theory of differentition of frctionl order, Mth. Sb., (3)868, pp -7. [4] J. Liouville, Sur le clcul des differentielles ` indices quelconques(in french), J. Ecole Polytechnique, (3)83, 7. [ 5] M. Cputo, Liner models of dissiption whose q is lmost frequency independent-ii, Geophysicl Journl of the Royl Astronomicl Society,, 967. vol. 3, no. 5, pp [6] Abhy Prvte, A. D. Gngl. Clculus of frctls subset of rel line: formultion-; World Scientific, Frctls Vol. 7, 9. [7] Abhy Prvte, Seem stin nd A.D.Gngl. Clculus on frctl curve in R n rxiv:96 oo76v 3.6.9; lso in Prmn-J-Phys. [8] Abhy prvte, A. D. Gngl. Frctl differentil eqution nd frctl time dynmic systems, Prmn-J-Phys, Vol 64, No. 3, 5 pp

31 [9]. E. Stin, Abhy Prvte, A. D. Gngl. Fokker-Plnk Eqution on Frctl Curves, Seem, Chos Solitons & Frctls-5 3, pp [] K M Kolwnkr nd A D. Gngl. Locl frctionl Fokker plnk eqution, Phys Rev Lett , Acknowledgement Acknowledgments re to Bord of Reserch in Nucler Science (BRNS), Deprtment of Atomic Energy Government of Indi for finncil ssistnce received through BRNS reserch project no. 37(3)/4/46/4-BRNS with BSC BRNS, title Chrcteriztion of unrechble (Holderin) functions vi Locl Frctionl Derivtive nd Devition Function. 3