The k-fractional Hilfer Derivative

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1 Int. Journal of Math. Analysis, Vol. 7, 213, no. 11, The -Fractional Hilfer Derivative Gustavo Abel Dorrego and Rubén A. Cerutti Faculty of Exact Sciences National University of Nordeste. Av. Libertad Corrientes, Argentina Abstract In this paper we study a -version of the fractional derivative of two parameter introduced by Hilfer in [4], we calculate its Laplace transform and calculate the derivative of some functions. Also study a new operator that contains in its ernel the -Mittag-Leffler function introduced by authors in [3]. Finally, we generalize and solve a fractional differential equation propuse by Srivastava-Tomovsy in [9]. Mathematics Subject Classification: 26A33, 42A38 Keywords: -Fractional calculus. -Fractional Riemann-Liouville integral. -Fractional Riemann-Liouville derivative. I Introduction and Preliminaries In this short article we will begin considering a definition introduced by Hilfer cf. [4] of a fractional derivative. Indeed, we have the following Definition 1 The fractional derivative of order <α<1 and type β 1 with respect to x is defined by D μ,ν a+ fx I ν1 μ d a+ a+ f x I.1 dx I1 ν1 μ for functions for which the expression on the right hand side exists. As particular case we obtain the Riemann-Liouville derivative D α a+ : D α, a+ corresponding to a> and type β. From the definition by Diaz and Pariguan of the -Pochhammer symbol and the -Gamma function are many wors devoted to the subject among which

2 544 G. A. Dorrego and R. A. Cerutti may be cited definitions of the -Bessel functions cf.[1],the -Wright function cf.[7] and also a -integral of Riemann-Liouville cf. [6] and a -Riemann- Liouville fractional derivative cf.[8]. It is interesting to note the definition of a -Mittag-Leffler function of four parameters introduced by autors cf.[3]. For further development of this wor we need to remember elements of fractional calculus as derivatives and integrals of arbitrary orders. Also remember the action of integral transforms such as Laplace and Euler transformation on fractional operators. Definition 2 The Riemann-Liouville integral of order ν of a function f is given by I ν ft : 1 Γν cf.[5],p.69. t t τ ν 1 fτdτ t R +,ν C, Reν >. I.2 Definition 3 The Riemann-Liouville derivative of order ν C, Reν > of a function f is given by n d D ν ft : I n ν ft, n [Reν] + 1,t> I.3 dt cf.[5]. Definition 4 Let f : R + R an exponential order function and piecewise continuous, then the Laplace transform of f is L{ft}s : e st ftdt. I.4 The integral exist for Res >. Proposition 1 Let f be a sufficiently well-behaved function and let α be a real number, <α 1. The Laplace transform of the -Riemann-Liouville fractional integral of the f function is given by L{I α ft}s L{ft}s, cf.[8]. I.5 s α Proposition 2 Let β be a real number, <β 1. The Laplace transform of the -Riemann-Liouville fractional derivative is given by L{D β ft}s s.s 1 β L{ft}s I 1 β f, cf.[8]. I.6

3 The -fractional Hilfer derivative 545 II Main results In this paragraph we introduce the definition of a generalizes derivative - fractional which the one introduced by Hilfer it can be obtained the -Riemann- Liouville derivative and -Caputo derivative. Definition 5 Let μ, ν R/ <μ<1; ν 1 we define where D μ,ν fx I ν1 μ I α fx 1 Γ α if ν, 1 we have d f dx I1 μ1 ν x II.1 x t α 1 ftdt II.2 D μ, fx d dx I1 μ fx II.3 that is the fractional Riemann-Liouville derivative. If ν 1, we obtain the -fractional Caputo derivative D μ,1 fx I 1 μ d dx I f x d I 1 μ dx f x II.4 For further development of our wor we need to evaluate D μ,ν [t a λ 1 ]x. Started calculating I 1 ν1 μ [t a λ 1 ]x Γ λ 1 ν1 μ + x a λ 1. Γ 1 ν1 μ+λ II.5 Differentiating with respect to the above equality x Γ λx a 1 ν1 μ + λ 2 1 ν1 μ + λ Γ 1 ν1 μ+λ 1 1 ν1 μ + Γ λ x a λ 2 1 ν1 μ + λ 1 ν1 μ + λ 1 Γ 1 ν1 μ + λ Γ λ x a + λ 2 Γ 1 ν1 μ + λ 1 1 ν1 μ Γ λx a 1 ν1 μ + λ 2 Γ 1 ν1 μ+λ 1 ν1 μ + λ 2

4 546 G. A. Dorrego and R. A. Cerutti Finally we calculate the -fractional integral of ν1 μ order of above result. For that, considering the values 1 ν σ, 1 μ ρ, we have: I 1 σρ [Γ λx a σρ+λ Γσρ + λ Finally we obtain 2 ] I 1 σρ [Γ λx a σρ+λ Γσρ + λ Γ λγ σρ + λ x a 1 σρ+σρ+λ 1 Γσρ + λ Γ 1 σρ + σρ + λ Γ λ ρ+λ 1 x a Γ ρ + λ 1 ] D μ,ν [t a λ 1 ]x Γ λ 1 μ+λ x a 1. II.6 Γ 1 μ + λ Another important case is given by D μ,ν [t a β 1 E γ,α,β wt a α ]x D μ,ν [ Finally n γ n, Γ αn + β n n w n n! αn+β 1 t a γ n, w n Γ αn + β n! γ n, w n Γ αn + β n! ] x ] D [t μ,ν a αn+β 1 x t a αn+β+1 μ 1 Γ αn + β Γ 1 μ + αn + β β+1 μ 1 x a E γ,α,β+1 μ wx a α D μ,ν [t a β 1 E γ,α,β wt a α x a β+1 μ 1 ]x E γ,α,β+1 μ wx a α Proposition 3 Let α, β, μ, γ, δ C, min{rα; Rβ; Rμ; Rν; Rγ; Rδ} >. Then x t β 1 t μ 1 E γ α,α,β 1 wx t α α E δ,α,μ 1 wt α dt x β+μ 1 E δ+γ α,α,β+μ 1 wx α II.7

5 The -fractional Hilfer derivative 547 Proof Applying Laplace transform to the left member and using II.2 of [3] result L{ x t β 1 t μ 1 E γ,α,β α 1 wx t α α E δ,α,μ 1 wt α dt}s s β 2 1 β s α +δ 2 s β+μ α s w γ s μ β+μ 1 w γ+δ s αδ 2 1 μ α s w δ Then by uniqueness of the inverse Laplace transform is desired. An important player in the fractional calculus applications in fractional solution of differential equations is given by the following Definition 6 Let α, β, γ, w C, R αn+β > 1 we define the following operator We show that E γ,α,β ϕ x x t β 1 E γ,α,β wx t α ϕtdt II.8 Theorem 1 The operator E γ,α,β ϕ x is bounded on the space La, b of the Lebesgue integrable functions. Proof. Following a procedure analogous to [9] we arrive to E γ,α,β ϕ x 1 ϕ 1 M II.9 where M b a R β n γ n, wb a R α n [nr α +R β ]Γ αn + βn! II.1 Proposition 4 Let <μ<1, and ν 1, then L{ D μ,ν fx}s sl{fx}s s 1 μ I1 μ1 ν f s 1 μν II.11

6 548 G. A. Dorrego and R. A. Cerutti Proof. Applying Laplace transform to the definition II.1 and using the result of Laplace transform of the -fractional integral of Riemann-Liouville calculate for [8] L{ D μ,ν fx}s L{I ν1 μ d dx L{ d dx sl{ I 1 μ1 ν I 1 μ1 ν f s 1 μν I 1 ν1 μ f s 1 μν sl{fx}s s 1 μν+1 ν1 μ sl{fx}s s 1 μ x}s x}s f x}s I1 ν1 μ f s 1 μν I1 ν1 μ f s 1 μν I1 ν1 μ f s 1 μν Proposition 5 Let α, β, γ C, Reα >, Reβ >, Reγ >, Res >, and 1 α ws α/ < 1. Then hold L{ E γ,α,β ϕ x}s 2 1 β L{ϕt}s s β s α 1 α w γ II.12 Proof. Applying Laplace transform and taing w α 1 a in II.19 of [3] we tae L{ E γ,α,β ϕ x}s L{ x t β 1 E γ,α,β L{x β 1 E γ,α,β wx α ϕt}s wx t α ϕtdt}s L{x β 1 E γ,α,β wx α }sl{ϕt}s 2 1 β L{ϕt}s s β s α 1 α w γ III Application Proposition 6 Let <μ<1, ν 1 the following fractional differential equation: D μ,ν yx λ E γ,α,β ϕ x+fx III.1

7 The -fractional Hilfer derivative 549 with the initial condition: I 1 μ1 ν y C, C real arbitatry constant. III.2 has a solution given by yx λ[x β E γ,α,β wx α I +μ 1 ϕx] + I +μ 1ϕ where fx+i 1 μ1 ν+ C III.3 I 1 μ1 ν+ C C 1 ν1 μ+ Γ 1 μ1 ν+ x 1 ν1 μ +1 III.4 Proof. Applying Laplace transform to both member of III.1 and using the condition III.2 sl{yx}s s 1 μ L{ D μ,ν yx}s λl{ E γ,α,β ϕ x}s+l{fx}s C s 1 μν λ 2 1 β L{ϕt}s+L{fx}s s β s α 1 α w γ L{yx}s λ 2 1 β s 1 μ 1 μ L{fx}ss L{ϕt}s+ ss β s α 1 α w γ s L{yx}s λ 2 1 β s 1 μ 1 μ L{fx}ss L{ϕt}s+ ss β s α 1 α w γ s L{yx}s λ 2 1 β L{ϕt}s + L{fx}s s β s α 1 α w γ s +μ 1 s μ C 1 μ Cs ss 1 μν + s 1 μ1 ν+ 1 μ Cs ss 1 μν L{yx}s λl{x β 1 E γ,α,β wx α }sl{i +μ 1 ϕx}s+l{i +μ 1 fx}s + L{I 1 μ1 ν+ C}s L{yx}s λl{x β 1 E γ,α,β wx α I +μ 1 ϕx}s+l{i +μ 1 fx}s + L{I 1 μ1 ν+ C}s Finally for uniqueness of inverse Laplace transform result

8 55 G. A. Dorrego and R. A. Cerutti yx λ[x β 1 E γ,α,β wx α I +μ 1 ϕx] + I +μ 1 fx+i 1 μ1 ν+ C The last term of the above solution can be calculate using II.2 and then arrive to I 1 μ1 ν+ C x 1 ν1 μ +1 C 1 ν1 μ+ Γ 1 μ1 ν+ References [1] R. Cerutti. On the -Bessel functions. Int. Math. Forum. Vol 7. no [2] R. Diaz and E. Pariguan. On hypergeometric functions and -Pochammer symbol. Divulgaciones Matematicas Vol [3] G. Dorrego; R. Cerutti. The -Mittag-Leffler function. Journal of Applied Math. Int. J. Contemp. Math. Sciences. Vol 7. no [4] R. Hilfer. Applications of Fractional Calculus in Phisics. World Scientific Publishing Co. Pte. Ltd. 2. [5] A. Kilbas, H. Srivastava and J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier. 26. [6] S. Mubeen; G. M. Habibullah. -Fractional Integrals and Application. Int. J. Contemp. Math, Science. Vol 7 No [7] G. Romero and R. Cerutti. Fractional Calculus of a -Wright type function. Int. Journal Contemp. Math. Sciences. Vol 7. no [8] G. Romero, L. Luque, G. Dorrego, R. Cerutti. On the -Riemann-Liouville Fractional Derivative. To appear. [9] H. M. Srivastava and Z. Tomosvi. Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the ernel. Appl. Math. Comput. doi:1.116/j.amc Received: October, 212