k-weyl Fractional Derivative, Integral and Integral Transform

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1 Int. J. Contemp. Math. Sciences, Vol. 8, 213, no. 6, HIKARI Ltd, -Weyl Fractional Derivative, Integral and Integral Transform Luis Guillermo Romero 1 and Luciano Leonardo Luque Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 554 (34) Corrientes, Argentina Dedicated to Professor Ruben A. Cerutti for the trust placed in us Copyright c 213 Luis Guillermo Romero and Luciano Leonardo Luque. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract In this paper we define a new fractional derivative in the -calculus context, the -Weyl fractional derivative. Also we study the action of Laplace and Stieltjes Transforms on the new fractional operator and the -Weyl Fractional Integral operator introduced by Romero, Cerutti, Dorrego (cf. 7]). Mathematics Subject Classification: 26A33, 42A38 Keywords: -Fractional Calculus, Laplace Transfrom, Generalized Stieltjes Transfrom I Introduction and Preliminaries At present the -calculus constitutes a type of generalization of the classical differential and integral Calculus, and therefore is addressed in different and numerous investigations. Several authors that were dedicated to study such operators and since in 27 Diaz and Pariguan defined by the -Gamma function and the -symbol 1 guille-romero@live.com.ar

2 264 L. G. Romero and L. L. Luque Pochhammer are more the functions of the Fractional Calculus is generalized to the context. In this article we evaluate the Laplace and Generalized Stieljes Transform of the a -Weyl Fractional Integral operator and the a -Weyl fractional derivative and in its development we will use the following important definitions. Definition 1 The Gamma Euler function Γ(z) is given by the integral by Γ(z) = e t t z 1 dt, z C, Re(z) >, cf. 2] (I.1) and the Γ (z) is the -generalization of the Gamma Euler function defined Γ (z) = t z 1 e t dt, z C, Re(z) > (cf. 1]) (I.2) Is necessary to remember the definition of the convolution introduced by P. Miana given by Definition 2 Let f and g be function belonging to L 1 (R + ). Miana (cf. 4]) introduce the convolution product as the integral (f g)(t) = t f(x t)g(x)dx, t (I.3) Romero, Cerutti and Dorrego (cf. 7]) has introduced the analogue to the Riemann-Liouville singular ernel at -calculus given by the following Definition 3 Let α be a real number, <α<1, >. The -Riemann- Liouville singular ernel is given by j α, (t) = t α 1 Γ (α) t> (I.4) where Γ (z) is the -Gamma Euler function. Taing into account (I.3) and (I.4) they defined the -Weyl Fractional Integral given by Definition 4 Let α be a real number, <α<1, >. The -Weyl Fractional Integral is defined (cf. 7]) as W α f(x) =j α,(t) f(t) = 1 Γ (α) where denote the convolution due to Miana (cf. 4]). x (t x) α 1 f(t)dt (I.5)

3 -Weyl fractional derivative 265 Now, we remembering the definition of Laplace and Stieltjes Transforms. In fact Let L f](s) be the Laplace transform of an exponential order function and piecewise continuous where L f](s) = e st f(t)dt (I.6) t R +, and s C.(cf. 3]) And S α f](y) be the generalized Stieltjes transform of the function f belonging to S(R) the Schwartzian space of functions that decay rapidly at infinity together with all derivatives, where S α f](y) = f(x) (x + y) α dx, y > (cf.3]) (I.7) II Main Result For development of our article we give the following Definition 5 Let α be a real number, <α 1, f belonging to S(R). The -Weyl fractional derivative is given by W α f(x) = d dx W 1 α f(t) (II.1) Now, in the first lemma we will obtain a relationship between the -Weyl Fractional Integral and the lower incomplete gamma function through Laplace Transform. Lemma 1 Let f be a function belonging to S(R). The Laplace Transform of the -Weyl Fractional Integral of the f function is L W α f](s) =L f. ( s)2 α ( α )] Γ (α) γ, sx (s) (II.2) where γ (s, x) denote the lower incomplete gamma function given by γ (s, x) = x t s 1 e t dt (II.3)

4 266 L. G. Romero and L. L. Luque Proof. Remembering the following property (cf. 4]) where L f g](s) =L {g L (f,.)] ( s)} (s) L (f,t) = t is the incomplete Laplace Transform. From definition (I.5) we have f(z)e sz dz, t (II.4) L W α f](s) =L f.l (j α,,.)( s)] (s) (II.5) Then, we need evaluate the incomplete Laplace Transform of the -Riemann- Liouville singular ernel j α, (t) in the point s. Then, we have L (j α,,x)( s) = x j α, (t)e st dt = = 1 x t α 1 e st dt Γ (α) (II.6) Maing the change of variables that then, replacing (II.7) in (II.6) it result u = st, du = s dt (II.7) L (j α,,x)( s) = 1 ( s) Γ (α) ( s) α 1 sx u α 1 e u du = = ( s)2 α Γ (α) sx u α 1 e u du (II.8) Taing into account the definition (II.3) it result L (j α,,x)( s) = ( s)2 α Γ (α) γ ( α, sx ) (II.9) From (II.9) and (II.5) we have

5 -Weyl fractional derivative 267 L W α f](s) =L f. ( s)2 α ( α )] Γ (α) γ, sx (s) which is the thesis. In the next Lemma we evaluate the Stieltjes Transform of the -Weyl Fractional Integral and we will obtain a Stieltjes Transform of other order. Lemma 2 Let f be a function belonging to S(R). The generalized Stieltjes Transform of the -Weyl Fractional Integral of the f function is S β W α f](y) =( 1) α 1 Γ (β α) S β α Γ (β) f(y) (II.1) Proof. From definition (I.5) and (I.7) we have S β W α f](y) = 1 Γ (α) x (t x) α 1 f(t)dt (x + y) β dx = = 1 (t x) α 1 (x + y) β f(t)dtdx (II.11) Γ (α) x Applying Fubini s theorem it result S β W α f](y) = 1 Γ (α) f(t) (t x) α 1 (x + y) β dx dt = t = ( 1 ) f(t) (t x) α 1 (x + y) β dx dt Γ (α) t (II.12) In (II.12) considering that 1 Γ (α) t (t x) α 1 (x + y) β dx = ( 1) α 1 Γ (α) t (x t) α 1 (x + y) β dx =( 1) α 1 W α (x + y) β (II.13) Romero, Cerutti, Dorrego (cf. 7]) calculate W α (x + y) β. In fact

6 268 L. G. Romero and L. L. Luque W α (x + y) β =(t + y) α β Γ (β α) Γ (β) (II.14) Replacing (II.13) and (II.14) in (II.12) we have S β W α f](y) =( 1) α 1 Γ (β α) Γ (β) f(t) dt = (t + y) β α ( 1) α 1 Γ (β α) S β α Γ (β) f(y) that is the thesis of Lemma 2. Using the following property of the Laplace Transform of the derivative ] d L dt f(t) (s) =sl f(t)] (s) f() (II.15) to obtain the Laplace Transform of the -Weyl fractional derivative. Lemma 3 Let f be a function belonging to S(R), and let α be a real number, <α 1. The Laplace transform of the -Weyl fractional derivative is given by L W α f(x) ] (s) = s L f. 1 α ( s)2 Γ (1 α) γ ( 1 α Proof. By definition (II.1) and property (II.15) we have L W α f(x) ] (s) =L d ] dx W 1 α f(x) (s) = ) ], sx (s) + W 1 α f() (II.16) From (II.2) and (II.17) we obtain s L W 1 α f(x) ] (s) + W 1 α f() (II.17) L W α f(x) ] (s) = s L f. 1 α ( s)2 Γ (1 α) γ ( 1 α ) ], sx (s) + W 1 α f() (II.18) We can prove an analogous property for Stieltjes Transform of the derivative. In fact

7 -Weyl fractional derivative 269 Proposition 1 Let be a function f belonging to the S(R). Then we have ] d S α dx f(x) (y) = f() + α S y α α+1 f](y) (II.19) Proof. From definition (I.7) and integrating by parts we obtain ] d + S α dx f(x) (y) = f(x) (x + y) α ] + + ( α) d f(x) dx (x + y) dx = α f(x) dx = (x + y) α+1 f() y α + α S α+1 f](y) (II.2) Remar: f(x) (x+y) α when x because f S(R) We will evaluate the Stieltjes Transform of the -Weyl fractional derivative Lemma 4 Let f be a function belonging to S(R). The generalized Stieltjes Transform of the -Weyl Fractional derivative of the f function is S β W α f ] (y) = f() y β α ( 1) 1 α 1 Γ ((β +1) 1+α) Γ ((β + 1)) Proof. From definition (II.1) and the property (II.19) we obtain S β W α f ] (y) =S β d ] dx W 1 α f (y) = f() y β By (II.1) and (II.22) we have S β+1 1 α f(y) (II.21) α S β+1 W 1 α f ] (y) (II.22) S β W α f ] (y) = f() y β α ( 1) 1 α 1 Γ ((β +1) 1+α) Γ ((β + 1)) S β+1 1 α f(y) (II.23)

8 27 L. G. Romero and L. L. Luque References 1] R. Diaz and E. Pariguan. On hypergeometric functions and -Pochammer symbol. Divulgaciones Matematicas. Vol ] N. Lebedev. Special functions and their applications. Dover ] A. C. Mc Bride. Fractional Calculus and integral transforms of generalized functions. Pitman ] P.J. Miana. Convolution Products in L 1 (R + ), Integral Transforms and Fractional Calculus. Fractional Calculus & Applied Analysis. An International Journal for Theory and Applications. Volume B Number 4 (25). 5] K. S. Miller; B. Ross. An introduction to the Fractional Differential Equations. J. Wiley and Sons ] I. Podlubny. Fractional Differential Equations. An introduction to fractional derivatives. Academic Press ] L. Romero. R. Cerutti. G. Dorrego. -Weyl Fractional Integral. Int. Journal of Math. Analysis. Vol Received: December, 212