BASIC DEVELOPMENTS OF FRACTIONAL CALCULUS AND ITS APPLICATIONS

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1 Bullein of he Marahwada Mahemaical Sociey Vol., No., Dec, Pages 7. BASIC DEVELOPMENTS OF FRACTIONAL CALCULUS AND ITS APPLICATIONS A. P. Bhadane Deparmen of Mahemaics, Sm. Puspaai Hiray Mahila Mahavidyalya Malegaon Camp, Malegaon 43 5, (M. S.), India. and K. C. Takale Deparmen of Mahemaics, RNC Ars, JDB Commerce and NSC Science College, Nashik-Road 4, (M. S.) India. Absrac The aim of his paper is o sudy he basic heory and applicaions of fracional calculus. We have obain fracional inegral and fracional derivaive of some funcions. The fracional inegral and fracional derivaive of hese funcions are simulaed by mahemaical sofware Mahemaica. INTRODUCTION In he lae seveneenh cenury, he philosopher and creaor of modern calculus G. W. Leibniz made some remarks on he meaning and possibiliy of fracional derivaive of order. In he year 695, he well known mahemaician L Hospial was asked he quesion as he meaning of dn f dx n, wha if n =?, ha is, wha is a fracional derivaive?. Leibniz (695) answered, This is an apparen paradox from which, one day, useful consequences will be drawn. Thereafer, L Hospial, Euler, Lagrange, Laplace, Riemann, Liouville developed he basic concep of fracional calculus. The rigorous invesigaion was firs carried ou by Keywords: Fracional Inegral, Fracional Derivaive, Laplace Transform, Mahemaica.

2 A. P. Bhadane & K. C. Takale Liouville in a series of papers (83 837), where he defined he firs oucas of an operaor of fracional inegraion. The fracional order inegral and differeniaion, which represen a rapidly growing field boh in heory and applicaions o real world problems. Afer his Euler ook he firs sep in he sudy of fracional inegraion [3]. In he year 865, Liouville and Swedish mahemaician Holmgren, made an imporan conribuions o he growing field of fracional calculus. Today here exis many differen forms of fracional inegral operaors, ranging from divided-difference ypes o infinie-sum ypes [8], bu he Riemann-Liouville operaor is sill he mos frequenly used when fracional inegraion is performed [, 5]. In he year 967, Capuo defined a useful formula o obain fracional derivaive of he funcion [4, 7]. I should be noed ha he appearance of he idea of fracional inegraion and differeniaion is no any accepable geomerical and physical inerpreaion of hese conceps for more han 3 years. A differen approach o geomeric inerpreaion of fracional inegraion and fracional differeniaion, based on he idea of he conac of α h order, has been suggesed by F. Ben Adda. However, i is difficul o speak abou an accepable geomerical inerpreaion wihou visualizaion. Finally, in he year I. Podlubny [6] represen he geomeric inerpreaion of he inegral and he fracional inegral. Recenly, he researchers and scieniss found he use of fracional calculus in science, engineering, hydrology and finance ec. We organize his paper as follows: In secion, we sudy he some definiions of fracional inegral and fracional derivaive and properies of Gamma funcion and Miag-Liffler funcion. The secion 3, is devoed for he fracional inegral and fracional derivaive of some funcions and hese are simulaed by mahemaical sofware Mahemaica. The secion 4, is devoed for he applicaions of fracional calculus o Abel inegral and differenial equaion. DEFINITIONS AND PROPERTIES In his secion we discuss some definiions of fracional derivaives and fracional inegrals of fracional calculus. Definiion. The gamma funcion is defined as follows Γ(x) = y x e y dy, x > where for convergence of he inegral, x >

3 Basic Developmens of Fracional Calculus and Is Applicaions 3 Definiion. (Gamma funcion is defined by Euler Limi):..3...(N ) Γ(n) = lim N n(n + )(n + )...(n + N ) N n Γ(n + ) = lim N Γ(n + ) = lim N = nγ(n) Properies of Γ-Funcion: (i) Γ(n + ) = nγ(n), n >..3...(N ) (n + )(n + )...(n + N) N n+ Nn n + N..3...(N ) n(n + )(n + )...(n + N ) N n (ii) Γ(n + ) = nγ(n) = n(n )Γ(n ) =... = n(n )...(n m + )Γ(n m+) (iii) Γ(n + ) = n!, n =,, 3, (iv) Γ(n) = Γ(n+) n, Γ() =, Γ( n) =, n > (v) Γ(n)Γ( n) = π sinnπ. Example. (i)γ( ) = π,(ii)γ( + n) = (n)! π 4 n n!,(iii)γ( n) = ( 4n )n! π (iv) Γ( 3 ) = 4 3 π, (v) Γ( ) = π, (vi) Γ( ) = ± (n)!, Definiion.3 Le f() be a funcion of a variable such ha he funcion e s f() is inegrable in [, ) for some domain of values of s. The Laplace ransform of he funcion f() is he inegral is defined as e s f()d; for above domain values of s.i is denoed by L{f()} = e s f()d. If L{f()} = ϕ(s) hen he inverse laplace ransform of ϕ(s) is defined as L {ϕ(s)} = f(). Example. Consider f() = n, n is a posiive ineger.by above definiion,we ge L{ n } = e s n d = [ n ( e s s ) nn ( e s ( s) ) e s ( )n n(n ) ( ( s) n+ ) ] = =, s > Afer simplificaion and using Gamma funcion he above formula can be wrien as we ge L{ n } = n! Γ(n + ) = sn+ s n+, s > This formula gives he idea of Laplace ransform of fracional calculus.

4 4 A. P. Bhadane & K. C. Takale Remark. If we ake α is non-ineger order n < α n hen he Laplace ransform of fracional calculus is L{ α } = Γ(α + ) s α+, n < α n, s > Example.3 If α =, we ge L{ } = Γ( 3 ) s 3 = π s s, s > Similarly, by he definiion.3, we have obained he Laplace ransform of some funcions which are given in he following able: Funcion Laplace Transform Funcion Inverse Laplace ransform s, s > s, s >, s > s n+ n n n! s n+, s > n! e a s a, s > a sina cosa a, s > s +a s, s > s +a s a, s > a a, s > s +a s, s > s +a ea sina cosa Definiion.4 The Laplace ransform of he differinegrable funcion f() is defined for (n < α n) as { } L D α f() = n e s D α f()d = s α F (s) k= s k [ D α k ] f() The Laplace ransform and inverse Laplace ransform of some funcions of fracional calculus are given in he following able: = Funcion Laplace Transform Funcion Inverse Laplace ransform sink π s, s > s π s s, s > s s k coh( πs s +k k ) k coh( πs s +k k ) π π sink Definiion.5 (i) The Miag-Liffler (93) funcion of one parameer is defined as follows z k E α (z) =, (α ε C, R(α) > ) Γ(αk + ) k=

5 Basic Developmens of Fracional Calculus and Is Applicaions 5 (ii) The Miag-Liffler (93) funcion of wo parameer is defined as follows E α,β (z) = k= z k, (α ε β εc, R(α) >, R(β) > ) Γ(αk + β) Properies:The following are he basic properies of Miag-Liffler funcion: (I) E α,β (z) = ze α,α+β (z) + Γ(β), (II) E α,β(z) = βe α,β+ (z) + αz d dz E α,β+(z) Definiion.6 (Grūnwald-Lenikov ) The Grūnwald-Lenikov definiion of fracional derivaive of a funcion generalize he noion of backward difference quoien of ineger order. In his case α = if he limi exiss he Grūnwald-Lenikov fracional derivaive is he lef derivaive of he funcion. The Grūnwald-Lenikov fracional derivaive of order α of he funcion f(x) is define as [5] ad α x f(x) = lim N { ( x a N ) α N Γ( α) j= } Γ(j α) a f(x j[x Γ(j + ) N ]) where α C. If α =, we have a Riemann sum which i he firs inegral. If α =, hen we have { f(x) f(x [ x a N lim ]) } N [ x a N ] which is lef derivaive of he funcion f a x. Definiion.7 (Riemann-Liouville ): Riemann (953), Liouville (83) (a) If f(x) C[a, b] and a < x < b hen Ia+f(x) α = f() d, where α (, ) Γ(α) a (x ) α is called he Riemann-Liouville fracional inegral of order α. (b) If f(x) C[a, b] and a < x < b hen D α a+f(x) = d Γ( α) dx a f() d, where α (, ) (x ) α is called he Riemann-Liouville fracional derivaive of order α [6].

6 6 A. P. Bhadane & K. C. Takale Definiion.8 (M. Capuo (967)): If f(x) C[a, b] and a < x < b hen he Capuo fracional derivaive of order α is defined as follows [6] ad α x f(x) = Γ(n α) a f (n) (τ) dτ, where n < α < n (x τ) α n+ In he nex secion we obain he fracional inegral and fracional derivaive of some sandard funcions and hese are simulaed by mahemaical sofware Mahemaica. 3 FRACTIONAL INTEGRAL AND FRACTIONAL DERIVA- TIVE OF SOME FUNCTIONS 3. Fracional Inegral of Some Funcions: As an applicaion of fracional inegraion, we can ake α = which is called he semi-inegral if i is used in Riemann-Liouville fracional inegral formula of order α. Consider he Riemann-Liouville fracional inegral formula I α a+f(x) = Γ(α) a f() d, where α (, ) (x ) α using his formula we have obain he fracional inegral of some funcions a follows. (i)ideniy Funcion Le f(x) = x, α = By puing x = u, we ge I x = Γ( ) I x = Γ( ) (x ) d (x u) du = 3 4x Γ( 5 = )x3 3 π u Similarly, we obain he fracional inegrals of ideniy funcion for differen values of α as follows I x = x π I 3 x = 8x 5 5 π I 3 x = πx The fracional inegral of ideniy funcion for α = mahemaical sofware Mahemaica as follows: is represened graphically by

7 Basic Developmens of Fracional Calculus and Is Applicaions 7 (ii)consan Funcion f(x) = K, K is consan, α = I K = Γ( ) K (x ) Uni Funcion: f(x) = d = K Γ( )[(x ) ] x = K Γ( x 3 ) I x () = π, I () = πx, I 3 () = 4x 3 3 π, I 3 () = π x 3 The fracional inegral of uni funcion for α = mahemaical sofware Mahemaica as follows: is represened graphically by Properies:The following are he basic wo properies of Miag-Liffler funcion: (i) E α,β (z) = ze α,α+β (z) + Γ(β) (ii) E α,β (z) = βe α,β+ (z) + αz d dz E α,β+(z) Definiion 3. (Grūnwald-Lenikov ) The Grūnwald-Lenikov definiion of fracional derivaive of a funcion generalize he noion of backward difference quoien of

8 8 A. P. Bhadane & K. C. Takale ineger order. In his case α = if he limi exiss he Grūnwald-Lenikov fracional derivaive is he lef derivaive of he funcion. The Grūnwald-Lenikov fracional derivaive of order α of he funcion f(x) is define as [5] ad α x f(x) = lim N { ( x a N ) α N Γ( α) j= } Γ(j α) a f(x j[x Γ(j + ) N ]) where α C. If α =, we have a Riemann sum which i he firs inegral. If α =, hen we have { f(x) f(x [ x a N lim ]) } N [ x a N ] which is lef derivaive of he funcion f a x. Definiion 3. (Riemann-Liouville ): Riemann (953), Liouville (83) (a) If f(x) C[a, b] and a < x < b hen Ia+f(x) α = f() d, where α (, ) Γ(α) a (x ) α is called he Riemann-Liouville fracional inegral of order α. (b) If f(x) C[a, b] and a < x < b hen D α a+f(x) = d Γ( α) dx a f() d, where α (, ) (x ) α is called he Riemann-Liouville fracional derivaive of order α [6]. Definiion 3.3 (M. Capuo (967)): If f(x) C[a, b] and a < x < b hen he Capuo fracional derivaive of order α is defined as follows [6] ad α x f(x) = x f (n) (τ) dτ, where n < α < n Γ(n α) a (x τ) α n+ 4 FRACTIONAL INTEGRAL AND DERIVATIVE In his secion we obain he fracional inegral and fracional derivaive of some sandard funcions and hese are simulaed by mahemaical sofware Mahemaica.

9 Basic Developmens of Fracional Calculus and Is Applicaions 9 4. Fracional Inegral of Some Funcions As an applicaion of fracional inegraion, we can ake α = which is called he semi-inegral if i is used in Riemann-Liouville fracional inegral formula of order α. Consider he Riemann-Liouville fracional inegral formula Ia+f(x) α = f() d, where α (, ) Γ(α) a (x ) α using his formula we have obain he fracional inegral of some funcions a follows. (i)ideniy Funcion Le f(x) = x, α = By puing x = u, we ge I x = Γ( ) (x ) d I x = (x u) Γ( ) du = 3 4x u Γ( 5 = )x3 3 π The fracional inegral of ideniy funcion for α = mahemaical sofware Mahemaica as follows: is represened graphically by Similarly, we obain he fracional inegrals of ideniy funcion for differen values of α as follows I x = x, I 3 π x = 8x 5 5 π, I 3 x = πx The fracional inegrals of ideniy funcion can be obained for α =, 3 respecively and cab be represened graphically using Mahemaica sofware. and 3

10 A. P. Bhadane & K. C. Takale (ii)consan Funcion f(x) = K, K is consan, α = I K = Γ( ) K (x ) d = K Γ( )[(x ) ] x = K Γ( 3 ) x Uni Funcion: f(x) =, I () = x π The fracional inegral of uni funcion for α = mahemaical sofware Mahemaica as follows: is represened graphically by The fracional inegral of uni funcion can be obained for α =, 3 respecively and cab be represened graphically using Mahemaica sofware: and 3 I () = πx, I 3 () = 4x 3 3 π, I 3 () = π x 3 (iii)exponenial Funcion:The general formula for fracional inegral of exponenial funcion f(x) = e bx is I α (e bx ) = b α e bx, a =, α >, Rel(b) >. The graphical represenaion of fracional inegral of exponenial funcion f(x) = e bx for α = 3 is as follows:

11 Basic Developmens of Fracional Calculus and Is Applicaions 4. Fracional Derivaive of Some Funcions (i)ideniy Funcion:(a) Apply he Grūnwald-Lenikov definiion o he ideniy funcion f(x) = x, we ge D α x (x) = lim N = x α [ [ = {( ) N α N x j= lim N {N α N ( ( ))} Γ(j α) x x j Γ( α)γ(j + ) N j= Γ( α) + α Γ( α) Γ(j α) Γ( α)γ(j + ) ] x α } N lim {N α N j= }] Γ(j α) j Γ( α)γ(j + ) Using Gamma propery (i) and afer simplificaion, we ge Dx α (x) = x α Γ( α). Therefore, Grūnwald-Lenikov fracional derivaive of f(x) = x for α = is D x x = π (4.) Similarly, we can obain he fracional derivaives for oher non-ineger values of α. (b) As an applicaion of fracional derivaive, we use Riemann-Liouville fracional derivaive formula of order α. D α a + f(x) = d Γ( α) dx a f() (x ) α d

12 A. P. Bhadane & K. C. Takale Ideniy Funcion f(x) = x, α = D d x = Γ(/) dx = d dx [ Γ(/) = d (x ) (x ) x 3 π ] = π dx [ 4x 3 d d = Γ(/) dx d] = d dx [I (x)] (x ) d (4.) Similarly, we obain D x = 4x 3 3 π The fracional derivaive of ideniy funcion for α = by mahemaical sofware Mahemaica as follows: is represened graphically The fracional derivaive of ideniy funcion for α = 3 can be obained as D 3 x = πx D 3 x = 8x 5 5 π α = 3 respecively and cab be represened graphically using Mahemaica sofware (c) Here we obain he firs fracional derivaive of he ideniy funcion f(x) = x by Capuo fracional derivaive formula for differen values of α. Consider he Capuo fracional derivaive formula ad α x f(x) = x f (n) (τ) dτ, where n < α < n Γ(n α) a (x τ) α n+

13 Basic Developmens of Fracional Calculus and Is Applicaions 3 Puing α =, we ge D x f(x) = Γ( ) = Γ( ) (x τ) dτ = Γ( ) dτ (x τ) x π (x τ) dτ = Γ( )[ (x τ) ] x = (4.3) Similarly, we obain D x = 4x 3 3 π, D 3 x =, πx D 3 x = 8x 5 5 π The fracional derivaives of he funcion f(x) = x obained by Grūnwald-Lenikov, Riemann-Liouville and Capuo fracional derivaive formula are same follow from equaions (3.) (3.3). (ii)consan Funcion f(x) = C, C is consan (a) Le f(x) =, using Grūnwald-Lenikov definiion we have obain he fracional derivaives of uni funcion as follows: D α x () = lim N {( ) N α N x j= } Γ(j α) Γ( α)γ(j + ) () Using properies (.3.4) and (.3.8) (page no., [5]), we ge D α x () = x α Γ( α) Puing he differen values of α, we ge D =, D x πx = π, D 3 = πx 3, D 3 = 4x 3 3 π (b) The fracional derivaive of he consan funcion is obained by Riemann- Liouville definiion as follows: D α d f(x) = Γ( α) dx C (x ) α d = C Γ( α) d dx (x ) α d C d α+ = [ x Γ( α) dx ( α + ) ] = C d Γ( α) ( α) dx [x α+ ] C = Γ( α) ( α) ( C α)x α = Γ( α) x α The fracional derivaive of uni funcion for α = mahemaical sofware Mahemaica as follows: is represened graphically by

14 4 A. P. Bhadane & K. C. Takale In paricular, le f(x) = (Uni funcion), he Riemann-Liouville fracional derivaives for differen values of α are obained as follows: D =, D x πx = π, D 3 = πx 3, D 3 = 4x 3 3 π (c) The fracional derivaive of he consan funcion is obained by Capuo definiion as follows: adx α x C = dτ =, n < α < n Γ(n α) a (x τ) α n+ Remark 4. The fracional derivaive of consan by Riemann-Liouville is no consan bu he Capuo fracional derivaive of consan is zero. Graphical represenaion of he Riemann-Liouville fracional derivaives of some oher sandard funcions for differen values of α such as α =, α =, α = 3 3 and α = are given as follows: (iii)sine Funcion If f(x) = sin(x) hen is fracional derivaive is D α f(x) = sin(x + απ ) and i is represened graphically for α = 3 by Mahemaica as follows:

15 Basic Developmens of Fracional Calculus and Is Applicaions 5 (iv)cosine Funcion If f(x) = cos(x) hen he fracional derivaive of cosine funcion is D α f(x) = cos(x + απ ) and i is represened graphically for α = 3 by Mahemaica as follows: 5 APPLICATIONS (i) Abel s Inegral: The Abel s fracional order inegral equaion is defined as [6] Γ(α) φ(τ)dτ = f(), ( > ) (5.) ( τ) α where < α <. The soluion is given by he well-known formula [] d φ() = Γ( α) d f(τ)dτ, ( > ) (5.) ( τ) α In erms of fracional order derivaives, equaions (5.) and (5.) akes on he form od α φ() = f(), ( > )od α f() = φ(), (5.3) Example 5. Le us consider he equaion φ( s + y )ds s + y = f(y) y (5.4) Afer simplificaion, we arrive a an equaion of he ype (5.), wih α = φ(τ)dτ ( τ) / = f( ) (5.5) The soluion of equaion (5.5) can be found wih help of formula (5.4) as follows φ() = Γ(/) D / f( ) (5.6)

16 6 A. P. Bhadane & K. C. Takale and performing backward subsiuion we obain he soluion of equaion (5.5) in erms of fracional derivaives φ( ) = π D / f( ) (5.7) (ii) Differenial Equaion Example 5. Consider he ordinary differenial equaion [ ] D α f() + af() =, ( > ), D α f() = C, (n < α n). = Applying he Laplace ransform on boh sides, we ge n s α F (s) k= s k [ L{ D α f()} + al{f()} = L{} ] f() + af (s) = D α k For < α, k is zero, herefore, we ge [ ] s α F (s) D α f() = = + af (s) = s α F (s) C + af (s) = F (s) = C s α + a Taking inverse Laplace ransform, we ge ] [ ] L [F (s) = CL s α + a ( f() = C α E α,α a ) (5.8) where E α,α is a Miag-Liffler funcion [6]. Tes Problem In paricular, if α =, hen we have he following differenial equaion f() + af() =, ( > ), [ D / f()] = = C D /

17 Basic Developmens of Fracional Calculus and Is Applicaions 7 Applying he Laplace ransform on boh sides, we ge L[ D / f()] + al[f() = s / F (s) [ D / f()] + af (s) = s / F (s) C + af (s) = C F (s) = s / + a Taking inverse Laplace ransform, we ge ] [ L [F (s) = CL f() = C / E, ] s / + a ( a ) Noe ha he formula (5.8), gives us direc soluion of above paricular problem. Acknowledgmen: We are very much hankful o Dr. D. B. Dhaigude, Professor and Coordinaor Applied Mahemaics, Deparmen of Mahemaics, Dr. Babasaheb Ambedkar Marahwada Universiy, Aurangabad for his valuable suggesions and guidance during preparaion of his paper. References [] N. H. Abel, Reśoluion d un Problem de Mécanique, Oeuvres Complées de Niels Henrik Abel, Vol., 97. [] A. Mc Bride, G. Roach (Eds), Fracional Calculus, Research Noes in Mahemaics, Vol.38, Piman, Boson-London-Melbourne, (985). [3] J. L. Lavoie, T. J. Osler, R. Tremblay, Fracional Derivaives and Special Funcions, SIAM Review, V.8, Issue, [4] K. Nishimoo (Ed.), Fracional Calculus and is Applicaions., Nihon Universiy, Koriyama, (99). [5] K. B. Oldham, J. Spanier, The Fracional Calculus, Academic Press, New York and Londao, (974). [6] I. Podlubny, Fracional Differenial Equaions, Academic Press, New York, (999). [7] B. Ross (Ed.), Fracional Calculus and is Applicaions, Lecure Noes in Mahemaics, Vol. 457, Springer-Verlag, New York, (975). [8] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fracional Inegraion and Derivaives: Theory and Applicaion, Gordon and Breach, Amserdam, (993).