Mittag-Leffler Functions and Fractional Calculus

Chpter 2 Mittg-Leffler Functions nd Frctionl Clculus [This chpter is bsed on the lectures of Professor R.K. Sxen of Ji Nrin Vys University, Jodhpur, Rjsthn, Indi.] 2. Introduction This section dels with Mittg-Leffler function nd its generliztions. Its importnce is relized during the lst one nd hlf decdes due to its direct involvement in the problems of physics, biology, engineering nd pplied sciences. Mittg-Leffler function nturlly occurs s the solution of frctionl order differentil equtions nd frctionl order integrl equtions. Vrious properties of Mittg-Leffler functions re described in this section. Among the vrious results presented by vrious reserchers, the importnt ones del with Lplce trnsform nd symptotic expnsions of these functions, which re directly pplicble in the solution of differentil equtions nd in the study of the behvior of the solution for smll nd lrge vlues of the rgument. Hille nd Tmrkin in 192 hve presented solution of Abel-Volterr type integrl eqution φx) λ Γα) x φt) dt = f x), < x < 1 x t) 1 α in terms of Mittg-Leffler function. Dzherbshyn 1966) hs shown tht both the functions defined by 2.1.1) nd 2.1.2) re entire functions of order p = 1 α nd type σ = 1. A detiled ccount of the bsic properties of these functions is given in the third volume of Btemnn Mnuscript Project written by Erdélyi et l 1955) under the heding Miscellneous Functions. 2.1 Mittg-Leffler Function Nottion 2.1.1. E α x) : Mittg-Leffler function Nottion 2.1.2. E α,β x) : Generlized Mittg-Leffler function 79

8 2 Mittg-Leffler Functions nd Frctionl Clculus Note 2.1.1: According to Erdélyi, et l 1955) both E α x) nd E α,β x) re clled Mittg-Leffler functions. Definition 2.1.1. Definition 2.1.2. E α,β z)= E α z)= k= k= z k z k, α C,Rα) >. 2.1.1) Γαk + 1), α,β C,Rα) >,Rβ) >. 2.1.2) Γαk + β) The function E α z) ws defined nd studied by Mittg-Leffler in the yer 193. It is direct generliztion of the exponentil series. For α = 1 we hve the exponentil series. The function defined by 2.1.2) gives generliztion of 2.1.1). This generliztion ws studied by Wimn in 195, Agrwl in 1953, Humbert nd Agrwl in 1953, nd others. Exmple 2.1.1. Solution 2.1.1: Prove tht E 1,2 z)= ez 1 z. We hve E 1,2 z)= k= z k z k Γk + 2) = k= k + 1)! = 1 z k= z k+1 k + 1)! = 1 z ez 1). Definition 2.1.3. Hyperbolic function of order n. h r z,n)= k= z nk+r 1 Definition 2.1.4. Trigonometric functions of order n. K r z,n)= nk + r 1)! = zr 1 E n,r z n ), r = 1,2,... 2.1.3) k= 1) k z kn+r 1 kn+ r 1)! = zr 1 E n,r z n ). 2.1.4) z k E 1 2,1 z)= k= Γ k 2 + 1 ) = ez2 erfc z), 2.1.5) where erfc is complementry to the error function erf. Definition 2.1.5. Error function. erfcz)= 2 π 1 2 z e u2 du = 1 erfz), z C. 2.1.6)

2.1 Mittg-Leffler Function 81 To derive 2.1.5), we see tht Dzherbshyn 1966, P.297, Eq.7.1.) reds s wheres Dzherbshyn 1966, P.297, Eq.7.1.8) is wz)=e z2 erfc iz) 2.1.7) wz)= n= iz) n Γ ). n 2.1.8) 2 + 1 From 2.1.7) nd 2.1.8) we esily obtin 2.1.5). In pssing, we note tht wz) is lso n error function Dzherbshyn 1966)). Definition 2.1.6. Mellin-Ross function. E t ν,)=t ν k= Definition 2.1.7. Robotov s function. R α β,t)=t α k= t) k Γν + k + 1) = tν E 1,ν+1 t). 2.1.9) β k t kα+1) Γ1 + α)k + 1)) = tα E α+1,α+1 βt α+1 ). 2.1.1) Exmple 2.1.2. Solution 2.1.2: Prove tht E 1,3 z)= ez z 1 z 2. We hve E 1,3 z)= k= z k Γk + 3) = 1 z 2 = 1 z 2 ez z 1). k= z k+2 k + 2)! Exmple 2.1.3. Prove tht E 1,r z)= 1 z r 1 { e z r 2 k= } z k, r = 1,2,... k! The proof is similr to tht in Exmple 2.1.2. Revision Exercises 2.1. 2.1.1. Prove tht [ H 1,1 1,2 x ],A) = A 1 1),A),,1) k x k+)/a Γ1 +k + )A), k=

82 2 Mittg-Leffler Functions nd Frctionl Clculus nd write the right side in terms of generlized Mittg-Leffler function. 2.1.2. Prove tht d [ dx H1,1 1,2 x ],A),A),,1) [ = H 1,1 1,2 x ] A,A). A,A),,1) 2.1.3. Prove tht H 1,1 2,1 [ 1 x 1,A),1,1) 1,A) ] = A 1 1) k 1 x ) k+1 A Γ1 k + 1 )/A). k= 2.2 Bsic Properties of Mittg-Leffler Function As consequence of the definitions 2.1.1) nd 2.1.2) the following results hold: Theorem 2.2.1. There hold the following reltions: i) E α,β z)=ze α,α+β z)+ 1 Γβ) 2.2.1) ii) E α,β z)=βe α,β+1 z)+αz d dz E α,β+1z) 2.2.2) ) d m [ iii) z β 1 E dz α,β z )] α = z β m 1 E α,β m z α ), 2.2.3) Rβ m) >, m =,1,... 2.2.4) Solutions 2.2.1: i) We hve E α,β z)= k= z k Γαk + β) = k= 1 z k+1 Γα + β + αk) = ze α,α+β z)+ 1, Rβ) >. Γβ) ii) We hve R.H.S. = βe α,β+1 z)+αz d dz k= k= z k Γαk + β + 1) αk + β)z k = Γαk + β + 1) = z k k= Γαk + β) = E α,β z)=l.h.s.

2.2 Bsic Properties of Mittg-Leffler Function 83 iii) since k= d m L.H.S. = dz) k= = k= ) d m z αk+β 1 )= dz z αk+β m 1 z αk+β 1 Γαk + β), Rβ m) >, Γαk + β m) k= Γαk + β) Γαk + β m) zαk+β m 1 = z β m 1 E α,β m z α ), m =,1,2,... = R.H.S. Following specil cses of 2.2.3) re worth mentioning. If we set α = m n, m,n = 1,2,... then ) d m [ ] z β 1 E m dz n,β z m n ) = z β m 1 E m n,β mz m n ) for Rβ m) >, replcing k by k + n) = z β m 1 = z β m 1 k= n z mk+n) n k= Γ β + mk n = z β 1 E m n,β z m n )+z β 1 n ) k=1 z mk n Γ mk n + β m ). z mk n Γ β mk n ), m,n = 1,2,3. 2.2.5) ) d m [ z β 1 E dz m,β z )] m = z β 1 E m,β z m )+ z m, Rβ m) >. 2.2.6) Γβ m) Putting z = t m n in 2.2.3) it yields m n t1 m n d ) m [t β 1) m n E m dt n,β t)] = t β 1) m n E m n,β t) +t β 1) n m n k=1 When m = 1, 2.2.7) reduces to t 1 n n t k Γβ mk Rβ m) >, m,n = 1,2,... 2.2.7) n ), d ] [t β 1)n E dt 1n,β t) = t β 1)n n E 1 n,β t)+tβ 1)n k=1 t k Γ β n k ),

84 2 Mittg-Leffler Functions nd Frctionl Clculus for Rβ) > 1, which cn be written s 1 d n dt [t β 1)n E 1n,β t) ] = t βn 1 E 1 n,β t)+tβn 1 n k=1 t k Γ β n k ), Rβ) > 1. 2.2.8) 2.2.1 Mittg-Leffler functions of rtionl order Now we consider the Mittg-Leffler functions of rtionl order α = p q with p,q = 1,2,... reltively prime. The following reltions redily follow from the definitions 2.1.1) nd 2.1.2). i) ii) ) d p E p z p )=E p z p ) dz 2.2.9) ) d p E p dz q )=Ep q 1 z q )+ q p q q Γk p q + 1 p), 2.2.1) k=1 q = 1,2,3,... We now derive the reltion iii) E 1 q z 1 q )=e z [1 + q 1 k=1 γ1 q k,z) ] Γ1 q k ), 2.2.11) where q = 2,3,... nd γα,z) is the incomplete gmm function, defined by z γα,z)= e u u α 1 du To prove 2.2.11), set p = 1 in 2.2.1) nd multiply both sides by e z nd use the definition of γα,z). Thus we hve d [ ] e z E 1 z 1 q 1 q z ) = e dz q k=1 Γ z k q Integrting 2.2.12) with respect to z, we obtin 2.2.11). ). 2.2.12) 1 q k 2.2.2 Euler trnsform of Mittg-Leffler function By virtue of bet function formul it is not difficult to show tht 1 [ z ρ 1 1 z) σ 1 E α,β xz γ )dz = Γσ) 2 ψ 2 x ] ρ,γ),1,1) β,α),σ+ρ,γ) 2.2.13)

2.2 Bsic Properties of Mittg-Leffler Function 85 where Rα) >,Rβ) >,Rσ) >,γ >. Here 2 ψ 2 is the generlized Wright function nd α,β,ρ,σ C. Specil cses of 2.2.13): i) When ρ = β,γ = α, 2.2.13) yields 1 z β 1 1 z) σ 1 E α,β xz α )dz = Γσ)E α,σ+β x), 2.2.14) where α > ;β,σ C,Rβ) >,Rσ) > nd, ii) 1 z σ 1 1 z) β 1 E α,β [x1 z) α ]dz = Γσ)E α,β+σ x), 2.2.15) where α > ;β,σ C,Rβ) >,Rσ) >. iii) When α = β = 1wehve 1 [ z ρ 1 1 z) σ 1 expxz γ )dz = Γσ) 2 ψ 2 x ] ρ,γ),1,1) 1,1),σ+ρ,γ) [ = Γσ) 1 ψ 1 x ] ρ,γ), 2.2.16) σ+ρ,γ) where γ >,ρ,σ C,Rρ) >,Rσ) >. 2.2.3 Lplce trnsform of Mittg-Leffler function Nottion 2.2.1. prmeter s. Nottion 2.2.2. Fs) =L{ f t); s} =L f)s) : Lplce trnsform of f t) with L 1 {Fs);t} : Inverse Lplce trnsform Definition 2.2.1. The Lplce trnsform of function f t), denoted by Fs), is defined by the eqution Fs)=Lf)s)=L{ f t);s} = e st f t)dt, 2.2.17) where Rs) >, which my be symboliclly written s Fs)=L{ f t);s} or f t)=l 1 {Fs);t}, provided tht the function f t) is continuous for t, it being tcitly ssumed tht the integrl in 2.2.17) exists.

86 2 Mittg-Leffler Functions nd Frctionl Clculus Exmple 2.2.1. Prove tht It follows from the Lplce integrl L 1 {s ρ } = tρ 1, Rs) >, Rρ) >. 2.2.18) Γρ) e st t ρ 1 dt = Γs), Rs) >, Rρ) >. 2.2.19) sρ Exmple 2.2.2. Find the inverse Lplce trnsform of Rs) >,Fs)=L{ f t);s}. Fs) +s α ;,α > ; where Solution 2.2.1: Let Gs)= 1 + s α = ) r s α αr, r= s α < 1. Therefore, { } L 1 {Gs)} = gt)=l 1 ) r s α αr r= = t α 1 E α,α t α ). 2.2.2) Appliction of convolution theorem of Lplce trnsform yields the result { } Fs) x L 1 + s α ;t = x t) α 1 E α,α x t) α ) f t)dt 2.2.21) where Rα) >. By the ppliction of Lplce integrl, it follows tht z ρ 1 e z E α,β xz γ )dz = 1 [ x ] ρ 2 ψ 1 1,1),ρ,γ) γ, 2.2.22) β,α) where ρ,,α,β C,Rα) >,Rβ) >,Rγ) >,R) >,Rρ) > nd z γ < 1. Specil cses of 2.2.22) re worth mentioning. i) For ρ = β,γ = α,rα) >, 2.2.22) gives where,α,β C,Rα) >,Rβ) >, x α < 1. e z z β 1 E α,β xz α )dz = α β α x, 2.2.23)

2.2 Bsic Properties of Mittg-Leffler Function 87 When = 1, 2.2.23) yields known result. e z z β 1 E α,β xz α )dz = 1, x < 1, 2.2.24) 1 x where Rα) >,Rβ) >. If we further tke β = 1, 2.2.24) reduces to e z E α xz α )dz = 1, x < 1. 1 x ii) When β = 1, 2.2.23) gives e z E α xz α )dz = α 1 α x, 2.2.25) where R) >,Rα) >, x α < 1. 2.2.4 Appliction of Llce trnsform From 2.2.23) we find tht L{x β 1 E α,β x α )} = sα β s α 2.2.26) where Rα) >,Rβ) >. We lso hve L{x γ 1 E α,γ x α )} = sα γ s α +. 2.2.27) Now [ ] [ s α β ] s α γ s α s α = s2α β+γ) + s 2α 2 for Rs2 ) > R). 2.2.28) By virtue of the convolution theorem of the Lplce trnsform, it redily follows tht t u β 1 E α,β u α )t u) γ 1 E α,γ t u) α )du = t β+γ 1 E 2α,β+γ 2 t 2α ), 2.2.29) where Rβ) >,Rγ) >. Further, if we use the identity 1 s 2 = sα β [ s α s β 2 s β α 2] 2.2.3) 1 nd the reltion L{t ρ 1 ;s} = Γρ)s ρ, 2.2.31)

88 2 Mittg-Leffler Functions nd Frctionl Clculus where Rρ) >, Rs) >, we obtin [ ] t u β 1 E α,β u α t u) 1 β t u)α β+1 ) du = t, 2.2.32) Γ2 β) Γα β + 2) where < β < 2,Rα) >. Next we note tht the following result 2.2.34) cn be derived by the ppliction of inverse Lplce trnsform to the identity [ ] s 2α β s 2α [s α ]= s2α β 1 s 2α 1 + sα β s α 1, Rsα ) > 1. 2.2.33) We hve 1 Γα) x x t) α 1 E 2α,β t 2α )t β 1 dt = x β 1 E 2α,β x 2α )+x β 1 E α,β x α ), 2.2.34) where Rα) >,Rβ) >. If we set β = 1 in 2.2.34), it reduces to 1 x x t) α 1 E 2α t 2α )dt = E α x α ) E 2α x 2α ) 2.2.35) Γα) where Rα) >. 2.2.5 Mittg-Leffler functions nd the H-function Both the Mittg-Leffler functions E α z) nd E α,β z) belong to H-function fmily. We derive their reltions with the H-function. Lemm 2.2.1: Let α R + =,). Then E α z) is represented by the Mellin- Brnes integrl E α z)= 1 Γs)Γ1 s) z) s ds, rgz < π, 2.2.36) 2πi L Γ1 αs) where the contour of integrtion L, beginning t c i nd ending t c+i,<c<1, seprtes ll poles s = k,k =,1,2,... to the left nd ll poles s = 1+n,n =,1,... to the right. Proof. We now evlute the integrl 2.2.36) s the sum of the residues t the points s =, 1, 2,.... We find tht

2.2 Bsic Properties of Mittg-Leffler Function 89 1 Γs)Γ1 s) z) s [ ] s + k)γs)γ1 s) z) s ds = 2πi L Γ1 αs) lim k= s k Γ1 αs) 2.2.37) 1) = Γ1 + k) k= k!γ1 + αk) z)k = E α z), which yields 2.2.36) in ccordnce with the definition 2.1.1). It redily follows from the definition of the H-function nd 2.2.36) tht E α z) cn be represented in the form [ E α z)=h 1,1 1,2 z ],1), 2.2.38),1),,α) where H 1,1 1,2 is the H-function, which is studied in Chpter 1. Lemm 2.2.2: Let α R + =,),β C, then E α,β z)= 1 Γs)Γ1 s) z) s ds. 2.2.39) 2πi L Γβ αs) The proof of 2.2.39) is similr to tht of 2.2.36). Hence the proof is omitted. From 2.2.39) nd the definition of the H-function we obtin the reltion [ E α,β z)=h 1,1 1,2 z ],1). 2.2.4),1),1 β,α) In prticulr, E α z) cn be expressed in terms of generlized Wright function in the form [ E α z)= 1 ψ 1 z ] 1,1). 2.2.41) Similrly, we hve [ E α,β z)= 1 ψ 1 z ] 1,1). 2.2.42) β,α) Next, if we clculte the residues t the poles of the gmm function Γ1 s) t the points s = 1 + n,n =,1,2,... it gives 1 Γs)Γ1 s) 2πi L Γ1 αs) z) s ds = lim n= s 1+n n= = = 1,α) s 1 n)γs)γ1 s) z) s Γ1 αs) 1) n Γ1 + n) z) n 1 n!γ1 α1 + n)) n=1 z n for α 1,2,. Similrly for α 1,2,,E α,β z),gives 1 Γs)Γ1 s) 2πi L Γβ αs) z) s ds = Γ1 αn), 2.2.43) n=1 z n Γβ αn). 2.2.44)

9 2 Mittg-Leffler Functions nd Frctionl Clculus 2.2.1. Let Exercises 2.2. U 1 t)=t β 1 E m n,β t m n ) U 2 t)=t β 1 E m,β t m ) U 3 t)=t β 1) m n E m n,β t) U 4 t)=t β 1)n E 1 n,β t). Then show tht these functions respectively stisfy the following differentil equtions of Mittg-Leffler functions when m,n re reltively prime. 2.2.2. Prove tht i) dm dt mu 1t) U 1 t)=t β 1 n k=1 Rβ) > m,m,n = 1,2,3,...); t m n k Γβ mk n ) ii) dm dt mu 2t) U 2 t)= t m+β 1, Rβ) > m,m = 1,2,...; Γβ m) m iii) n t1 m n d ) m U 3 t) U 3 t)=t β 1) n n t m dt k k=1 Γβ mk n ) m,n = 1,2,3,...,Rβ) > m; iv) 1 [ ] d n dt U 4t) t n 1 U 4 t)=t nβ 1 n t k k=1 Γβ n k ) n = 1,2,3,...,Rβ) > 1. λ x E α λt α ) Γα) x t) 1 α dt = E αλx α ) 1,Rα) >. 2.2.3. Prove tht d dx [xγ 1 E α,β x α )] = x γ 2 E α,β 1 x α )+γ β)x γ 2 E α,β x α ),β γ. 2.2.4. Prove tht 1 z t β 1 z t) ν 1 E Γν) α,β λt α )dt = z β+ν 1 E α,β+ν λz α ), Rβ) >,Rν) >,Rα) >. 2.2.5. Prove tht 1 z z t) α 1 cosh λt)dt = z α E 2,α+1 λz 2 ),Rα) >. Γα)

2.3 Generlized Mittg-Leffler Function 91 2.2.6. Prove tht 1 z e λt z t) α 1 dt = z α E 1,α+1 λz),rα) >. Γα) 2.2.7. Prove tht 1 z z t) α 1 sinh λt) dt = z α+1 E 2,α+2 λz 2 ),Rα) >. Γα) λ 2.2.8. Prove tht 2.2.9. Prove tht 2.2.1. Prove tht e sx x β 1 E α,β x α )dx = sα β s α,rs) > 1. 1 e st E α t α )dt = 1,Rs) > 1. s s1 α x where y,z C; y z,γ >,β >. u γ 1 E α,γ yu α )x u) β 1 E α,β [zx u) α ]du = ye α,β+γyx α ) ze α,β+γ zx α ) x β+γ 1, y z 2.3 Generlized Mittg-Leffler Function Nottion 2.3.1. Eβ,γ δ z): Generlized Mittg-Leffler function Definition 2.3.1. Eβ,γ δ z)= δ) n z n n= Γβn + γ)n!, 2.3.1) where β,γ,δ C with Rβ) >. For δ = 1, it reduces to Mittg-Leffler function 2.1.2). This function ws introduced by T.R. Prbhkr in 1971. It is n entire function of order ρ =[Rβ)] 1.

92 2 Mittg-Leffler Functions nd Frctionl Clculus 2.3.1 Specil cses of E δ β,γ z) i) E β z)=eβ,1 1 z) 2.3.2) ii) E β,γ z)=e 1 β,γ z) 2.3.3) iii) φγ,δ;z)= 1 F 1 γ;δ;z)=γδ)e γ 1,δ z), 2.3.4) where φγ,δ;z) is Kummer s confluent hypergeometric function. 2.3.2 Mellin-Brnes integrl representtion Lemm 2.3.1: Let β R + =,);γ,δ C,γ,Rδ) >. Then Eβ,γ δ z) is represented by the Mellin-Brnes integrl Eβ,γ δ z)= 1 1 Γs)Γδ s) Γδ) 2πi Γγ βs) z) s ds, 2.3.5) L where rgz) < π; the contour of integrtion beginning t c i nd ending t c + i, < c < Rδ), seprtes ll the poles t s = k,k =,1,... to the left nd ll the poles t s = n + δ,n =,1,... to the right. Proof. We will evlute the integrl on the R.H.S. of 2.3.5) s the sum of the residues t the poles s =, 1, 2,....Wehve [ ] 1 Γs)Γδ s) s + k)γs)γδ s) z) s 2πi L Γγ βs) z) s ds = lim s k Γγ βs) which proves 2.3.5). = k= k= = Γδ) 1) k k! k= Γδ + k) Γγ + βk) z)k δ) k Γβk + γ) k! = Γδ)Eδ β,γ z) z k 2.3.3 Reltions with the H-function nd Wright function It follows from 2.3.5) tht Eβ,γ δ z) cn be represented in the form Eβ,γ δ z)= 1 [ Γδ) H1,1 1,2 z ] 1 δ,1),1),1 γ,β) 2.3.6)

2.3 Generlized Mittg-Leffler Function 93 where H 1,1 1,2 z) is the H-function, the theory of which cn be found in Chpter 1. This function cn lso be represented by Eβ,γ δ z)= 1 [ Γδ) 1 ψ 1 z ] δ,1) 2.3.7) γ,β) where 1 ψ 1 is the Wright hypergeometric function p ψ q z). 2.3.4 Cses of reducibility In this subsection we present some interesting cses of reducibility of the function Eβ,γ δ z). The results re given in the form of five theorems. The results re useful in the investigtion of the solutions of certin frctionl order differentil nd integrl equtions.the proofs of the following theorems cn be developed on similr lines to tht of eqution 2.2.1). Theorem 2.3.1. If β,γ,δ C with Rβ) >,Rγ) >,Rγ β) >, then there holds the reltion zeβ,γ δ z)=eδ β,γ β z) Eδ 1 β,γ β z). 2.3.8) Corollry 2.3.1: If β,γ C,Rγ) > Rβ) >, then we hve ze 1 β,γ z)=e β,γ β z) 1 Γγ β). 2.3.9) Theorem 2.3.2. If β,γ,δ C,Rβ) >,Rγ) > 1, then there holds the formul βe 2 β,γ z)=e β,γ 1z)+1 + β γ)e β,γ z). 2.3.1) Theorem 2.3.3. If Rβ) >, Rγ) > 2 + Rβ), then there holds the formul ze 3 β,γ z)= 1 2β 2 [E β,γ β 2z) 2γ 3β 3)E β,γ β 1 z) +2β 2 + γ 2 3βγ+ 3β 2γ + 1)E β,γ β z)]. 2.3.11) Theorem 2.3.4. IfRβ) >, Rγ) > 2, then there holds the formul E 3 β,γ z)= 1 2β 2 [E β,γ 2z)+3 + 3β 2γ)E β,γ 1 z) +2β 2 + γ 2 + 3β 3βγ 2γ + 1)E β,γ z)]. 2.3.12)

94 2 Mittg-Leffler Functions nd Frctionl Clculus 2.3.5 Differentition of generlized Mittg-Leffler function Theorem 2.3.5. Let β,γ,δ,ρ,w C. Then for ny n = 1,2,... there holds the formul, for Rγ) > n, ) d n [z γ 1 Eβ,γ δ dz wzβ )] = z γ n 1 Eβ,γ n δ wzβ ). 2.3.13) In prticulr, for Rγ) > n, ) d n [z γ 1 E dz β,γwz β )] = z γ n 1 E β,γ nwz β ) 2.3.14) nd for Rγ) > n, ) d n [z γ 1 φδ;γ;wz)] = Γγ) dz Γγ n) zγ n 1 φδ;γ n;wz). 2.3.15) Proof. Using 2.3.1) nd tking term by term differentition under the summtion sign, which is possible in ccordnce with uniform convergence of the series in 2.3.1) in ny compct set of C, we obtin ) d n [z γ 1 E δ dz β,γ wzβ )] = k= δ) k Γβk + γ) d dz ) ] n [w k z βx+γ 1 k! = z γ n 1 E δ β,γ n wzβ ), Rγ) > n, which estblishes 2.3.13). Note tht 2.3.14) follows from 2.3.13) when δ = 1 due to 2.3.3), nd 2.3.15) follows from 2.3.13) when β = 1 on ccount of 2.3.4). 2.3.6 Integrl property of generlized Mittg-Leffler function Corollry 2.3.2: Let β,γ,δ,w C,Rγ) >,Rβ) >,Rδ) >. Then z t γ 1 Eβ,γ δ wtβ )dt = z γ Eβ,γ+1 δ wzβ ) 2.3.16) nd 2.3.16) follows from 2.3.13). In prticulr, z t γ 1 E β,γ wt β )dt = z γ E β,γ+1 wz β ) 2.3.17) nd z t δ 1 φγ,δ;wt)dt = 1 δ zδ φγ,δ + 1;wz) 2.3.18) Remrk 2.3.1: The reltions 2.3.15) nd 2.3.18) re well known.

2.3 Generlized Mittg-Leffler Function 95 2.3.7 Integrl trnsform of E δ β,γ z) By ppeling to the Mellin inversion formul, 2.3.5) yields the Mellin trnsform of the generlized Mittg-Leffler function. t s 1 Eβ,γ δ Γs)Γδ s)w s wt)dt = Γδ)Γγ sβ). 2.3.19) If we mke use of the integrl t ν 1 e t 2W λ,µ t)dt = Γ 1 2 + µ + ν ) Γ 1 2 µ + ν ) Γ1 λ + ν) 2.3.2) where Rν ± µ) > 1 2, we obtin the Whittker trnsform of the Mittg-Leffler function t ρ 1 e 1 2 pt W λ,µ pt)e δ β,γ wtα )dt = p ρ Γδ) 3 ψ 2 [ ] w δ,1), 1 2 ±µ+ρ,α) p α γ,β),1 λ+ρ,α) 2.3.21) where 3 ψ 2 is the generlized Wright function, nd Rρ) > Rµ) 1 2,Rp) >, p w α < 1. When λ = nd µ = 1 2, then by virtue of the identity W ± 1 2, t)=exp t ), 2.3.22) 2 the Lplce trnsform of the generlized Mittg-Leffler function is obtined. [ ] t ρ 1 e pt Eβ,γ δ wtα )dt = p ρ w Γδ) 2 ψ 1 δ,1),ρ,α) p α 2.3.23) γ,β) where Rβ) >,Rγ) >,Rρ) >,Rp) >, p > w Rα). In prticulr, for ρ = γ nd α = β we obtin result given by Prbhkr 1971, Eq.2.5). t γ 1 e pt E δ β,γ wtβ )dt = p γ 1 wp β ) δ 2.3.24) where Rβ) >,Rγ) >,Rp) > nd p > w 1 Rβ). The Euler trnsform of the generlized Mittg-Leffler function follows from the bet function. 1 t 1 1 t) b 1 Eβ,γ δ xtα )dt = Γb) [ Γδ) 2 ψ 2 x ] δ,1),,α), 2.3.25) γ,β),+b,α) where R) >,Rb) >,Rδ) >,Rβ) >,Rγ) >,Rα) >. 1

96 2 Mittg-Leffler Functions nd Frctionl Clculus Theorem 2.3.6. We hve where Rp) > 1 e pt t αk+β 1 E k) α,β tα )dt = k!pα β p α, 2.3.26) ) k+1 Rα),Rα) >,Rβ) >, nd E k) dk α,β y)= E dy k α,β y). Proof: We will use the following result: The given integrl e t t β 1 E α,β zt α )dt = 1, z < 1. 2.3.27) 1 z = dk d k e pt t β 1 E α,β ±t α )dt = dk p α β d k p α ) = k!pα β p α,rβ) >. ) k+1 Corollry 2.3.3: e pt t k 1 2 E k) 1 k! t)dt = 2, 2 1 p ) k+1 2.3.28) where Rp) > 2. Exercises 2.3. 2.3.1. Prove tht 1 1 u γ 1 1 u) α 1 Eβ,γ δ Γα) zuβ )du = Eβ,γ+α δ z),rα) >,Rβ) >,Rγ) >. 2.3.2. Prove tht 1 x x u) α 1 u t) γ 1 Eβ,γ δ Γα) [λu t)β ]du =x t) γ+α 1 Eβ,γ+α δ [λx t)β ] t where Rα) >,Rγ) >,Rβ) >. 2.3.3. Prove tht for n = 1,2,... E δ n,γz)= 1 Γγ) 1 F n δ; n;γ);n n z), where n;γ) represents the sequence of n prmeters γ n, γ+1 n 2.3.4. Show tht for Rβ) >, Rγ) >, γ+n 1,..., n.

2.4 Frctionl Integrls 97 ) d m Eβ,γ δ dz z)=δ) me δ+m β,γ+mβ z). 2.3.5. Prove tht for Rβ) >, Rγ) >, z ddz ) + δ Eβ,γ δ z)=δeδ+1 β,γ z). 2.3.6. Prove tht for Rγ) > 1, γ βδ 1)Eβ,γ δ z)=eδ β,γ 1 z) βδeδ+1 β,γ z). 2.3.7. Prove tht x t ν 1 x t) µ 1 Eρ,µ[wx γ t) ρ ]Eρ,νwt σ ρ )dt = x µ+ν 1 Eρ,µ+νwx γ+σ ρ ), where ρ, µ,γ,ν,σ,w C;Rρ),Rµ),Rν) >. 2.3.8. Find [ L 1 s λ 1 z ) ] α s ρ nd give the conditions of vlidity. 2.3.9. Prove tht [ L 1 s λ 1 z ) 1 α1 1 z ) ] 2 α2 = tλ 1 s s Γλ) Φ 2α 1,α 2 ;λ;z 1 t,z 2 t), where Rλ) >,Rs) > mx[,rz 1 ),Rz 2 )] nd Φ 2 is the confluent hypergeometric function of two vribles defined by Φ 2 b,b ;c;u,z)= k, j= 2.3.1. From the bove result deduce the formul b) k b ) j u k z j. 2.3.29) c) k+ j k! j! [ L 1 s λ 1 z s ) α] = tλ 1 φα,λ;zt), 2.3.3) Γλ) where Rλ) >,Rs) > mx[, z ]. 2.4 Frctionl Integrls This section dels with the definition nd properties of vrious opertors of frctionl integrtion nd frctionl differentition of rbitrry order. Among the vrious opertors studied re the Riemnn-Liouville frctionl integrl opertors, Riemnn- Liouville frctionl differentil opertors, Weyl opertors, Kober opertors etc. Besides the bsic properties of these opertors, their behviors under Lplce, Fourier nd Mellin trnsforms re lso presented. Appliction of Riemnn-Liouville opertors in the solution of frctionl order differentil nd frctionl order integrl equtions is demonstrted.

98 2 Mittg-Leffler Functions nd Frctionl Clculus 2.4.1 Riemnn-Liouville frctionl integrls of rbitrry order Nottion 2.4.1. Ix n, D n x, n N : Frctionl integrl of integer order n Definition 2.4.1. Ix n f x)= D n x f x)= 1 x x t) n 1 f t)dt 2.4.1) Γn) where n N. We begin our study of frctionl clculus by introducing frctionl integrl of integer order n in the form of Cuchy formul. D n x f x)= 1 x x t) n 1 f t)dt. 2.4.2) Γn) It will be shown tht the bove integrl cn be expressed in terms of n-fold integrl, tht is, Proof. x x1 D n x f x)= dx 1 dx 2 x2 xn 1 dx 3... f t)dt. 2.4.3) When n = 2, by using the well-known Dirichlet formul, nmely 2.4.3) becomes b x x b b dx f x,y)dy = dy f x, y)dx 2.4.4) y dx 1 x1 x x f t)dt = dtft) dx 1 t x = x t) f t)dt. 2.4.5) This shows tht the two-fold integrl cn be reduced to single integrl with the help of Dirichlet formul. For n = 3, the integrl in 2.4.3) gives D 3 x x f x)= = x x1 x2 dx 1 dx 2 f t)dt [ x1 x2 dx 1 dx 2 f t)dt ]. 2.4.6) By using the result in 2.4.5) the integrls within big brckets simplify to yield x [ x1 D 3 x f x)= dx 1 If we use 2.4.4), then the bove expression reduces to x 1 t) f t)dt ]. 2.4.7)

2.4 Frctionl Integrls 99 x t x D 3 x t) 2 x f x)= dtft) x 1 t)dx 1 = f t)dt. 2.4.8) x 2! Continuing this process, we finlly obtin D n 1 x x f x)= x t) n 1 f t)dt. 2.4.9) n 1)! It is evident tht the integrl in 2.4.9) is meningful for ny number n provided its rel prt is greter thn zero. 2.4.2 Riemnn-Liouville frctionl integrls of order α Nottion 2.4.2. integrl of order α. xib α, xdb α,ib α : Riemnn-Liouville right-sided frctionl Definition 2.4.2. Let f x) L,b),α C,Rα) >, then Ix α f x)= Dx α f x)=i+ α f x)= 1 x f t) dt,x > 2.4.1) Γα) x t) 1 α is clled Riemnn-Liouville left-sided frctionl integrl of order α. Definition 2.4.3. Let f x) L,b),α C,Rα) >, then xib α f x)= xdb α f x)=ib α f x)= 1 b f t) dt,x < b 2.4.11) Γα) x t x) 1 α is clled Riemnn-Liouville right-sided frctionl integrl of order α. Exmple 2.4.1. Solution 2.4.1: If f x)=x ) β 1, then find the vlue of I α x f x). We hve Ix α f x)= 1 x x t) α 1 t ) β 1 dt. Γα) If we substitute t = + yx ) in the bove integrl, it reduces to where Rβ) >. Thus I α x f x)= Γβ) x )α+β 1 Γα + β) Γβ) Γα + β) x )α+β 1. 2.4.12)

1 2 Mittg-Leffler Functions nd Frctionl Clculus Exmple 2.4.2. It cn be similrly shown tht xi α b gx)= where Rβ) > nd gx)=b x) β 1. Γβ) Γα + β) b x)α+β 1,x < b 2.4.13) Note 2.4.1: It my be noted tht 2.4.12) nd 2.4.13) give the Riemnn- Liouville integrls of the power functions f x) =x ) β 1 nd gx) =b x) β 1,Rβ) >. 2.4.3 Bsic properties of frctionl integrls Property: Frctionl integrls obey the following properties: I α x I β x φ = I α+β x φ = I β x I α x φ, xib α xi β b φ = xi α+β b φ = x I β b xib α φ. 2.4.14) Proof: By virtue of the definition 2.4.1), it follows tht x Ix α Ix β φ = 1 dt 1 Γα) x t) 1 α Γβ) 1 x x = duφu) Γα)Γβ) u t φu)du t u) 1 β dt x t) 1 α. 2.4.15) t u) 1 β If we use the substitution y = x u t u, the vlue of the second integrl is 1 1 Γα)Γβ)x u) 1 α β y β 1 1 y) α 1 dy = x u)α+β 1, Γα + β) which, when substituted in 2.4.15) yields the first prt of 2.4.14). The second prt cn be similrly estblished. In prticulr, I n+α x f = I n x I α x f,n N,Rα) > 2.4.16) which shows tht the n-fold differentition d n dx n Ix n+α f x)= Ix α f x),n N,Rα) > 2.4.17) for ll x. When α =, we obtin Ix f x)= f x); Ix n f x)= dn dx n f x)= f n) x). 2.4.18) Note 2.4.2: The property given in 2.4.14) is clled semigroup property of frctionl integrtion.

2.4 Frctionl Integrls 11 L, b): spce of Lebesgue mesurble rel or complex vl- Nottion 2.4.3. ued functions. Definition 2.4.4. L, b), consists of Lebesgue mesurble rel or complex vlued functions f x) on [,b]: b L,b)={ f : f 1 f t) dt < }. 2.4.19) Note 2.4.3: The opertors I α x nd x I α b re defined on the spce L,b). Property: The following results hold: b b f x) Ix α g)dx = gx) x Ib α f )dx. 2.4.2) 2.4.2) cn be estblished by interchnging the order of integrtion in the integrl on the left-hnd side of 2.4.2) nd then using the Dirichlet formul 2.4.4). The bove property is clled the property of integrtion by prts for frctionl integrls. 2.4.4 A useful integrl We now evlute the following integrl given by Sxen nd Nishimoto [Journl of Frctionl Clculus, Vol. 6, 1994, 65-75]. b t ) α 1 b t) β 1 ct + d) γ dt =c + d) γ b ) α+β 1 [ ] b)c Bα,β) 2 F 1 α, γ;α + β;, c + d) 2.4.21) where Rα) >,Rβ) >, rg d+bc) d+c) < π,,c nd d re constnts. Solution Let b I = t ) α 1 b t) β 1 ct + d) γ dt =c + d) γ 1) k γ) k c k b k= c + d) k t ) α+k 1 b t) β 1 dt ) =c + d) γ b ) α+β 1 b)c Bα,β) 2 F 1 γ,α;α + β;. c + d)

12 2 Mittg-Leffler Functions nd Frctionl Clculus In evluting the inner integrl the modified form of the bet function, nmely b where Rα) >,Rβ) >, is used. t ) α 1 b t) β 1 dt =b ) α+β 1 Bα,β), 2.4.22) Exmple 2.4.3. As consequence of 2.4.21) it follows tht Ix α [x ) β 1 cx + d) γ ]=c + d) γ x ) α+β 1 Γβ) Γα + β) ) x)c 2 F 1 β, γ;α + β;, 2.4.23) c + d) where Rα) >,Rβ) >, rg x)c c+d) < π,,c nd d being constnts. In similr mnner we obtin the following result: Exmple 2.4.4. We lso hve xib α [b x)β 1 cx + d) γ =cx + d) γ ]b x) α+β 1 γβ) Γα + β) where Rα) >,Rβ) >, rg x b)c cx+d) < π. Exmple 2.4.5. tht ) x b)c 2 F 1 α, γ;α + β;, 2.4.24) cx + d) On the other hnd if we set γ = α β in 2.4.21) it is found Dx α [x ) β 1 cx + d) α β ]= Γβ) Γα + β) c + d) α x ) α+β 1 d + cx) β, 2.4.25) where Rα) >,Rβ) >. Exmple 2.4.6. Similrly, we hve xi α b [b x)β 1 cx + d) α β ]= where Rα) >,Rβ) >. Γβ) Γα + β) cx + d) β bc + d) α b x) α+β 1 2.4.26)

2.4 Frctionl Integrls 13 2.4.5 The Weyl integrl Nottion 2.4.4. xw α, x I α : Weyl integrl of order α. Definition 2.4.5. The Weyl frctionl integrl of f x) of order α, denoted by xw α, is defined by xw α f x)= 1 t x) α 1 f t)dt, < x < 2.4.27) Γα) x where α C,Rα) >. 2.4.27) is lso denoted by x I α f x). Exmple 2.4.7. Prove tht Solution: Wehve Nottion 2.4.5. xw α e λx = e λx where Rα) >. 2.4.28) λ α xw α e λx = 1 Γα) x = e λx Γα)λ α t x) α 1 e λt dt, λ > u α 1 e u du = e λx, Rα) >. λ α xd, α D α Weyl frctionl derivtive. Definition 2.4.6. The Weyl frctionl derivtive of order α, denoted by x D, α is defined by ) d m xd α f x)=d f x)= 1) m xw m α f x) ) dx ) d m = 1) m 1 f t) dt, < x < 2.4.29) dx Γm α) x t x) 1+α m where m 1 α < m, α C,m =,1,2,.... Exmple 2.4.8. Find x D α e λx, λ >. Solution: Wehve xd α e λx = 1) m d dx ) m xw m α e λx = 1) m d dx) m λ m α) e λx 2.4.3) = λ α e λx.

14 2 Mittg-Leffler Functions nd Frctionl Clculus 2.4.6 Bsic properties of Weyl integrl Property: The following reltion holds: φx) Ix α ψx))dx = x W α φx))ψx)dx. 2.4.31) 2.4.31) is clled the formul for frctionl integrtion by prts. It is lso clled Prsevl equlity. 2.4.31) cn be estblished by interchnging the order of integrtion. Property: Weyl frctionl integrl obeys the semigroup property. Tht is, ) ) ) xw α xw β f = xw α+β f = xw β xw α f. 2.4.32) Proof: We hve xw α xw β f x)= 1 dtt x) α 1 Γα) x 1 u t) β 1 f u)du. Γβ) By using the modified form of the Dirichlet formul 2.4.4), nmely dtt x) α 1 u t) β 1 f u)du = Bα,β) u t) α+β 1 f u)du, 2.4.33) x t nd letting, 2.4.33) yields the desired result: ) ) xw α xw β f = xw α+β f. 2.4.34) x t Nottion 2.4.6. W α x, I α + : Weyl integrl with lower limit. Definition 2.4.7. Another compnion to the opertor 2.4.27) is the following: Wx α f x)=i+ α f x)= 1 x x t) α 1 f t)dt, < x < 2.4.35) Γα) where Rα) >. Note 2.4.4: The opertor defined by 2.4.35) is useful in frctionl diffusion problems in strophysics nd relted res. Exmple 2.4.9. Prove tht W α x e x = ex α. 2.4.36)

2.4 Frctionl Integrls 15 Solution: We hve the result by setting x t = u. Note 2.4.5: An lterntive form of 2.4.35) in terms of convolution is given by Wx α f x)= 1 t+ α 1 f x t)dt 2.4.37) Γα) where Exmple 2.4.1. t α 1 + = { t α 1, t >, t < Prove tht xw ν cosx)= ν cos x + 12 ) πν 2.4.38) where >, < Rν) < 1. Solution: The result follows from the known integrl x u) ν 1 cosx dx = Γν) ν cos u + νπ 2 u ). 2.4.39) Exmple 2.4.11. Prove tht xw ν sinx)= ν sin x + 12 πν ). 2.4.4) Hint: Usetheintegrl x u) ν 1 sinx dx = Γν) ν sin u + 12 ) πν u 2.4.41) where >, < Rν) < 1. Exercises 2.4. 2.4.1. Prove tht Ix α x ) β 1) = Γβ) Γα + β) x )α+β 1, Rβ) >. 2.4.2. Prove tht I x α x ± c) γ 1) = where Rβ) >,γ C, c, x < 1. ± c)γ 1 Γα + 1) x )α 2F 1 1,1 γ;α + 1; x ± c ±c )

16 2 Mittg-Leffler Functions nd Frctionl Clculus 2.4.3. Prove tht ) Ix α [x ) β 1 b x) γ 1 ] = Γβ) x ) α+β 1 Γα + β) b ) 1 γ 2 F 1 β,1 γ;α + β; x ) b where Rβ) >,γ C, < x < b. 2.4.4. Prove tht [ ]) Ix α x ) β 1 b x) α+β = Γβ) x ) α+β 1 Γα + β) b ) α b x) β where Rβ) >, < x < b. 2.4.5. Prove tht [ Ix α x ) β 1 x ± c) γ 1]) = Γβ) x ) α+β 1 Γα + β) ± c) 1 γ 2 F 1 β,1 γ;α + β; ) x), ± c where Rβ) >, γ C, c, x ±c < 1. 2.4.6. Prove tht for Rβ) >, [ ]) Ix α x ) β 1 x ± c) α+β = Γβ) x ) α+β 1 Γα + β) ± c) α x ± c) β, x < 1. ± c 2.4.7. Prove tht ) Ix α [e λx ] = e λ x ) α E 1,α+1 λx λ). 2.4.8. Prove tht ) Ix α [e λx x ) β 1 ] = Γβ) Γα + β) eλ x ) α+β 1 1F 1 β;α + β;λx λ), where Rβ) >,Rα) >. 2.4.9. Prove tht [ Ix α x ) 2 ν J ν λ ]) x ) = where Rν) > 1. ) 2 α x ) α+ν 2 J α+ν λ x ), λ

2.4 Frctionl Integrls 17 2.4.1. Prove tht [ ] ) Ix ν x ) β 1 2F 1 µ,ν;β;λx )) where Rβ) >. = Γβ) Γν + β) x )ν+β 1 2F 1 µ,ν;ν + β;λx λ), 2.4.7 Lplce trnsform of the frctionl integrl We hve Ix ν f x)= 1 x x t) ν 1 f t)dt, 2.4.42) Γν) where Rν) >. Appliction of convolution theorem of the Lplce trnsform gives { } t L{ Ix ν ν 1 f x)};s = L L{ f t);s} Γν) = s ν Fs), 2.4.43) where Rs) >, Rν) >. 2.4.8 Lplce trnsform of the frctionl derivtive If n N, then by the theory of the Lplce trnsform, we know tht { } d n L dx n f ;s = s n n 1 Fs) k= s n k 1 f k) +) 2.4.44) = s n n 1 Fs) s k f n k 1) +), n 1 α < n) 2.4.45) k= where Rs) > nd Fs) is the Lplce trnsform of f t). By virtue of the definition of the derivtive, we find tht { } d L{ Dx α n f ;s} = L dx n Ix n α f ;s = s n L { I n α x f ;s } n 1 k= s k dn k 1 dx n k 1 I n α x f +)

18 2 Mittg-Leffler Functions nd Frctionl Clculus where Rs) >. = s α n 1 Fs) s k D α k 1 f +), D = d ) dx = s α Fs) k= n k=1 2.4.46) s k 1 D α k f +) 2.4.47) 2.4.9 Lplce trnsform of Cputo derivtive Nottion 2.4.7. C D α x Definition 2.4.8. The Cputo derivtive of csul function f t) tht is f t)= for t < ) with α > ws defined by Cputo 1969) in the form C D α x f x)= I n α x = d n 1 Γn α) dx n f x)= D n α) t f n) t) 2.4.48) x where n N. From 2.4.43) nd 2.4.49), it follows tht x t) n α 1 f n) t)dt,n 1 < α < n) 2.4.49) L{C D α t f t);s} = s n α) L{ f n) t)}. 2.4.5) On using 2.4.44), we see tht ] L{C Dt α f t);s} = s [s n α) n n 1 Fs) s n k 1 f k) +) k= = s α n 1 Fs) s α k 1 f k) +), n 1 < α n), 2.4.51) k= where Rs) > nd Rα) >. Note 2.4.6: From 2.4.48), it cn be seen tht C D α t A =, where A is constnt, wheres the Riemnn-Liouville derivtive D α t A = which is surprising result. At α, α 1,2, ), 2.4.52) Γ1 α)

2.5 Mellin Trnsform 19 Exercises 2.4. 2.4.11. Prove tht where Rν) >. I ν x f x)) = L 1 s ν L{ f x);s}, 2.4.53) 2.4.12. Prove tht the solution of Abel integrl eqution of the second kind α >, is given by φx) λ x φt)dt = f x), < x < 1 Γα) x t) 1 α x φx)= d E α [λx t) α ] f t) dt, 2.4.54) dx where E α x) is the Mittg-Leffler function defined by eqution 2.1.1). 2.4.13. Show tht λ x E α λt α ) Γα) x t) 1 α dt = E αλx α ) 1, α >. 2.4.55) 2.5 Mellin Trnsform of the Frctionl Integrls nd the Frctionl Derivtives 2.5.1 Mellin trnsform Nottion 2.5.1. Nottion 2.5.2. m{ f x); s}, f s): The Mellin trnsform m 1 { f s); x}: Inverse Mellin trnsform Definition 2.5.1. The Mellin trnsform of function f x), denoted by f s), is defined by f s)=m{ f x); s} = x s 1 f x)dx, x >. 2.5.1) The inverse Mellin trnsform is given by the contour integrl f x)=m 1 { f s); x} = 1 γ+i 2πi γ i f s)x s ds, i = 1 2.5.2) where γ is rel.

11 2 Mittg-Leffler Functions nd Frctionl Clculus 2.5.2 Mellin trnsform of the frctionl integrl Theorem 2.5.1. The following result holds true. m Ix α Γ1 α s) f )s)= Γ1 α) where Rα) > nd Rα + s) < 1. f s + α), 2.5.3) Proof 2.5.1: We hve z m Ix α f )s)= z s 1 1 z t) α 1 f t) dtdz Γα) = 1 f t) dt z s 1 z t) α 1 dz. 2.5.4) Γα) t On setting z = u t,thez-integrl becomes 1 t α+s 1 u α s 1 u) α 1 du = t α+s 1 Bα,1 α s), 2.5.5) where Rα) >, Rα + s) < 1. Putting the bove vlue of z-integrl, the result follows. Similrly we cn estblish Theorem 2.5.2. The following result holds true. where Rα) >, Rs) >. m x I α f )s)= Γs) Γs + α) m {tα f t); s} = Γs) Γs + α) f s + α), 2.5.6) Note 2.5.1: If we set f x)=x α φx), then using the property of the Mellin trnsform x α φx) φ s + α), 2.5.7) the results 2.5.3) nd 2.5.6) become I α x x α f x))s)= where Rα) >, Rα + s) < 1 nd Γ1 α s) Γ1 s) f s), 2.5.8) x I α x α f x))s)= Γs) Γs + α) f s), 2.5.9) where Rα) >, nd Rs) >, respectively.

2.6 Kober Opertors 111 2.5.3 Mellin trnsform of the frctionl derivtive Theorem 2.5.3. If n N nd lim t t s 1 f ν) t)=, ν =,1,,n, then m{ f n) t); s)} = 1) n Γs) m{ f t); s n}, 2.5.1) Γs n) where Rs) >,Rs n) >. Proof 2.5.2: Integrte by prts nd using the definition of the Mellin trnsform, the result follows. Exmple 2.5.1. Find the Mellin trnsform of the frctionl derivtive. Solution 2.5.1: Therefore, We hve D α x m D α x f )s)= 1)n Γs) Γs n) f = D n x D α n x = 1)n Γs)Γ1 s α)) Γs n)γ1 s + n) where Rs) >,Rs) < 1 + Rα). f = D n x I n α x f. m { I n α x f } s n),n 1 Rα) < n) 2.5.11) m{ f t);s α}, 2.5.12) Remrk 2.5.1: An lterntive form of 2.5.12) is given in Exercise 2.5.2. 2.5.1. Prove Theorem 2.5.2. Exercises 2.5. 2.5.2. Prove tht the Mellin trnsform of frctionl derivtive is given by m Dx α f )s)= 1)n Γs)sin[πs n)] m{ f t);s α}, 2.5.13) Γs α)sin[πs α)] where Rs) >,Rα s) > 1. 2.5.3. Find the Mellin trnsform of 1 + x ) b ;,b >. 2.6 Kober Opertors Kober opertors re the generliztion of Riemnn-Liouville nd Weyl opertors. These opertors hve been used by mny uthors in deriving the solution of single, dul nd triple integrl equtions possessing specil functions of mthemticl physics, s their kernels.

112 2 Mittg-Leffler Functions nd Frctionl Clculus Nottion 2.6.1. Kober opertor of the first kind I[ f x)], I[α,η : f x)], Iα,η) f x),e α,η,x f, I n,α x f. Nottion 2.6.2. Kober opertor of the second kind R[ f x)], R[α,ζ : f x)], Rα,ζ ) f x),k α,ζ x, f,k ζ,α x f. Definition 2.6.1. where Rα) >. Definition 2.6.2. where Rα) >. I[ f x)] = I[α,η : f x)] = Iα,η) f x)=e α,η = Ix η,α f = x η α Γα) x,x f R[ f x)] = R[α,ζ : f x)] = Rα,ζ ) f x)=k α,ζ = K ζ,α x f = xζ Γα) x x t) α 1 t η f t)dt, 2.6.1) x, f 2.6.1) nd 2.6.2) hold true under the following conditions: t x) α 1 t ζ α f t)dt, 2.6.2) f L p,),rα) >,Rη) > 1 q,rζ ) > 1 p, 1 p + 1 = 1, p 1. q When η =, 2.6.1) reduces to Riemnn-Liouville opertor. Tht is, I,α x f = x α I α x f. 2.6.3) For ζ =, 2.6.2) yields the Weyl opertor of t α f t). Tht is, Theorem 2.6.1. [Kober 194)]. K,α x f = x W α t α f t). 2.6.4) If Rα) >,Rη s) > 1, f L p o,),1 p 2 or f M p o,), subspce of L p o,) nd p > 2 ), Rη) > 1 q, 1 p + 1 q = 1, then there holds the formul Γ1 + η s) m{iα,η) f }s)= m{ f x);s}. 2.6.5) Γα + η + 1 s)

2.6 Kober Opertors 113 Proof 2.6.1: It is similr to the proof of Theorem 2.6.1. In similr mnner, we cn estblish Theorem 2.6.2. [Kober 194)]. If Rα) >,Rs + ζ ) >, f L p o,),1 p 2 or f M p o,), subspce of L p o,) nd p > 2 ) then, Rζ ) > 1 p, 1 p + 1 q = 1, m{rα,ζ ) f }s)= Γζ + s) m{ f x);s}. 2.6.6) Γα + ζ + s) Semigroup property of the Kober opertors hs been given in the form of Theorem 2.6.3. If f L p o,),g L q o,), 1 p + 1 q = 1,Rη) > 1 q,rζ ) > 1 p,1 p 2, or f M po,), subspce of L p o,) nd p > 2 ), then gx)iα,η : f ))x)dx = f x)rα,η : g))x)dx. 2.6.7) Proof 2.6.2: Interchnge the order of integrtion. Remrk 2.6.1: Kober opertors. Opertors defined by 2.6.1.) nd 2.6.2) re lso clled Erdélyi- Exercises 2.6. 2.6.1. Prove Theorem 2.6.1. 2.6.2. For the modified Erdélyi-Kober opertors, defined by the following equtions for m > : nd Iα,η : m) f x)=i f x) : α,η,m) = m x Γα) x η mα+m 1 t η x m t m ) α 1 f t)dt, 2.6.8) o Rα,ζ : m) f x)=r f x) : α,ζ,m) = mxζ Γα) x t ζ mα+m 1 t m x m ) α 1 f t)dt, 2.6.9)

114 2 Mittg-Leffler Functions nd Frctionl Clculus where f L p,),rα) >,Rη) > 1 q,rζ ) > 1 p, 1 p + 1 q = 1, find the Mellin trnsforms of i) Iα, η : m) f x) nd ii) Rα, ζ : m) f x), giving the conditions of vlidity. 2.6.3. For the opertors defined by 2.6.8) nd 2.6.9.), show tht R f x) : α,η,m)gx)dx = f x)igx) : α, η, m)dx, 2.6.1) where the prmeters α,η,m re the sme in both the opertors I nd R. Give conditions of vlidity of 2.6.1). 2.6.4. For the Erdélyi-Kober opertor, defined by I η,α f x)= 2x 2α 2η Γα) x x 2 t 2 ) α 1 t 2η+1 f t)dt, 2.6.11) where Rα) >, estblish the following results Sneddon 1975)): i) I η,α x 2β f x) =x 2β I η+β,α f x) 2.6.12) ii) I η,α I η+α,β = I η,α+β = I η+α,β, I η,α 2.6.13) iii) I 1 η,α = I η+α, α. 2.6.14) Remrk 2.6.2: The results of Exercise 2.6.4 lso hold for the opertor, defined by where Rα) >. K η,α f x)= 2x2η t 2 x 2 ) α 1 t 2α 2η+1 f t)dt, 2.6.15) Γα) x Remrk 2.6.3: Opertors more generl thn the opertors defined by 2.6.11) nd 2.6.15) re recently defined by Glué et l [Integrl Trnsform & Spec. Funct. Vol. 9 2), No. 3, pp. 185-196] in the form Ix η,α f x)= x η α x x t) α 1 t η f t)dt, 2.6.16) Γα) where Rα) >. 2.7 Generlized Kober Opertors Nottion 2.7.1. Nottion 2.7.2. I[α,β,γ : m, µ,η, : f x)],i[ f x)] I[α,β,γ : m, µ,δ, : f x)],i[ f x)]

2.7 Generlized Kober Opertors 115 [ ] α,β,γ; Nottion 2.7.3. R[ f x)], R σ,ρ,; : f x) Nottion 2.7.4. [ ] α,β,γ; K[ f x)], K δ,ρ,; : f x) Nottion 2.7.5. Nottion 2.7.6. I α,β,η;,x f x) Sigo, 1978) J α,β,η; x,α f x) Sigo, 1978) Definition 2.7.1. I[ f x)] = I[α,β,γ : m, µ,η, : f x)] = µx η 1 Γ1 α) x where 2 F 1 ) is the Guss hypergeometric function. Definition 2.7.2. I[ f x)] = I[α,β,γ : m, µ,δ, : f x)] = µxδ Γ1 α) x 2F 1 α,β + m,γ; t µ ) t η f t)dt, 2.7.1) x µ ) 2F 1 α,β + m;γ; xµ t µ t δ 1 f t)dt. 2.7.2) Opertors defined by 2.7.1) nd 2.7.2) exist under the following conditions: i) 1 p, q <, p 1 + q 1 = 1, rg1 ) < π ii) R1 α) > m,rη) > 1 q,rδ) > 1 p,rγ α β m) > 1,m N ; γ, 1, 2, iii) f L p,) Equtions 2.7.1) nd 2.7.2) re introduced by Kll nd Sxen 1969). For γ = β, 2.7.1) nd 2.7.2) reduce to generlized Kober opertors, given by Sxen 1967). Definition 2.7.3. [ ] α,β,γ; R[ f x)] = R σ,ρ,; f x) = x σ ρ Γρ) x Definition 2.7.4. [ ] α,β,γ; K[ f x)] = K δ,ρ,; f x) = xδ Γρ) x t σ x t) ρ 1 2F 1 [α,β;γ; 1 t )] f t)dt. 2.7.3) x t δ ρ t x) ρ 1 2F 1 [α,β;γ; 1 x )] f t)dt. 2.7.4) t

116 2 Mittg-Leffler Functions nd Frctionl Clculus The conditions of vlidity of the opertors 2.7.3) nd 2.7.4) re given below: i) p 1, q <, p 1 + q 1 = 1, rg1 ) < π. ii) Rσ) > 1 q,rδ) > 1 p,rρ) >. iii) γ, 1, 2, ;Rγ α β) >. iv) f L p,). The opertors defined by 2.7.3) nd 2.7.4) re given by Sxen nd Kumbht 1973). When is replced by α nd α tends to infinity, the opertors defined by 2.7.3) nd 2.7.4) reduce to the following opertors ssocited with confluent hypergeometric functions. Definition 2.7.5. [ ] [ ] β,γ; R σ,ρ,; f x) = lim R α,β,γ; α σ,ρ, α ; f x) = x σ ρ Γρ) x Definition 2.7.6. [ ] [ ] β,γ; K σ,ρ,; f x) = lim K α,β,γ; α δ,ρ, α ; f x) = xδ Γρ) x where Rρ) >,Rδ) >. [ Φ β,γ; 1 t )] t σ x t) ρ 1 f t)dt. 2.7.5) x [ Φ β,γ; 1 x )] t δ ρ t x) ρ 1 f t)dt, 2.7.6) t Remrk 2.7.1: Mny interesting nd useful properties of the opertors defined by 2.7.3) nd 2.7.4) re investigted by Sxen nd Kumbht 1975), which del with reltions of these opertors with well-known integrl trnsforms, such s Lplce, Mellin nd Hnkel trnsforms. Eqution 2.7.3) ws first considered by Love 1967). Remrk 2.7.2: In the specil cse, when α is replced by α + β,γ by α,σ by zero, ρ by α nd β by η, then 2.7.3) reduces to the opertor 2.7.7) considered by Sigo 1978). Similrly, 2.7.4) reduces to nother opertor 2.7.9) introduced by Sigo 1978). Definition 2.7.7. Let α,β,η C, nd let x R + the frctionl integrl Rα) > ) nd the frctionl derivtive Rα) < ) of the first kind of function f x) on R + re defined by Sigo 1978) in the form I α,β,η,x f x)= x α β x x t) α 1 Γα) 2 F 1 α + β, η;α;1 t ) f t)dt, Rα) > 2.7.7) x = dn dx n Iα+n,β n,η n,x f x), < Rα)+n 1, n N ). 2.7.8)

2.7 Generlized Kober Opertors 117 Definition 2.7.8. The frctionl integrl Rα) > ) nd frctionl derivtive Rα) < ) of the second kind of function f x) on R + re given by Sigo 1978) in the form Jx, α,β,η f x)= 1 t x) α 1 t α β Γα) x 2 F 1 α + β, η;α;1 x ) f t)dt, Rα) > 2.7.9) t = 1) n dn dx n Jα+n,β n,η x, f x), < Rα)+n 1, n N ). 2.7.1) Exmple 2.7.1. Find the vlue of I α,β,η {,x x σ 1 2 F 1,b;c; x) }. Solution 2.7.1: We hve K = I α,β,η { x σ 1 2 F 1,b;c; x) } =,x r= ) r b) r 1) r ) r c) r r! I α,β,η,x x r+σ 1. Applying the result of Exercise 2.7.1, we obtin K = x σ β 1 1) r ) rb) r Γσ + r)γσ β + η + r) ) r x r r= c) r r! Γσ β + r)γα + η + σ + r) = x σ β 1 Γσ)Γσ + η β) Γσ β)γσ + α + η) 4 F 3,b,σ,σ + η β;c,σ β,σ + α + η; x), where Rα) >,Rσ) >,Rσ + η β) >,c, 1, 2, ; x < 1. Exmple 2.7.2. Find the vlue of J α,β,η x, x λ 2F 1,b;c; )). x Solution 2.7.2: 2.7.3, it gives J α,β,η x, Following similr procedure nd using the result of Exercise x λ 2F 1,b;c; )) = x Γβ λ)γη λ) Γ λ)γα + β + η λ) xλ β ) 4 F 3,b,β λ,η λ;c, λ,α + β + η λ;, x where Rα) >,Rβ λ) >,Rη λ) >,x >,c, 1, 2, ; x >.

118 2 Mittg-Leffler Functions nd Frctionl Clculus Remrk 2.7.3: Specil cses of the opertors I α,β,η,x nd Jx, α,β,η re the opertors of Riemnn -Liouville: I α, α,η,x f x)= Dx α f x)= 1 x x t) α 1 f t)dt, Rα) > ) 2.7.11) Γα) the Weyl: Jx, α, α,η f x)= x W α f x)= 1 t x) α 1 f t)dt, Rα) > ) 2.7.12) Γα) x nd the Erdélyi-Kober opertors: I α,,η,x f x)=e α,η,x f x)= x α η x x t) α 1 t η f t)dt, Rα) > ) 2.7.13) Γα) nd J α,,η x, f x)=kx, α,η f x)= xη t x) α 1 t α η f t)dt, Rα) > ) Γα) x 2.7.14) Exmple 2.7.3. Prove the following theorem. If Rα) > nd Rs) < 1 + min[, Rη β)], then the following formul holds for f x) L p,) with 1 p 2or f x) M p,) with p > 2: { } m x β I α,β,η Γ1 s)γη β + 1 s),x f = m{ f x)}. 2.7.15) Γ1 s β)γα + η + 1 s) Solution 2.7.3: Use the integrl u σ γ u x) γ 1 2F 1 α,β;γ;1 x Γγ)Γσ)Γγ + σ α β) )du = x u Γγ + σ α)γγ + σ β), 2.7.16) where Rγ) >, Rσ) >, Rγ + σ α β) >. Exercises 2.7. 2.7.1. Prove tht I α,β,η,x x λ Γ1 + λ)γ1 + λ + η β) = Γ1 + λ β)γ1 + λ + α + η) xλ β, 2.7.17) nd give the conditions of vlidity.

2.7 Generlized Kober Opertors 119 2.7.2. Find the Mellin trnsform of x β J α,β,η x, f x), giving conditions of its vlidity. 2.7.3. Prove tht Jx, α,β,η x λ Γβ λ)γη λ) = Γ λ)γα + β + η λ) xλ β 2.7.18) nd give the conditions of vlidity. 2.7.4. Prove tht I α,β,η,x x k e λx Γk + 1)Γη + k β + 1) )= Γk β + 1)Γα + η + k + 1) xk β nd give the conditions of vlidity. 2.7.5. Prove tht 2 F 2 k + 1,η + k β + 1; k β + 1,α + η + k + 1; λx), 2.7.19) Jx, α,β,η e sx = s η η β Γβ η) x Φ1 α β,1 + η β; sx) Γα + β) β Γη β) + s Φ1 α η,1 + β η; sx), 2.7.2) Γα + η) nd give the conditions of its vlidity. Deduce the results for L[ x W α L[ Kx, α,η f ]s). f ]s) nd 2.7.6. Prove tht [Sxen nd Nishimoto 22)] I α,β,η,x [x σ 1 + bx) c ]= c Γσ)Γσ + η β) Γσ β)γσ + α + η) xσ β 1 3 F 2 σ,σ + η β, c; σ β,σ + α + η; bx ), 2.7.21) where Rσ) > mx[,rβ η)], bx < 1. 2.7.7. Evlute { [ ]} I α,β,η,x x σ 1 Hp,q m,n x λ p,a p ) b q,b q, λ >, 2.7.22) ) nd give the conditions of its vlidity. 2.7.8. Evlute J α,β,η x, { [ ]} x σ 1 Hp,q m,n x λ p,a p ) b q,b q, λ >, 2.7.23) ) nd give the conditions of its vlidity.

12 2 Mittg-Leffler Functions nd Frctionl Clculus 2.7.9. Estblish the following property of Sigo opertors clled Integrtion by prts. ) ) f x) I α,β,η,x g x)dx = gx) Jx, α,β,η f x)dx. 2.7.1. From Exercise 2.7.6, deduce the formul for given by B. Ross 1993). 2.7.11. Prove tht where Rα) >, Rk) > 1, 2.7.12. Prove tht I α x x k = I α, α,η,x + bx) c, 2.7.24) W α x,x k = where Rα) >, Rk) < Rα). 2.7.13. Show tht Γk + 1) Γα + k + 1) xk+α, 2.7.25) Γ α k) x k+α, 2.7.26) Γ k) [ ] Jx, α,β,η x λ e px )=x λ β G 3, 2,3 px λ,α+β+η λ,β λ,η λ, 2.7.27) where G 3, 2,3 ) is the Meijer s G-function, Rpx) >, Rα) >. Hint: Use the integrl e px = 1 Γ s)px) s ds. 2.7.28) 2πi L 2.7.14. Evlute [ ] I α,β,η,x x σ 1 Hp,q m,n x λ p,a p ) b q,b q ), λ >, 2.7.29) giving the conditions of its vlidity. 2.7.15. Evlute [ ] Jx, α,β,η x σ 1 Hp,q m,n x λ p,a p ) b q,b q ), λ > 2.7.3) nd give the conditions of vlidity of the result. 2.7.16. With the help of the following chin rules for Sigo opertors Sigo, 1985)

2.8 Riemnn-Liouville Frctionl Clculus 121 I α,β,η,x I γ,δ,α+η,x f = I α+γ,β+δ,η,x f, 2.7.31) nd J α,β,η x, Jx, γ,δ,α+η f = J α+γ,β+δ,η x, f, 2.7.32) derive the inverses I α,β,η,x ) 1 = I α, β,α+η,x. 2.7.33) nd Jx, α,β,η ) 1 = Jx, α, β,α+η. 2.7.34) 2.8 Compositions of Riemnn-Liouville Frctionl Clculus Opertors nd Generlized Mittg-Leffler Functions In this section, composition reltions between Riemnn-Liouville frctionl clculus opertors nd generlized Mittg-Leffler functions re derived. These reltions my be useful in the solution of frctionl differintegrl equtions. For detils, one cn refer to the work of Sxen nd Sigo 25). For redy reference some of the definitions re repeted here. 2.8.1 Composition Reltions Between R-L Opertors nd E β, γ δ z) Nottion 2.8.1. Nottion 2.8.2. Nottion 2.8.3. Nottion 2.8.4. Nottion 2.8.5. Nottion 2.8.6. E α x) : Mittg-Leffler function. E α,β x) : Generlized Mittg-Leffler function. I+ α f : Riemnn-Liouville left-sided integrl. I α f : Riemnn-Liouville right-sided integrl. D+ α f : Riemnn-Liouville left-sided derivtive. D α f : Riemnn-Liouville right-sided derivtive. Nottion 2.8.7. Eβ,γ δ z) : Generlized Mittg-Leffler function Prbhkr, 1971).

122 2 Mittg-Leffler Functions nd Frctionl Clculus Definition 2.8.1. Definition 2.8.2. E α,β z)= E α z)= k= k= z k z k, α C,Rα) > ). 2.8.1) Γαk + 1), α,β C,Rα) >,Rβ) > ). 2.8.2) Γαk + β) Definition 2.8.3. I+ α f )x)= 1 x f t) dt, Rα) >. 2.8.3) Γα) x t) 1 α Definition 2.8.4. I α f )x)= 1 f t) dt, Rα) >. 2.8.4) Γα) x t x) 1 α Definition 2.8.5. D α + f )x)= d dx = Definition 2.8.6. ) [α]+1 ) I 1 {α} + x); Rα) > 2.8.5) ) 1 d [α]+1 x f t) dt, Rα) >. 2.8.6) Γ1 {α}) dx x t) {α} ) d [α]+1 D α f )x)= I 1 {α} f )x), Rα) > 2.8.7) dx = 1 Γ1 {α}) d ) [α]+1 dx x f t) dt, Rα) >. 2.8.8) t x) {α} Remrk 2.8.1: Here [α] mens the mximl integer not exceeding α nd {α} is the frctionl prt of α. Note tht Γ1 {α}) =Γm α),[α]+1 = m,{α} = 1 + α m. Definition 2.8.7. Eβ,γ δ z)= δ) k z k, β,γ,δ C;Rγ) >,Rβ) > ). 2.8.9) k= Γβk + γ)k! For δ = 1, 2.8.9) reduces to 2.8.2).

2.8 Riemnn-Liouville Frctionl Clculus 123 Theorem 2.8.1. Let α >, β >, γ > nd α R. Let I+ α be the left-sided opertor of Riemnn-Liouville frctionl integrl 2.8.3). Then there holds the formul I+ α [tγ 1 Eβ,γ δ tβ )])x)=x α+γ 1 Eβ,α+γ δ xβ ). 2.8.1) Proof 2.8.1: By virtue of 2.8.3) nd 2.8.9), we hve K I+ α [tγ 1 Eβ,γ δ tβ )])x)= 1 x x t) α 1 δ) n Γα) n t nβ+γ 1 n= Γβn + γ)n! dt. Interchnging the order of integrtion nd summtion nd evluting the inner integrl by mens of bet-function formul, it gives K x α+γ 1 δ) n x β ) n Γα + βn + γ)n)! = xα+γ 1 Eβ,α+γ δ xβ ). n= This completes the proof of Theorem 2.8.1. Corollry 2.8.1: For α >, β >, γ > nd α R, there holds the formul I α + [tγ 1 E β,γ t β )])x)=x α+γ 1 E β,α+γ x β ). 2.8.11) Remrk 2.8.2: For β = α, 2.8.11) reduces to I α + [tγ 1 E α,γ t α )])x)= xγ 1 by virtue of the identity [ E α,γ x α ) 1 ], ) 2.8.12) Γγ) E α,γ x)= 1 Γγ) + xe α,α+γx), ). 2.8.13) Theorem 2.8.2. Let α >,β >,γ > nd α R, ) nd let I+ α be the leftsided opertor of Riemnn-Liouville frctionl integrl 2.8.3). Then there holds the formul I+ α [tγ 1 Eβ,γ δ tβ )])x)= 1 xα+γ β 1 [Eβ,α+γ β δ xβ ) E δ 1 β,α+γ β xβ )]. 2.8.14) Proof. Use Theorem 2.8.1. The following two theorems cn be estblished in the sme wy.

124 2 Mittg-Leffler Functions nd Frctionl Clculus Theorem 2.8.3. Let α >,β >,γ > nd α R nd let I α be the right-sided opertor of Riemnn-Liouville frctionl integrl 2.8.4). Then we rrive t the following result: I α [t α γ E δ β,γ t β )])x)=x γ [E δ β,α+γ x β )] 2.8.15) Corollry 2.8.2: For α >, β >, γ > nd α R, there holds the formuls: I α [t α γ E β,γ t β )])x)=x γ [E β,α+γ x β )] 2.8.16) nd I α t α 1 E β t β ))x)=x 1 [E β,α+1 x β )]. 2.8.17) Theorem 2.8.4. Let α >,β >,γ >,α R, ),α + γ > β nd let I α be the right-sided opertor of Riemnn-Liouville frctionl integrl 2.8.4). Then there holds the formul I [t α α γ Eβ,γ δ t β )])x)= 1 xβ γ [Eβ,α+γ β δ x β ) E δ 1 β,α+γ β x β )]. 2.8.18) Corollry 2.8.3: For α >,β >,γ > with α +γ > β nd for α R, ), there holds the formul I [t α α γ E β,γ t β )])x)= 1 ] [E xβ γ β,α+γ β x β 1 ). Γα + γ β) 2.8.19) Remrk 2.8.3: Kilbs nd Sigo, 1998) ) I [t α α γ E α,γ t α )])x)= xα γ I [t α α 1 E α t α )])x)= xα 1 [ E α,γ x α ) 1 Γγ) ], ) 2.8.2) [ Eα x α ) 1 ], ). 2.8.21) Theorem 2.8.5. Let α >,β >,γ >,γ > α,α R nd let D+ α be the leftsided opertor of Riemnn -Liouville frctionl derivtive 2.8.6). Then there holds the formul. D+ α [tγ 1 Eβ,γ δ tβ )])x)=x γ α 1 Eβ,γ α δ xβ ). 2.8.22) Proof 2.8.2: By virtue of 2.8.9) nd 2.8.6), we hve

2.8 Riemnn-Liouville Frctionl Clculus 125 ) d [α]+1 K D+ α [tγ 1 Eβ,γ δ tβ )])x)= dx = = = n= n= n= which proves the theorem. I 1 {α} + [ t γ 1 E δ β,γ tβ )] ) x) n ) δ) n d [α]+1 x t nβ+γ 1 x t) {α} dt Γγ + nβ)γ1 {})n! dx n ) δ) n d [α]+1 x nβ+γ {α} Γγ + nβ + 1 {α})n! dx n δ) n x γ+nβ α 1 Γnβ + γ α)n! = xγ α 1 E δ β,γ α xβ ), By using similr procedure, we rrive t the following theorem. Theorem 2.8.6. Let α >,γ > β >,α R, ), γ > α + β nd let D+ α be the left-sided opertor of Riemnn-Liouville frctionl derivtive 2.8.6). Then there holds the formul ) D+ α [tγ 1 Eβ,γ δ tβ )] x)= 1 [ ] xγ α β 1 Eβ,γ α β δ xβ ) E δ 1 β,γ α β xβ ). 2.8.23) Corollry 2.8.4: Let α >,γ > β >,α R, ), γ > α + β, then there holds the formul. D+ α [tγ 1 E β,γ t β )] )x)= 1 [ ] xγ α β 1 E β,γ α β x β 1 ). Γγ α β) 2.8.24) Theorem 2.8.7. Let α >,γ >,γ α > with γ α + {α} > 1,α R, nd let D α be the right-sided opertor of Riemnn-Liouville frctionl derivtive 2.8.8). Then there holds the formul. ) D [t α α γ Eβ,γ δ t β )] x)=x γ Eβ,γ α δ x β ). 2.8.25) Theorem 2.8.8. Let α >,β > with γ {α} > 1, α R, γ > α + β, ) nd let D α be the right-sided opertor of Riemnn-Liouville frctionl derivtive 2.8.8). Then there holds the formul D [t α α γ Eβ,γ δ t β )] )x)= xβ γ [ ] Eβ,γ α β δ x β ) E δ 1 β,γ α β x β ). 2.8.26)

126 2 Mittg-Leffler Functions nd Frctionl Clculus Exercises 2.8. 2.8.1. Show tht x β E δ β,γ xβ )=E δ β,γ β xβ ) E δ 1 β,γ β xβ ), ) 2.8.27) 2.8.2. Show tht I α + [t γ 1 E α,γ t α )] ) x)= xγ 1 2.8.3. Prove Theorem 2.8.3. 2.8.4. Prove Theorem 2.8.4. 2.8.5. Prove Theorem 2.8.6. 2.8.6. Prove Theorem 2.8.7. 2.8.7. Prove Theorem 2.8.8. 2.8.8. Prove tht [ I+ α tω Hp,q m,n t σ p,a p ) bq,bq) ]) giving conditions of vlidity. 2.8.9. Evlute I t α ω Hp,q m,n nd give the conditions of vlidity. [ E α,γ x α ) 1 ], ). 2.8.28) Γγ) [ ] x)=x ω+α H m,n+1 p+1,q+1 x σ ω,σ), p,a p ), 2.8.29) bq,bq), ω α,σ) [ ]) t σ p,a p ) bq,bq) x), 2.8.3) 2.9 Frctionl Differentil Equtions Differentil equtions contin integer order derivtives, wheres frctionl differentil equtions involve frctionl derivtives, like dx dα α, which re defined for α >. Here α is not necessrily n integer nd cn be rtionl, irrtionl or even complexvlued. Tody, frctionl clculus models find pplictions in physicl, biologicl, engineering, biomedicl nd erth sciences. Most of the problems discussed involve relxtion nd diffusion models in the so clled complex or disordered systems. Thus, it gives rise to the generliztion of initil vlue problems involving ordinry differentil equtions to generlized frctionl-order differentil equtions nd Cuchy problems involving prtil differentil equtions to frctionl rection, frctionl diffusion nd frctionl rection-diffusion equtions. Frctionl clculus plys dominnt role in the solution of ll these physicl problems.

2.9 Frctionl Differentil Equtions 127 2.9.1 Frctionl relxtion In order to formulte relxtion process, we require physicl lw, sy the relxtion eqution d dt f t)+1 f t)=,t >,c >, 2.9.1) c to be solved for the initil vlue f t = ) = f. The unique solution of 2.9.1) is given by f t)= f e c t,t,c >. 2.9.2) Now the problem is s to how we cn generlize the initil-vlue problem 2.9.1) into frctionl vlue problem with physicl motivtion. If we incorporte the initil vlue f into the integrted relxtion eqution 2.9.1), we find tht where D 1 1 c α Dt α f t) f = 1 c D 1 t f t), 2.9.3) t is the stndrd Riemnn integrl of f t). On replcing 1 c Dt 1 f t), it yields the frctionl integrl eqution f t) by ) 1 f t) f = c α Dt α f t),α > 2.9.4) with initil vlue f = f t = ). Applying the Riemnn-Liouville differentil opertor Dt α use of the formul 2.4.16), we rrive t from the left nd mking with initil condition f = f t = ). D α t [ f x) f ]= c α f t), α >,c >, 2.9.5) Theorem 2.9.1. The solution of the frctionl differentil eqution 2.9.4) is given by [ f t)= f H 1,1 t ) α ],1) 1,2, 2.9.6) c,1),,α) where α >,c >. Proof 2.9.1: If we pply the Lplce trnsform to eqution 2.9.4), it gives Fs) f s 1 = 1 c α s α Fs), 2.9.7) where we hve used the result 2.4.7) nd Fs) is the Lplce trnsform of f t). Solving for Fs),wehve

128 2 Mittg-Leffler Functions nd Frctionl Clculus [ s 1 ] Fs)=L{ f t)} = f 1 +cs) α. 2.9.8) Tking inverse Lplce trnsform, 2.9.8) gives f t)=l 1 {Fs)} = f L 1 [ s 1 1 +cs) α [ ] = f L 1 1) k c αk s αk 1 k= ] 1) = f k c t )αk k= Γαk + 1) t α ] = f E α [, 2.9.9) c) where E α ) is the Mittg-Leffler function. 2.9.9) cn be written in terms of the H-function s [ f t)= f H 1,1 t ) α ],1) 1,2, 2.9.1) c,1),,α) where c >,α >. This completes the proof of the Theorem 2.9.1. Alterntive form of the solution. By virtue of the identity [ Hp,q m,n x µ p,a p ) b q,b q ) ] = 1 µ Hm,n p,q the solution 2.9.1) cn be written s f t)= f [ t α H1,1 1,2 c where α >,c >. [x p, Ap µ ) b q, Bq µ ) ], 1 α ), 1 α ),,1) ], µ > ) 2.9.11), 2.9.12) Remrk 2.9.1: In the limit s α 1, one recovers the result 2.9.2) f t)= f exp t ) t ) = f E 1. 2.9.13) c c Remrk 2.9.2: In terms of Wright s function, the solution 2.9.1) cn be expressed in the form [ 1,1) f t)= f 1 ψ 1 ; t c )α], 2.9.14) 1,α) where α >,c >. In similr mnner, we cn estblish Theorems 2.9.2 nd 2.9.3 given below.

2.9 Frctionl Differentil Equtions 129 Theorem 2.9.2. The solution of the frctionl integrl eqution Nt) N t µ 1 = c ν D ν t Nt), 2.9.15) is given by Nt)=N Γµ)t µ 1 E ν,µ c ν t µ ), 2.9.16) where E ν,µ ) is the generlized Mittg-Leffler function 2.1.2), ν >, µ >. Remrk 2.9.3: 2). When µ = 1, we obtin the result given by Hubold nd Mthi Theorem 2.9.3. If c >,ν >, µ >, then for the solution of the integrl eqution Nt) N t µ 1 Eν,µ[ ct) γ ν ]= c ν Dt ν Nt), 2.9.17) there holds the formul Hint: Use the formul Nt)=N t µ 1 E γ+1 ν,µ [ ct) ν ]. 2.9.18) L 1 { s β 1 s α ) γ} = t β 1 E γ α,β tα ), 2.9.19) where Rα) >,Rβ) >,Rs) > Rα),Rs) >. Corollry 2.9.1: 1 Ifc>, µ >, ν >, then for the solution of Nt) N t µ 1 E ν,µ [ c ν t ν ]= c ν Dt ν Nt), 2.9.2) there holds the reltion Nt)= N ν tµ 1 [ E ν,µ 1 c ν t ν )+1 + ν µ)e ν,µ c ν t ν ) ]. 2.9.21) Theorem 2.9.4. The Cuchy problem for the integro-differentil eqution D x µ f x)+λ Dx ν f x)=hx), λ, µ,ν C) 2.9.22) with the initil condition D x µ k 1 f )= k,k =,1,,[µ], 2.9.23) where Rν) >,Rµ) > nd hx) is ny integrble function on the finite intervl [,b] hs the unique solution, given by f x)= x x t) µ 1 E µ+ν,µ [ λx t) µ+ν ]ht)dt n 1 k= + k x µ k 1 E µ+ν,µ k λx µ+ν ) 2.9.24)

13 2 Mittg-Leffler Functions nd Frctionl Clculus Proof 2.9.2: Exercise. Theorem 2.9.5. The solution of the eqution [ D 1 2 t f t)+bft)=; where C is constnt is given by f t)=ct 1 2 E 1 2, 1 2 where E 1 2, 1 2 ) is the Mittg-Leffler function. D 1 2 t ] f t) = C, 2.9.25) t= ) bt 1 2, 2.9.26) Proof 2.9.3: Exercise see 2.4.47). Remrk 2.9.4: Theorem 2.9.5 gives the generlized form of the eqution solved by Oldhm nd Spnier 1974). Exercises 2.9. 2.9.1. Prove tht if c >,ν >, µ >, then the solution of Nt) N t µ 1 E 2 ν,µc ν t ν )= c ν D ν t Nt), 2.9.27) is given by Nt)=N t µ 1 Eν,µ c 3 ν t ν )= N t µ 1 [ 2ν 2 E ν,µ 2 c ν t ν ) + {3ν + 1) 2µ}E ν,µ 1 c ν t ν ) + { 2ν 2 + µ 2 + 3ν 2µ 3νµ+ 1 } ] E ν,µ c ν t ν ), 2.9.28) where Rν) >, Rµ) > 2. 2.9.2. Prove tht if ν >,c >,d >, µ >,c d, then for the solution of the eqution Nt) N t µ 1 E ν,µ d ν t ν )= c ν Dt ν Nt), 2.9.29) there holds the formul. t µ ν 1 [ Nt)=N Eν,µ ν c ν d ν d ν t ν ) E ν,µ ν c ν t ν ) ]. 2.9.3) 2.9.3. Prove tht if c >,ν >, µ >, then for the solution of the eqution

2.9 Frctionl Differentil Equtions 131 the following result holds: Nt) N t µ 1 E ν,µ c ν t ν )= c ν D ν t Nt), 2.9.31) Nt)= N [ ν tµ 1 E ν,µ 1 c ν t ν )+1 + ν µ)e ν,µ c ν t ν ) ]. 2.9.32) 2.9.4. Solve the eqution D Q t f t)+ D q t f t)=gt), where q Q is not n integer or hlf integer nd the initil condition is [ ] f t)+ Dt Q 1 f t) = C 2.9.33) where C is constnt. 2.9.5. Solve the eqution D q 1 t t= D α t xt) λxt)=ht), t > ), 2.9.34) subject to the initil conditions [ ] Dt α k ht) = b k, k = 1,,n) 2.9.35) t= where n 1 < α < n. 2.9.6. Prove Theorem 2.9.4. 2.9.7. Prove Theorem 2.9.5. 2.9.2 Frctionl diffusion Theorem 2.9.6. The solution of the following initil vlue problem for the frctionl diffusion eqution in one dimension D α t Ux,t)=λ 2 2 Ux,t) x 2, t >, < x < ) 2.9.36) with initil conditions : lim Ux,t)=;[ D α 1 x ± t Ux,t) ] = φx) 2.9.37) t= is given by

132 2 Mittg-Leffler Functions nd Frctionl Clculus Ux,t)= Gx ζ,t)φζ )dζ, 2.9.38) where Gx,t)= 1 t α 1 E α,α k 2 λ 2 t α )coskx dk. 2.9.39) π Solution 2.9.1: Let < α < 1. Using the boundry conditions 2.9.37), the Fourier trnsform of 2.9.36) with respect to vrible x gives Dx α Ūk,t)+λ 2 k 2 Ūk,t)= 2.9.4) [D t α 1 Ūk,t) ] t= = φk), 2.9.41) where k is Fourier trnsform prmeter nd indictes Fourier trnsform. Applying the Lplce trnsform to 2.9.4) nd using 2.9.41), it gives φk) Uk,s)= s α + k 2 λ 2, 2.9.42) where indictes Lplce trnsform. The inverse Lplce trnsform of 2.9.42) yields Ūk,t)=t α 1 φk)e α,α λ 2 k 2 t 2 ), 2.9.43) nd then the solution is obtined by tking inverse Fourier trnsform. By tking inverse Fourier trnsform of 2.9.43) nd using the formul 1 e ikx f k)dk = 1 f k) coskx)dk 2.9.44) 2π π we hve where with Rα) >,k >. Ux,t)= Gx ζ,t)φζ )dζ, 2.9.45) Gx,t)= 1 π t α 1 E α,α k 2 λ 2 t α )coskx)dk 2.9.46) Exercises 2.9. 2.9.8. Evlute the integrl in 2.9.46). 2.9.9. Find the solution of the Fick s diffusion eqution t Px,t)=λ 2 x 2 Px,t), with the initil condition Px,t = )=δx), where δx) is the Dirc delt function.

References 133 References Agrwl, R. P. 1953). A Propos d une note de M. Pierre Humbert, C. R. Acd. Sci. Pris, 296, 231-232. Agrwl, R. P. 1963). Generlized Hypergeometric Series, Asi Publishing House, Bomby, London nd New York. Cputo, M. 1969). Elsticitá e Dissipzione, Znichelli, Bologn. Dzherbshyn, M.M. 1966). Integrl Trnsforms nd Representtion of Functions in Complex Domin in Russin), Nuk, Moscow. Erdélyi, A. 195-51). On some functionl trnsformtions, Univ. Politec. Torino, Rend. Sem. Mt. 1, 217-234. Erdélyi, A., Mgnus, W., Oberhettinger, F. nd Tricomi, F. G. 1953). Higher Trnscendentl Functions, Vol. 1, McGrw - Hill, New York, Toronto nd London. Erdélyi, A., Mgnus, W., Oberhettinger, F. nd Tricomi, F. G. 1954). Tbles of Integrl Trnsforms, Vol. 1, McGrw - Hill, New York, Toronto nd London. Erdélyi, A., Mgnus, W., Oberhettinger, F. nd Tricomi, F. G. 1954). Tbles of Integrl Trnsforms, Vol. 2, McGrw - Hill, New York, Toronto nd London. Erdélyi, A., Mgnus, W., Oberhettinger, F. nd Tricomi, F. G. 1955). Higher Trnscendentl Functions, Vol. 3, McGrw - Hill, New York, Toronto nd London. Fox, C. 1963). Integrl trnsforms bsed upon frctionl integrtion, Proc. Cmbridge Philos. Soc., 59, 63-71. Hubold, H. J. nd Mthi, A. M. 2). The frctionl kinetic eqution nd thermonuc1er functions, Astrophysics nd Spce Science, 273, 53-63. Hilfer, R. Ed.). 2). Applictions of Frctionl Clculus in Physics, World Scientific, Singpore. Kll, S. L. nd Sxen, R. K. 1969). Integrl opertors involving hypergeometric functions, Mth. Zeitschr., 18, 231-234. Kilbs. A. A. nd Sigo, M. 1998). Frctionl clculus of the H-function, Fukuok Univ. Science Reports, 28, 41-51. Kilbs, A. A. nd Sigo, M. 1996). On Mittg- Leffler type function, frctionl clculus opertors nd solutions of integrl equtions, Integrl Trnsforms nd Specil Functions, 4, 355-37. Kilbs, A. A, Sigo, M. nd Sxen, R. K. 22). Solution of Volterr integrodifferentil equtions with generlized Mittg-Leffler function in the kernels, J. Integrl Equtions nd Applictions, 14, 377-396. Kilbs, A. A., Sigo, M. nd Sxen, R. K. 24). Generlized Mittg-Leffler function nd generlized frctionl clculus opertors, Integrl Trnsforms nd Specil Functions, 15, 31-49. Kober, H. 194). On frctionl integrls nd derivtives, Qurt. J. Mth. Oxford, Ser. ll, 193-211. Love, E. R 1967). Some integrl equtions involving hypergeometric functions, Proc. Edin. Mth. Soc., 152), 169-198. Mthi, A. M. nd Sxen, R. K. 1973). Generlized Hypergeometric Functions with Applictions in Sttistics nd Physicl Sciences, Lecture Notes in Mthemtics, 348, Springer- Verlg, Berlin, Heidelberg. Mthi, A. M. nd Sxen, R. K. 1978). The H-function with Applictions in Sttistics nd Other Disciplines, John Wiley nd Sons, New York - London - Sydney. Miller, K. S. nd Ross, B. 1993). An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions, Wiley, New York. Mittg-Leffler, G. M. 193). Sur l nouvelle fonction E α x), C. R. Acd. Sci. Pris, Ser.II)137, 554-558. Oldhm, K. B. nd Spnier, J. 1974). The Frctionl Clculus: Theory nd Applictions of Differentition nd Integrtion to Arbitrry Order, Acdemic Press, New York. Podlubny, 1. 1999). Frctionl Differentil Equtions, Acdemic Press, Sn Diego.

134 2 Mittg-Leffler Functions nd Frctionl Clculus Podlubny, 1. 22). Geometric nd physicl interprettions of frctionl integrtion nd frctionl differentition, Frc. Clc. Appl. Anl., 54), 367-386. Prbhkr, T. R. 1971). A singulr integrl eqution with generlized Mittg- Leffler function in the kernel, Yokohm Mth. J., 19, 7-15. Ross, B. 1994). A formul for the frctionl integrtion nd differentition of + bx) c, J. Frct. Clc., 5, 87-89. Sigo, M. 1978). A remrk on integrl opertors involving the Guss hypergeometric function, Mth. Reports of College of Gen. Edu., Kyushu University, 11, 135-143. Sigo, M. nd Rin, R. K. 1988). Frctionl clculus opertors ssocited with generl clss of polynomils, Fukuok Univ. Science Reports, 18, 15-22. Smko, S. G., Kilbs, A. A. nd Mrichev,. 1. 1993). Frctionl Integrls nd Derivtives, Theory nd Applictions, Gordon nd Brech, Reding. Sxen, R. K. 1967). On frctionl integrtion opertors, Mth. Zeitsch., 96, 288-291. Sxen, R. K. 22). Certin properties of generlized Mittg-Leffler function, Proceedings of the Third Annul Conference of the Society for Specil Functions nd Their Applictions, Vrnsi, Mrch 4-6, 75-81. Sxen, R. K. 23). Alterntive derivtion of the solution of certin integro-differentil equtions of Volterr-type, Gnit Sndesh, 171), 51-56. Sxen,R. K. 24). On unified frctionl generliztion of free electron lser eqution, Vijnn Prishd Anusndhn Ptrik, 47l), 17-27. Sxen, R. K. nd Kumbht, R. K. 1973). A generliztion of Kober opertors, Vijnn Prishd Anusndhn Ptrik, 16, 31-36. Sxen, R. K. nd Kumbht, R. K. 1974). Integrl opertors involving H-function, Indin J. Pure ppl. Mth., 5, 1-6. Sxen, R. K. nd Kumbht, R. K. 1975). Some properties of generlized Kober opertors, Vijnn Prishd Anusndhn Ptrik, 18, 139-15. Sxen, R. K, Mthi, A. M nd Hubold, H. J. 22). On frctionl kinetic equtions, Astrophysics nd Spce Science, 282, 281-287. Sxen, R. K, Mthi, A. M nd Hubold, H. J. 24). On generlized frctionl kinetic equtions, Physic A, 344, 657-664. Sxen, R. K, Mthi, A. M. nd Hubold, H. J. 24). Unified frctionl kinetic equtions nd frctionl diffusion eqution, Astrophysics nd Spce Science, 29, 241-245. Sxen, R. K. nd Nishimoto, K. 22). On frctionl integrl formul of Sigo opertor, J. Frct. Clc., 22, 57-58. Sxen, R. K. nd Sigo, M. 25). Certin properties of frctionl clculus opertors ssocited with generlized Mittg-Leffler function. Frc. Clc. Appl. An1., 82), 141-154. Sneddon, I. N. 1975). The Use in Mthemticl Physics of Erdélyi- Kober Opertors nd Some of Their Applictions, Lecture Notes in Mthemtics Edited by B. Ross), 457, 37-79. Srivstv, H. M. nd Sxen, R. K. 21). Opertors of frctionl integrtion nd their pplictions, Appl. Mth. Comput., 118, 1-52. Srivstv, H. M. nd Krlsson, P. W. 1985). Multiple Gussin Hypergeometric Series, Ellis Horwood, Chichester, U.K. Stein, E. M. 197). Singulr Integrls nd Differentil Properties of Functions, Princeton University Press, New Jersey. Wimn, A. 195). Uber den Fundmentl stz in der Theorie de Funktionen E α x). Act Mth., 29, 191-21. Weyl, H. 1917). Bemerkungen zum Begriff des Differentilquotienten gebrochener Ordnung, Vierteljhresschr. Nturforsch. Gen. Zurich, 62, 296-32.

http://www.springer.com/978--387-75893-