katugampola fractional operators
katugampola, un: new approach to generalized fractional integral. appl. math. comput .
this manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. we introduce new fractional integration and differentiation operators. we define spaces and present some theorems related to these operators .
hua chen, and udita n. katugampola, “hermite–hadamard and hermite–hadamard–fejér type inequalities for generalized fractional integrals, ”
katugampola, un: a new approach to generalized fractional derivatives. bull. math. anal. appl .
giorgio s. taverna, and delfim f. m. torres, “generalized fractional operators for nonstandard lagrangians, ”
in this paper we present a generalization to two existing fractional integrals and derivatives, namely, the riemann - liouville and hadamard fractional operators. the existence and uniqueness results for single term fractional differential equations (fde) have also been established. we also obtain the mellin transforms of such generalized fractional operators which are important in solving fractional differential equations .
abdeljawad, t, baleanu, d: monotonicity results for fractional difference operators with discrete exponential kernels. adv. differ. equ .
ryan m. evans, udita n. katugampola, and david a. edwards, “applications of fractional calculus in solving abel - type integral equations: surface–volume reaction problem, ”
this paper was devoted to an investigation of the fractional derivatives and integrals obtained by iterating conformable integrals. we obtained left - and right - fractional conformable integrals. with a standard fractional procedure we found left - and right - fractional conformable derivatives in the sense of riemann - liouville and caputo. we proved that these fractional integrals and derivatives have properties similar to the standard fractional integrals and derivatives. we also define the fractional derivatives of functions belonging to specific spaces in order to find the relation between these new fractional differential operators. the presented left - and right - fractional integrals are different from those defined by katugampola since their kernels depend on the end points a and b and hence need a different convolution theory when the conformable laplace is applied .
gao, f, yang, xj: fractional maxwell fluid with fractional derivative without singular kernel. therm. sci .
respectively. we suggest that iterating the above integral will end up with new fractional operators with two parameters and kernels different from the usual kernels of usual fractional derivatives and integrals. from the data analysis point of view we suppose that this new type of calculus will provide better understanding of the complexity of the dynamics of the phenomena from porous media .
]. as is well known, systems possessing a memory effect often appear in real world phenomena. however, for each type of data we always ask what is the optimal corresponding nonlocal model to be applied. moreover, many authors studied new fractional operators with local, nonlocal, singular and non - singular kernels (see [
for the right - fractional conformable derivatives, we can state a similar lemma .
podlubny, i: fractional differential equations. academic press, san diego (1999 )
kilbas, aa: hadamard type fractional calculus. j. korean math. soc .
, we define the left - and right - fractional conformable integrals and derivatives. in section
magin, rl: fractional calculus in bioengineering. begell house publishers, redding (2006 )
abdeljawad, t: on conformable fractional calculus. j. comput. appl. math .
we study incommensurate fractional variational problems in terms of a generalized fractional integral with lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. we obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized noether type theorem .
], in what follows, we recall some basic definitions and tools about classical fractional calculus .
hilfer, r: applications of fractional calculus in physics. word scientific, singapore (2000 )
, we define the fractional conformable derivatives of functions belonging to certain spaces and state their properties. in section
matheus j. lazo, and delfim f. m. torres, “variational calculus with conformable fractional derivatives, ”
in this section, we consider the fractional conformable derivatives of functions belonging to spaces stated in the following definitions .
] and the references therein). the standard fractional calculus may not provide us the required kernel in order to extract important information from these types of systems. at this stage, we ask the following question. can we generalize the standard fractional riemann - liouville integrals in a way such that we obtain unification to riemann - liouville, hadamard and other fractional derivatives [
before we state the inverse properties and the composition of two caputo fractional conformable derivatives, we shall consider the following lemmas .
el - nabulsi, ra, torres, dfm: fractional action - like variational problems. j. math. phys .
in this section we define the left - and right - fractional conformable derivatives in the sense of caputo and present their properties .
a. b. malinowska, and d. f. m. torres, “fractional calculus of variations of several independent variables, ”
shakoor pooseh, ricardo almeida, and delfim f. m. torres, “discrete direct methods in the fractional calculus of variations, ”
dina tavares, ricardo almeida, and delfim f. m. torres, “caputo derivatives of fractional variable order: numerical approximations, ”
atangana, a, baleanu, d: new fractional derivative with non - local and non - singular kernel. therm. sci .
a generalized fractional calculus of variations - control and cybernetics - tom vol. 42, no. 2 (2013) - baztech - yadda
tatiana odzijewicz, agnieszka b. malinowska, and delfim f. m. torres, “variable order fractional variational calculus for double integrals, ”
tatiana odzijewicz, agnieszka b. malinowska, and delfim f. m. torres, “green' s theorem for generalized fractional derivatives, ”
michael a. zhuravkov, and natalie s. romanova, “review of methods and approaches for mechanical problem solutions based on fractional calculus, ”
roberto garra, giorgio s. taverna, and delfim f. m. torres, “fractional herglotz variational principles with generalized caputo derivatives, ”
we present the fractional conformable derivatives in the caputo setting and state their properties. finally, the last section is devoted to our conclusion .
caputo, m, fabrizio, m: a new definition of fractional derivative without singular kernel. prog. fract. differ. appl .
losada, j, nieto, jj: properties of a new fractional derivative without singular kernel. prog. fract. differ. appl .
in the area of fractional calculus and its applications in many branches of science and engineering, several fractional derivatives were mainly utilized. the most common used were caputo and riemann - liouville derivatives, which were successfully utilized in modeling complex dynamics appearing in physics, biology, engineering and many other fields [
jarad, f, abdeljawad, t, baleanu, d: caputo - type modification of the hadamard fractional derivative. adv. differ. equ .
gastao s. f. frederico, and delfim f. m. torres, “fractional noether' s theorem with classical and riemann - liouville derivatives, ”
samko, sg, kilbas, aa, marichev, oi: fractional integrals and derivatives: theory and applications. gordon & breach, yverdon (1993 )
yang, xj, gao, f, machado, jat, baleanu, d: a new fractional derivative involving the normalized sinc function without singular kernel .
jarad, f, abdeljawad, t, baleanu, d: on the generalized fractional derivatives and their caputo modification. j. nonlinear sci. appl .
mohammad maleki, ishak hashim, saeid abbasbandy, and a. alsaedi, “direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method, ”
abdeljawad, t, baleanu, d: on fractional derivatives with exponential kernel and their discrete versions. j. rep. math. phys. (to appear )
gambo, yy, jarad, f, abdeljawad, t, baleanu, d: on caputo modification of the hadamard fractional derivative. adv. differ. equ .
kilbas, a, srivastava, hm, trujillo, jj: theory and application of fractional differential equations. north - holland mathematics studies, vol. 204 (2006 )
hegagi m. ali, fernando lobo pereira, and sã­lvio m. a. gama, “a new approach to the pontryagin maximum principle for nonlinear fractional optimal control problems, ”
adjabi, y, jarad, f, baleanu, d, abdeljawad, t: on cauchy problems with caputo hadamard fractional derivatives. j. comput. anal. appl .
matheus j. lazo, and delfim f. m. torres, “the dubois–reymond fundamental lemma of the fractional calculus of variations and an euler–lagrange equation involving only derivatives of caputo, ”
loic bourdin, tatiana odzijewicz, and delfim f. m. torres, “existence of minimizers for generalized lagrangian functionals and a necessary optimality condition - application to fractional variational problems, ”
abdeljawad, t, baleanu, d: integration by parts and its applications of a new nonlocal fractional derivative with mittag - leffler nonsingular kernel. j. nonlinear sci. appl .
s. s. ezz - eldien, e. h. doha, a. h. bhrawy, a. a. el - kalaawy, and j. a. t. machado, “a new operational approach for solving fractional variational problems depending on indefinite integrals, ”
]. the core of this procedure is to decide which differentiation operator should be used as a starting point for the iteration procedure. for the standard fractional calculus, we iterate the usual integral of a function and using the cauchy formula we obtain the integral of higher integer orders and then replace this integer by any complex number. in [
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faculty of computer science, bialystok university of technology, 15 - 351 białystok, poland, a. malinowska @ urltoken
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tatiana odzijewicz, 1 agnieszka b. malinowska, 2 and delfim f. m. torres 1
copyright © 2012 tatiana odzijewicz et al. 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 work is properly cited .
the following is the list of published articles that have cited the current article .
delfim f. m. torres, and agnieszka b. malinowskapp. 1–275, 2012 .
rajesh k. pandey, and om p. agrawal, “numerical scheme for generalized isoperimetric constraint variational problems with a - operator, ”
rajesh k. pandey, and om p. agrawal, “numerical scheme for a quadratic type generalized isoperimetric constraint variational problems with a - operator, ”
], it was suggested that the conformable integral should be fractionalized properly. we recall that an integral type like the one from [
$ $ \ bigl (\ mathbf { i } _ { b } ^ { \ alpha } f \ bigr) (t) = \ frac { 1 } { \ gamma (\ alpha) } \ int _ { t } ^ { b } (y - t) ^ { \ alpha - 1 } f (y) \, dy. $ $
$ $ \ bigl ({ } _ { a } d ^ { \ alpha } f \ bigr) (t) = \ biggl (\ frac { d } { dt } \ biggr) ^ { n } \ bigl ({ } _ { a } \ mathbf { i } ^ { n - \ alpha } f \ bigr) (t), \ quad n = [ \ alpha ] + 1. $ $
$ $ \ bigl (d _ { b } ^ { \ alpha } f \ bigr) (t) = \ biggl (- \ frac { d } { dt } \ biggr) ^ { n } \ bigl (\ mathbf { i } _ { b } ^ { n - \ alpha } f \ bigr) (t). $ $
$ $ \ bigl ({ } ^ { c } d _ { b } ^ { \ alpha } f \ bigr) (t) = \ bigl (\ mathbf { i } _ { b } ^ { n - \ alpha } (- 1) ^ { n } f ^ { (n) } \ bigr) (t). $ $
$ $ \ bigl (\ mathfrak { j } _ { b } ^ { \ alpha } f \ bigr) (t) = \ frac { 1 } { \ gamma (\ alpha) } \ int _ { t } ^ { b } (\ ln y - \ ln t) ^ { \ alpha - 1 } f (y) \ frac { dy } { y }. $ $
$ $ \ bigl (\ mathfrak { d } _ { b } ^ { \ alpha } f \ bigr) (t) = \ biggl (- t \ frac { d } { dt } \ biggr) ^ { n } \ bigl (i _ { b } ^ { n - \ alpha } f \ bigr) (t). $ $
$ $ \ bigl ({ } ^ { c } \ mathfrak { d } _ { b } ^ { \ alpha } f \ bigr) (t) = \ biggl (\ mathfrak { j } _ { b } ^ { n - \ alpha } \ biggl (- t \ frac { d } { dt } \ biggr) ^ { n } f \ biggr) (t). $ $
$ $ x _ { c } ^ { p } (a, b) = \ biggl \ { f: [ a, b ] \ rightarrow \ mathbb { r }: \ | f \ | _ { x _ { c } ^ { p } } = \ biggl (\ int _ { a } ^ { b } \ bigl \ vert t ^ { c } f (t) \ bigr \ vert ^ { p } \ frac { dt } { t } \ biggr) ^ { 1 / p } < \ infty \ biggr \ }. $ $
$ $ \ bigl ({ } ^ { c } d _ { b } ^ { \ alpha, \ rho } f \ bigr) (t) = \ bigl ({ } _ { a } i ^ { n - \ alpha, \ rho } (- \ gamma) ^ { n } f \ bigr) (t) = \ frac { 1 } { \ gamma (n - \ alpha) } \ int _ { t } ^ { b } \ biggl (\ frac { y ^ { \ rho } - t ^ { \ rho } } { \ rho } \ biggr) ^ { n - \ alpha - 1 } (- \ gamma) ^ { n } f (y) \ frac { dy } { y ^ { 1 - \ rho } }. $ $
$ $ \ begin { aligned } \ begin { aligned } [ b ] { } _ { a } i ^ { n, \ alpha } f (x) & = \ int _ { a } ^ { x } \ frac { dt _ { 1 } } { (t _ { 1 } - a) ^ { 1 - \ alpha } } \ int _ { a } ^ { t _ { 1 } } \ frac { dt _ { 2 } } { (t _ { 2 } - a) ^ { 1 - \ alpha } } \ cdots \ int _ { a } ^ { t _ { n - 1 } } \ frac { f (t _ { n }) \, dt _ { n } } { (t _ { n } - a) ^ { 1 - \ alpha } } \ \ & = \ frac { 1 } { \ gamma (n) } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n - 1 } f (t) \ frac { dt } { (t - a) ^ { 1 - \ alpha } }. \ end { aligned } \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) = \ frac { 1 } { \ gamma (\ beta) } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { \ beta - 1 } f (t) \ frac { dt } { (t - a) ^ { 1 - \ alpha } }. $ $
$ $ { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f (x) = \ frac { 1 } { \ gamma (\ beta) } \ int _ { x } ^ { b } \ biggl (\ frac { (b - x) ^ { \ alpha } - (b - t) ^ { \ alpha } } { \ alpha } \ biggr) ^ { \ beta - 1 } f (t) \ frac { dt } { (b - t) ^ { 1 - \ alpha } }. $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ gamma } \ mathfrak { i } ^ { \ alpha } \ bigr) f (x) = { } _ { a } ^ { \ beta + \ gamma } \ mathfrak { i } ^ { \ alpha } f (x), \ qquad { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ({ } ^ { \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigr) f (x) = { } ^ { \ beta + \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } f (x). $ $
$ $ z = \ frac { (t - a) ^ { \ alpha } - (u - a) ^ { \ alpha } } { (x - a) ^ { \ alpha } - (u - a) ^ { \ alpha } }. $ $
the second formula can be proved in a similar way or by using the action of the q - operator. □
$ $ \ begin { aligned } & \ bigl ({ } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } (b - t) ^ { \ alpha \ nu - \ alpha } \ bigr) (x) = \ frac { 1 } { \ alpha ^ { \ beta } } \ frac { \ gamma (\ nu) } { \ gamma (\ beta + \ nu) } (b - x) ^ { \ alpha (\ beta + \ nu - 1) }. \ end { aligned } $ $
$ $ \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } (t - a) ^ { \ nu - 1 } \ bigr) (x) = \ frac { 1 } { \ gamma (\ beta) } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { \ beta - 1 } (t - a) ^ { \ alpha \ nu - \ alpha } \ frac { dt } { (t - a) ^ { 1 - \ alpha } }. $ $
$ $ \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } (t - a) ^ { \ nu - 1 } \ bigr) (x) = \ frac { (x - a) ^ { \ alpha (\ beta + \ nu - 1) } } { \ gamma (\ beta) \ alpha ^ { \ beta } } \ int _ { 0 } ^ { 1 } (1 - u) ^ { \ beta - 1 } u ^ { \ nu - 1 } \, du = \ frac { \ gamma (\ nu) } { \ alpha ^ { \ beta } \ gamma (\ beta + \ nu) } (x - a) ^ { \ alpha (\ beta + \ nu - 1) }. $ $
$ $ \ begin { aligned } & \ bigl [ { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } (b - t) ^ { \ alpha \ nu - \ alpha } \ bigr ] (x) = \ alpha ^ { \ beta } \ frac { \ gamma (\ nu) } { \ gamma (\ nu - \ beta) } (b - x) ^ { \ alpha (\ nu - \ beta - 1) }. \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f = { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { - \ alpha } f, \ qquad { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } = { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { - \ alpha } f. $ $
$ $ \ frac { d } { dx } \ bigl ({ } _ { a } ^ { n - 2 } t ^ { \ alpha } f (x) \ bigr) = \ frac { 1 } { (x - a) ^ { 1 - \ alpha } } \ int _ { a } ^ { x } \ psi (t) \ frac { dt } { (t - a) ^ { 1 - \ alpha } } + \ frac { { } _ { a } ^ { n - 1 } t ^ { \ alpha } f (a) } { (x - a) ^ { 1 - \ alpha } }. $ $
$ $ \ begin { aligned } { } _ { a } ^ { n - 2 } t ^ { \ alpha } f (x) = & \ int _ { a } ^ { x } \ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ psi (t) \ frac { dt } { (t - a) ^ { 1 - \ alpha } } \ \ & { } + { } _ { a } ^ { n - 1 } t ^ { \ alpha } f (a) \ frac { (x - a) ^ { \ alpha } } { \ alpha } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f (a). \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { n - 3 } t ^ { \ alpha } f (x) = & \ frac { 1 } { 2! } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { 2 } \ psi (t) \ frac { dt } { (t - a) ^ { 1 - \ alpha } } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f (a) \ frac { (x - a) ^ { 2 \ alpha } } { 2 \ alpha ^ { 2 } } \ \ & { } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f (a) \ frac { (x - a) ^ { \ alpha } } { \ alpha } + { } _ { a } ^ { n - 3 } t ^ { \ alpha } f (a). \ end { aligned } $ $
$ $ \ begin { aligned } f (x) = & \ frac { 1 } { (n - 1)! } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n - 1 } \ psi (t) \ frac { dt } { (t - a) ^ { 1 - \ alpha } } \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { k } t ^ { \ alpha } f (a) } { \ alpha ^ { k } k! } (x - a) ^ { \ alpha k }. \ end { aligned } $ $
$ $ \ begin { aligned } f (x) = & \ frac { 1 } { (n - 1)! } \ int _ { x } ^ { b } \ biggl (\ frac { (b - x) ^ { \ alpha } - (b - t) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n - 1 } \ frac { ({ } ^ { n } t _ { b } ^ { \ alpha } f) (t) } { (b - t) ^ { 1 - \ alpha } } \, dt \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { (- 1) ^ { k } \, { } ^ { k } t _ { b } ^ { \ alpha } f (b) } { k! } \ frac { (b - x) ^ { \ alpha k } } { \ alpha ^ { k } }. \ end { aligned } $ $
$ $ \ begin { aligned } { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) = & \ frac { 1 } { \ gamma (n - \ beta) } \ int _ { x } ^ { b } \ biggl (\ frac { (b - x) ^ { \ alpha } - (b - t) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n - \ beta - 1 } \ frac { ^ { n } t _ { b } ^ { \ alpha } f (t) } { (b - t) ^ { 1 - \ alpha } } \, dt \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { (( - 1) ^ { k } \, { } ^ { k } t _ { b } ^ { \ alpha } f (b) ) (b - x) ^ { \ alpha (k - \ beta) } } { \ alpha ^ { k - \ beta } \ gamma (k - \ beta + 1) }. \ end { aligned } $ $
$ $ f (x) = \ frac { 1 } { (n - 1)! } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n - 1 } \ frac { ({ } _ { a } ^ { n } t ^ { \ alpha } f) (t) } { (t - a) ^ { 1 - \ alpha } } \, dt + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f (a) } { k! } \ frac { (x - a) ^ { \ alpha k } } { \ alpha ^ { k } }. $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f (x) = & \ frac { { } _ { a } ^ { n } t ^ { \ alpha } } { (n - 1)! \ gamma (n - \ beta) } \ int _ { a } ^ { x } \ int _ { a } ^ { t } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n - \ beta - 1 } \ biggl (\ frac { (t - a) ^ { \ alpha } - (u - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n } \ \ & { } \ times \ bigl ({ } _ { a } ^ { n } t ^ { \ alpha } f (u) \ bigr) \ frac { du } { (u - a) ^ { 1 - \ alpha } } \ frac { dt } { (t - a) ^ { 1 - \ alpha } } \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f (a) } { k! \ alpha ^ { k } } \ alpha ^ { \ beta } \ frac { \ gamma (k + 1) } { \ gamma (k + 1 - \ beta) } (x - a) ^ { \ alpha (k - \ beta) }. \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f (x) = & \ biggl (\ frac { \ gamma (n - \ beta) \ gamma (n) { } _ { a } ^ { n } t ^ { \ alpha } } { (n - 1)! \ gamma (n - \ beta) \ gamma (2n - \ beta) } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (u - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { 2n - \ beta - 1 } \ frac { ({ } _ { a } ^ { n } t ^ { \ alpha } f) (u) \, du } { (u - a) ^ { 1 - \ alpha } } \ biggr) \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f (a) } { \ alpha ^ { k - \ beta } \ gamma (k + 1 - \ beta) } (x - a) ^ { \ alpha (k - \ beta) }. \ end { aligned } $ $
$ $ { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) = { } _ { a } ^ { \ beta - m } \ mathfrak { i } ^ { \ alpha } f (x); \ qquad { } ^ { m } t _ { b } ^ { \ alpha } \ bigl ({ } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f (x) \ bigr) = { } ^ { \ beta - m } \ mathfrak { i } _ { b } ^ { \ alpha } f (x). $ $
$ $ { } _ { a } ^ { \ gamma } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) = { } _ { a } ^ { \ beta - \ gamma } \ mathfrak { i } ^ { \ alpha } f (x); \ qquad { } ^ { \ gamma } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ({ } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f (x) \ bigr) = ^ { \ beta - \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } f (x). $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ gamma } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) & = { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ({ } _ { a } ^ { m - \ gamma } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) \ bigr) \ \ & = { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta + m - \ gamma } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) \ \ & = { } _ { a } ^ { \ beta - \ gamma } \ mathfrak { i } ^ { \ alpha } f (x). \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) = f (x); \ qquad { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ({ } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f (x) \ bigr) = f (x). $ $
and utilized the properties of the gamma and beta functions. the last step in the proof is to use lemma 2. 1 in [
$ $ { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ({ } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) \ bigr) = f (x) - \ sum _ { j = 1 } ^ { n } \ frac { (- 1) ^ { n - j } \, { } ^ { \ beta - j } \ mathfrak { d } _ { b } ^ { \ alpha } f (b) } { \ alpha ^ { \ beta - j } \ gamma (\ beta - j + 1) } (b - x) ^ { \ alpha \ beta - \ alpha j }. $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f (x) \ bigr) = & \ frac { 1 } { \ gamma (\ beta) } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { \ beta - 1 } \ bigl ({ } _ { a } ^ { n } t ^ { \ alpha } \ bigl ({ } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f (t) \ bigr) \ bigr) \ frac { dt } { (t - a) ^ { 1 - \ alpha } } \ \ = & \ frac { { } _ { a } ^ { 1 } t ^ { \ alpha } } { \ gamma (\ beta + 1) } \ biggl [ \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ alpha } \ biggr) ^ { \ beta } \ bigl ({ } _ { a } ^ { n } t ^ { \ alpha } \ bigl ({ } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f (t) \ bigr) \ bigr) \ frac { dt } { (t - a) ^ { 1 - \ alpha } } \ biggr ]. \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f (x) \ bigr) = & { } _ { a } ^ { 1 } t \ biggl [ { } _ { a } ^ { 1 } \ mathfrak { i } ^ { \ alpha } f (x) - \ sum _ { j = 1 } ^ { n } \ frac { ({ } _ { a } ^ { n - j } t ^ { \ alpha } ({ } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f (a) )) } { \ gamma (\ beta + 2 - j) \ alpha ^ { \ beta - j + 1 } } (x - a) ^ { \ alpha \ beta - \ alpha j + \ alpha } \ biggr ] \ \ = & f (x) - \ sum _ { j = 1 } ^ { n } \ frac { { } _ { a } ^ { \ beta - j } \ mathfrak { d } ^ { \ alpha } f (a) } { \ alpha ^ { \ beta - j } \ gamma (\ beta - j + 1) } (x - a) ^ { \ alpha \ beta - \ alpha j }. \ end { aligned } $ $
$ $ { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) = { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ biggl [ f (t) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { (- 1) ^ { k } \, { } ^ { k } t _ { b } ^ { \ alpha } f (b) } { k! \ alpha ^ { k } } (b - t) ^ { \ alpha k } \ biggr ] (x). $ $
$ $ \ begin { aligned } { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) & = \ frac { (- 1) ^ { n } } { \ gamma (n - \ beta) } \ int _ { x } ^ { b } \ biggl (\ frac { (b - x) ^ { \ alpha } - (b - t) ^ { \ alpha } } { \ alpha } \ biggr) ^ { n - \ beta - 1 } \ frac { ^ { n } t _ { b } ^ { \ alpha } f (t) } { (b - t) ^ { 1 - \ alpha } } \, dt \ \ & = { } ^ { n - \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl (^ { n } t _ { b } ^ { \ alpha } f (x) \ bigr). \ end { aligned } $ $
$ $ \ begin { aligned } { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) & = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f (x) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ^ { k } t _ { a } ^ { \ alpha } f (a) } { \ alpha ^ { k - \ beta } k! } \ frac { \ gamma (k + 1) } { \ gamma (k - \ beta + 1) } (x - a) ^ { k \ alpha - \ beta \ alpha } \ \ & = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f (x) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ^ { k } t _ { a } ^ { \ alpha } f (a) } { \ alpha ^ { k - \ beta } \ gamma (k - \ beta + 1) } (x - a) ^ { k \ alpha - \ beta \ alpha } \ \ & = \ frac { 1 } { \ gamma (n - \ beta) } \ int _ { a } ^ { x } \ biggl (\ frac { (x - a) ^ { \ alpha } - (t - a) ^ { \ alpha } } { \ rho } \ biggr) ^ { n - \ beta - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f (t) } { (t - a) ^ { 1 - \ alpha } } \, dt \ \ & = { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { n } t ^ { \ alpha } f (x) \ bigr). \ end { aligned } $ $
$ $ \ bigl \ vert { } _ { a } ^ { \ beta - k } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr \ vert \ leq \ frac { \ vert f \ vert _ { c } } { \ vert \ gamma (\ beta - k) \ vert (\ operatorname { re } (\ beta) - k) } \ frac { (x - a) ^ { \ alpha (\ operatorname { re } (\ beta) - k) } } { \ alpha ^ { \ operatorname { re } (\ beta) - k } }. $ $
$ $ \ bigl \ vert ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) \ bigr \ vert \ leq \ frac { \ vert ^ { n } t _ { b } ^ { \ alpha } f \ vert _ { c } } { \ vert \ gamma (n - \ beta) \ vert (n - \ operatorname { re } (\ beta) ) } \ frac { (b - x) ^ { \ alpha (n - \ operatorname { re } (\ beta) ) } } { \ alpha ^ { n - \ operatorname { re } (\ beta) } }. $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) = f (x); \ qquad { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ({ } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f (x) \ bigr) = f (x). $ $
$ $ \ begin { aligned } & { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ({ } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f (x) \ bigr) = f (x) - \ frac { { } ^ { \ beta + 1 - n } \ mathfrak { i } _ { b } ^ { \ alpha } f (b) } { \ alpha ^ { n - \ beta } \ gamma (n - \ beta) } (b - x) ^ { \ alpha n - \ alpha \ beta }. \ end { aligned } $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { k } t ^ { \ alpha } ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (a) ) (x - a) } { \ alpha ^ { k - \ beta } \ gamma (k - \ beta + 1) }. $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f (x) \ bigr) = f (x) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { \ beta - k } \ mathfrak { i } ^ { \ alpha } f (a) (x - a) ^ { \ alpha - k - \ beta \ alpha } } { \ alpha ^ { k - \ beta } \ gamma (k - \ beta + 1) }. $ $
) is proved. the second identity can be proved by using the same arguments .
$ $ \ begin { aligned } & { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl (^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) \ bigr) = f (x) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { (- 1) ^ { k } \, { } ^ { k } t _ { b } ^ { \ alpha } f (b) (b - x) ^ { \ alpha k } } { \ alpha ^ { k } k! }. \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } f (x) \ bigr) & = { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { n } t ^ { \ alpha } f (x) \ bigr) \ bigr) \ \ & = { } _ { a } ^ { n } \ mathfrak { i } ^ { \ alpha } \ bigl ({ } _ { a } ^ { n } t ^ { \ alpha } f (x) \ bigr) \ \ & = f (x) - \ sum _ { j = 1 } ^ { n } \ frac { { } _ { a } ^ { n - j } d ^ { \ alpha } f (a) } { \ alpha ^ { n - j } \ gamma (n - j + 1) } (x - a) ^ { (n - j) \ alpha } \ \ & = f (x) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f (a) } { \ alpha ^ { k } k! } (x - a) ^ { k \ alpha }. \ end { aligned } $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ({ } _ { a } ^ { c \ gamma } \ mathfrak { d } ^ { \ alpha } f (x) \ bigr) = { } _ { a } ^ { c (\ beta + \ gamma) } \ mathfrak { d } ^ { \ alpha } f (x); \ qquad { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl (^ { c \ gamma } \ mathfrak { d } _ { b } ^ { \ alpha } f (x) \ bigr) = ^ { c (\ beta + \ gamma) } \ mathfrak { d } _ { b } ^ { \ alpha } f (x). $ $
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in mathematics , katugampola fractional operators are integral operators that generalize the riemann – liouville and the hadamard fractional operators into a unique form . the katugampola fractional integral generalizes both the riemann – liouville fractional integral and the hadamard fractional integral into a single form and it is also closely related to the erdelyi – kober operator that generalizes the riemann – liouville fractional integral . katugampola fractional derivative has been defined using the katugampola fractional integral and as with any other fractional differential operator , it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators . 
in mathematics, katugampola fractional operators are integral operators that generalize the riemann – liouville and the hadamard fractional operators into a unique form. the katugampola fractional integral generalizes both the riemann – liouville fractional integral and the hadamard fractional integral into a single form and it is also closely related to the erdelyi – kober operator that generalizes the riemann – liouville fractional integral. katugampola fractional derivative has been defined using the katugampola fractional integral and as with any other fractional differential operator, it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators.
in mathematics, katugampola fractional operators are integral operators that generalize the riemann – liouville and the hadamard fractional operators into a unique form. the katugampola fractional integral generalizes both the riemann – liouville fractional integral and the hadamard fractional integral into a single form and it is also closely related to the erdelyi – kober operator that generalizes the riemann – liouville fractional integral. katugampola fractional derivative has been defined using the katugampola fractional integral and as with any other fractional differential operator, it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators.