On the generalized fractional derivatives and their Caputo modification

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1 Aville online t J. Nonliner Sci. Appl., 0 207), Reserch Article Journl Homepge: On the generlized frctionl derivtives nd their puto modifiction Fhd Jrd, Thet Adeljwd, Dumitru Blenu,c, Mthemtics Deprtment, Fculty of Arts nd Sciences, Çnky University, 06790, Etimesgut, Ankr, Turkey. Deprtment of Mthemtics nd Physicl Sciences, Prince Sultn University, P. O. Bo 66833, 586 Riydh, Sudi Ari. c Institute of Spce Sciences, Mgurele-Buchrest, Romni. ommunicted y X. J. Yng Astrct In this mnuscript, we define the generlized frctionl derivtive on A n γ[, ], the spce of functions defined on [, ] such tht γ f A[, ], where γ d d. We present some of the properties of generlized frctionl derivtives of these functions nd then we define their puto version. c 207 All rights reserved. Keywords: Riemnn-Liouville frctionl derivtives, puto frctionl derivtives, Hdmrd frctionl derivtives, puto-hdmrd frctionl derivtives, generlized frctionl integrl, generlized puto frctionl derivtives. 200 MS: 26A33, 34A08.. Introduction nd preliminries The frctionl clculus is n importnt developing field in oth pure nd pplied mthemtics [6, 20, 2]. Mny rel world prolems hve een investigted within the frctionl derivtives, prticulrly puto frctionl derivtive is etensively nd successfully used in mny rnches of sciences nd engineering [, 8 20]. We recll tht the system hving memory effect re etter descried within frctionl differentil opertors minly due to the non-loclity of these opertors [, 6, 9 2]. However, the non-loclity hs vrious forms. Therefore, the reserchers try to generlize the frctionl opertors to cpture the hidden spects of the rel non-locl phenomen. On the other hnd, mny reserchers work on frctionl integrls nd derivtives with non-locl nd non-singulr kernels [7, 9, 7, 22, 23]. One of the trends in frctionl is the discrete frctionl opertors which re proved to hve good pplictions in vrious fields [ 4, 6, 0]. From the clssicl frctionl clculus, we recll [6, 20, 2]. The left Riemnn-Liouville frctionl integrl of order α > 0 strting from hs the following form I α f)) Γα) t) α ft)dt. orresponding uthor Emil ddresses: fhd@cnky.edu.tr Fhd Jrd), tdeljwd@psu.edu.s Thet Adeljwd), dumitru@cnky.edu.tr Dumitru Blenu) doi: /jns Received

2 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), The right Riemnn-Liouville frctionl integrl of order α > 0 ending t > is defined y I α f)) Γα) t ) α ft)dt. The left Riemnn-Liouville frctionl derivtive of order α > 0 strting t is given elow D α f)) d d )n I n α f)), n [α] +. The right Riemnn-Liouville frctionl derivtive of order α > 0 ending t ecomes D α f)) d d )n I n α f)). The left puto frctionl of order α > 0 strting from hs the following form D α f)) I n α f n) )), n [α] +. The right puto frctionl derivtive of order α > 0 ending t ecomes D α f)) In α ) n f n) )). The Hdmrd type frctionl integrls nd derivtives were introduced in [5] s: The left Hdmrd frctionl integrl of order α > 0 strting from hs the following form J α f)) Γα) ln ln t) α ft)dt. The right Hdmrd frctionl integrl of order α > 0 ending t > is defined y J α f)) Γα) ln t ln ) α ft)dt. The left Hdmrd frctionl derivtive of order α > 0 strting t is given elow D α f)) d d )n I n α f)), n [α] +. The right Hdmrd frctionl derivtive of order α > 0 ending t ecomes D α f)) d d )n I n α f)). The uthors in [8, 2] defined the puto-hdmrd frctionl derivtives s: The left puto-hdmrd frctionl of order α > 0 strting from hs the following form D α f)) D α [ft) δ k f) log t )k ]), δ d d, nd in the spce A n δ [, ] {g : [, ] : δ [g)] A[, ]} equivlently y D α f)) J n α d d )n f)), n [α] +.

3 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), The right frctionl derivtive of order α > 0 ending t ws defined y D α f)) D α [ft) nd the spce A n δ [, ] equivlently y ) k δ k f) log t )k ]), D α f)) Jn α d d )n f)). For <, c R nd p <, define the function spce X p c, ) { ) /p f : [, ] R : f X p c t c ft) p dt } <. t For p, f X p c ess sup t [t c ft) ]. The generlized left nd right frctionl integrls in the sense of Ktugmpol) re defined y [3] nd I α, f)) Γα) I α, f)) Γα) t t ) α ft) dt t, ) α ft) dt t, respectively. In [3] it ws shown tht the integrl opertor I α, is ounded on the function spce X p c, ), c. Indeed, Also, the semigroup property nd I α, f X p c K f X p c, K α Γα) / u u c α ) α du, 0. I α, I µ, f I α+µ, f, f X p c, ), α > 0, µ > 0, p <, 0, ),, c R, c. The left nd right generlized frctionl derivtives of order α re defined y [4] D α, f)) γ n I n α, f)) γ n D α, f)) γ)n I n α, f)) γ)n t t ) n α ft) dt t, ) n α ft) dt t, respectively, where > 0. The uthors in [5] did define the puto version of the generlized frctionl derivtives. From the mthemticl view, we hve to consider the frctionl derivtives of functions elonging to specific spces. In this spect, this will help us to tret efficiently the numericl solutions of differentil equtions involving the generlized frctionl derivtives. The min purpose of this rticle is to present the generlized frctionl derivtives of solute differentile continuous nd differentile continuous functions nd consider some of their properties tht will led us to define the puto modifiction of these derivtives. The orgniztion of the pper is s follows. In Section 2, we present the generlized frctionl derivtives of functions in the spces A n γ[, ] nd n γ[, ]. In Section 3, we define the puto version of generlized frctionl derivtives. Section 4 contins the conclusion.

4 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), The generlized frctionl derivtives on the spce A n γ[, ] nd n γ[, ] From sic nlysis we recll tht f A[, ] is the spce of solutely continuous functions on [, ] if nd only if f) c + ϕt)dt, ϕt) L, ) see lso [6,..5]). Let s denote the spce of ll continuous rel Bnch-) vlued functions on [, ] y [, ] endowed with the norm f sup [,] f),. is the solute vlue in the rel line or the norm in the Bnch spce. Net we introduce spce of weighted continuous functions in which continuity t is not stressed. For 0 ɛ <, we define ɛ, [, ] {f :, ] R : ) ɛ f) [, ]}, 0, endowed with the norm f ɛ, ) ɛ f). ɛ, [, ] {f :, ] R : log log ) ɛ f) [, ]}, 0, endowed with the norm f ɛ, f ɛ,log log log ) ɛ f). The convention tht 0, [, ] [, ] is used. Definition 2.. Let [, ] e finite intervl, 0 ɛ < nd A[, ] e the set of solute continuous functions on [, ]. Then, we define A n γ[, ] n γ,ɛ[, ] { f : [, ] nd γ f A[, ], γ d }, A d γ[, ] A[, ]. { f : [, ] nd γ f [, ], γ n f ɛ, [, ], γ d }, d endowed with the norm f n γ,ɛ γk f + γ n f ɛ,. The convention n γ,0 [, ] n γ[, ] endowed with the norm f n γ n γk f is used. Lemm 2.2. Assume 0. A function f A n γ[, ] if nd only if f is presented in the form f) n )! t ) γ n f)t) t dt + γ k f)) Proof. Since f A n γ[, ], from Definition 2., γ f A[, ]. Hence one cn write γ f) for some function g L [, ] nd c 0 is constnt. Dividing oth sides of 2.2) y nd then integrting gives γ n 2 f) t t gu)du + c 0 )dt Dividing oth sides of 2.3) y nd then integrting once more yields γ n 3 f) t Repeting the sme procedure n 3 times, one gets f) t ) k. 2.) gt)dt + c 0, 2.2) ) 2 gt) 2 dt + c 0 2 t )gt)dt ) + c 0 + c. 2.3) ) 2 + c ) + c 2. ) gt) n )! dt + c k ) n k. 2.4) n k )!

5 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), It is cler from 2.) tht γ n f) g) nd from the proof tht c k γ n f), k 0,,, n. This ws the proof of the necessity. To proof the sufficiency it is enough to pply the opertor γ n to oth sides of 2.). We should mention tht Lemm 2.2 cn e dpted to the cse of the right integrtion s f) n )! t ) ) n γ n f)t) t dt + ) k γ k f)) ) k. Notice tht if we let 0 in the representtion 2.) then we get the representtion 3.) in [5] with µ 0. An nlogous lemm cn e written for the spce n γ,ɛ[, ] s follows. Lemm 2.3. Assume 0. A function f n γ,ɛ[, ] if nd only if f is presented in the form f) n )! t ) gt)dt + ) k, c k 2.5) where gt) ɛ, [, ]. Moreover, g) γn f)) nd c k γk f)). In prticulr, f n γ[, ] if nd only if f is presented in the form of 2.5), where gt) [, ]. Proof. The proof is similr to the proof of Lemm 2.2. Notice tht if we let 0 in the representtion 2.5) then we get the representtion..29) in [6]. Also, if we let in the representtions 2.) nd 2.5), respectively, then, we get the representtions..8) nd..23) in [6], respectively. Now we present formul for the generlized frctionl derivtives of functions in the spce f A n γ[, ]. Theorem 2.4. Let Reα) > 0, n Reα) + nd f A n γ[, ] or f n γ[, ]. Then the generlized frctionl derivtives of f eist lmost everywhere nd cn e represented in the form D α, f) D α, )n f) t ) n α γ n g)t)dt γ k f)) ) k α, t + 2.6) Γk α + ) t ) n α γ n g)t)dt t + ) k γ k f)) Γk α + ) ) k α. 2.7) Proof. Here we prove 2.6). Eqution 2.7) cn e proved similrly. Apply D α, to oth sides of eqution 2.4), then using property 2.) one gets t D α, f) n )! γn{ t ) n α t u ) γ n f)u)du u ) k α. + γ k f)) Γk α + ) Reversing the order of integrtion one gets D α, f) n )! γn{ t ) n α t u ) γ n f)u)dt u t ) k α. + γ k f)) Γk α + ) dt } t du } u

6 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), Using the chnge of vrile v t u, one otins u D α, f) n )! γn{ u ) 2n α γ n f)u)du } u v) n α v dv 0 γ k f)) ) 2.8) k α. + Γk α + ) Evluting the second integrl in 2.8), one gets D α, Γn) f) Γ2n α)n )! γn{ u ) 2n α γ n f)u)du } u γ k f)) ) 2.9) k α. + Γk α + ) Now pplying the opertor γ n on the integrl, 2.9) ecomes D α, f) u ) n α γ n f)u)du γ k f)) u + Γk α + ) ) k α. Theorem 2.5. Let α > β > 0, p nd c R. Then for f X p c, ) we hve D β, I α, f I α β, f, nd D β, Proof. If β m positive integer, then we hve Now, if m < β m we hve D m, I α, f) γ m[ Γα) γ m [ Γα ) γ m 2[ Γα 2). Γα m) I α m, f). I α, f Iα β, f. t ) α ft)dt ] t t ) α 2 ft)dt ] t t ) α 3 ft)dt ] t t ) α m ft)dt t D β, I α, f γ m I m β, I α, f γ m I α+m β, f I α β, f. This ws the end of the proof of the first formul. The second cn e proved in similr wy. Theorem 2.6 [4]). Let α > 0, p nd c R. Then for f X p c, ) where > 0, > 0, we hve D α, I α, f f, nd D α, I α, f f. Theorem 2.7. Let Reα) > 0, n [ Reα)], f L, ) nd I α, f A n γ[, ] I α, f An γ[, ]). Then n I α, D α, D α j, f) ) α j, )f) f) Γα j + ) j

7 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), Proof. I α, Dα, n )f) f) ) j D α j, f) ) α j. 2.0) Γα j + ) j I α, D α, )f) t ) α D α, ft) dt Γα) t t ) α γ n I n α, f)t) dt Γα) t [ t ) αγ γ n I n α, f)t) dt ] Γα + ) t [ t ) α d ) ] γ γ Γα + ) dt I n α, f)t) dt. Now integrting y prts repetedly leds to [ I α, D α, )f) γ Γα n + ) n j t ) α n I n α, ft) dt γ n j I n α, ))f) Γα + 2 j) [ γ I α n+, I n α, f) ) α j+ ] By using the semigroup property Theorem 4..) in [3], one otins n j t γ n j I n α, f))) Γα + 2 j) ) α j+ ]. [ I α, D α, )f) γ I, f) n j D α j, f) Γα + 2 j) ) α j+ ]. The result is reched fter pplying the opertor γ to the integrl. 2.0) is proved nlogously Lemm 2.8. Let Reα) 0 nd Reβ) > 0. Then, D α, t ) β ) ) D α, t D α, t t D α, ) β ) ) Γβ) Γβ α) Γβ) Γβ α) ) β α, 2.) ) β α, ) α i ) ) 0, i, 2,, [Reα)] +, 2.2) ) α i ) ) 0, i, 2,, [Reα)] +. Proof. Here we prove 2.) nd 2.2). The rest of the results re proved nlogously. D α, t ) β ) d ) n [ t ) n α t ) β dt ] d t

8 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), d ) n [ ) n α+β u) n α u β du ], where u t d 0 ) Γβ) d ) n [ ) n α+β ] Γβ + n α) using properties of the et function) d Γβ) Γβ + n α) ) β α Γβ) ) β α. Γβ + n α) Γβ α) Γβ α) This ws the end of the proof of 2.). Now, D α, t ) α i) d ) n [ d d d Γα i + ) Γn i + ) t ) n α t ) α i dt ] t ) n [ ) n i u) n α u α i du 0 d ) n [ ) n i ] 0. d The limiting cse 0 in Lemm 2.8 will led to the Hdmrd frctionl formuls with replced with ln/) nd ) replced with ln/). Also, the cse will result in the Riemnn- Liouville s formuls. ) 3. puto modifiction of the generlized frctionl derivtive Below we present the definition of the generlized puto frctionl derivtive of ny order which is different from the definition stted in [3]. Definition 3.. Let Reα) 0 nd n [Reα)] +. If f A n γ[, ], where 0 < < <, we define the left nd right generlized puto frctionl derivtives of f of order α y D α, f) D α,[ ft) D α, [ f) Dα, ft) γ k f) ) k γ k f) t t respectively. In cse 0 < Reα) <, we hve D α, f) D α,[ ] ft) f) ), ) k ] ), 3.) ) k ] ), nd D α, [ ] f) Dα, ft) f) ). Theorem 3.2. Let Reα) 0, n [Reα)] + nd f A n γ[, ], where 0 < < <. Then,. If α / N 0, D α, f) D α, f) t t ) n α γ n f)t)dt t I n α, γ n f)), 3.2) ) n α ) n γ n f)t)dt I n α, γ n f)). 3.3) t

9 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), If α N Prticulrly, D α, f γ n f, D 0, f f, D α, f )n γ n f. 3.4) D 0, f f. Proof. 3.2) nd 3.3) re consequences of Theorem 2.4. Now, when α n we hve f) I n, γ n f)) + γ k f) t ) k. From Lemm 2.2, one gets D α, f γ n f. The second prt of 3.4) cn e proved likewise. Theorem 3.3. Let Reα) 0, n [Reα)] + nd f n γ[, ], where 0 < < <. Then, D α, f nd D α, f cn e represented s in 3.2) nd 3.3), respectively if α / N 0. If α N, 3.4) holds. Moreover D α, f nd D α, f re continuous on [, ] nd Proof. The representtion of since we hve D α, f) 0, D α, f ) ) D α, f nd D α, f cn e proved s in the proof of Theorem 2.4. Now, D α, f) t ) n α γ n f)t)dt t, D α, γ n f f n Reα). n Reα)) Thus the continuity is proved. The identities in 3.5) hold since D α, γ n f f n Reα), 3.6) n Reα)) nd D α, f γ n f n Reα). 3.7) n Reα)) Theorem 3.4. Let Reα) 0, n [Reα)] +. If α / N, D α, is ounded from the spce n γ[, ] to the spce [, ] {g [, ] : g) 0} nd D α, is ounded from the the spce n γ[, ] to the spce [, ] {g [, ] : g) 0} nd nd If α N, D α, nd D α, D α, γ n f n γ f n Reα), 3.8) n Reα)) D α, f γ n f n γ n Reα). 3.9) n Reα)) re ounded from the spce n γ[, ] to the spce [, ] nd D n, f f n γ, D n, f f n γ. 3.0) Proof. Equtions 3.8) nd 3.9) follow from 3.6) nd 3.7), respectively. strightforwrd. The inequlities in 3.0) re

10 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), Below we stte the inverse properties. Theorem 3.5. Let Reα) 0, n [Reα)] + nd f [, ].. If Reα) 0 or α N, then 2. If Reα) 0 nd Reα) N, then D α, I α, f) f), D α, Iα, f) f). 3.) D α, I α, f) f) I α+ n, f) ) n α, 3.2) Proof. From 3.), one hs D α, Iα, Iα+ n, f) f) f) ) n α. 3.3) D α, I α, f) D α, I α, γ k I α, f)) ) k. f) 3.4) Γk α + ) From Theorem 2.7 nd Theorem 3.2 one hs γ k I α, )f I α k, f nd D α, I α, f f, respectively. Thus 3.4) reds D α, I α, I α k, )f) ) k. f) f) 3.5) Γk α + ) On the other hnd, it should e esy to verify tht I α k, )f) 0, ecuse of the following estimte I α k, f ) Reα) k. f) 3.6) Γα k)reα) k) This is the end of the proof of the first identity in 3.). The second identity is proved in similr wy. Now if α m + iβ, β 0, then we hve n m + 2 nd γ k I α, )f I α k, f is vlid when k 0,, 2,, m. Becuse of the estimte 3.6) we hve I α k, )f) 0, k 0,, 2,, m. Sustituting in 3.5), we get 3.2). 3.3) is proved similrly. Theorem 3.6. Let f A n γ[, ] or n γ[, ] nd α. Then I α, D α, f) f) I α, D α, In prticulr, if 0 < α, we hve f) f) γ k f)) ) k γ k f)) I α, D α, f) f) f), I α, ) k, ) k. D α, f) f) f). 3.7) Proof. The proof is done y using the semigroup property [3, Theorem 4.] nd Theorem 2.7 α n) I α, D α, f) I α, I n α, γ n f I n, γ n f) Eqution 3.7) cn e proved nlogously. γ k f)) ) k.

11 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), Net we present the composition rule for two generlized puto frctionl derivtives. Theorem 3.7. Let f A m+n γ [, ], 0 < < <, α 0 nd β 0 such tht n < α n nd m < β m. Then D α, D β, f) D α+β, f). Proof. Let us ssume tht m n. Thus m n + k, k 0,, 2,, m n. Then the proof cn e done y using Theorem 2.7, Theorem 3.2 nd [3, Theorem 4.]. In fct, D α, D β, f) I n α, γ n I m β, γ m f) I n α, γ n I n+k β, γ n+k f) I n α, γ n I n β, I k, γ n+k f) I n α, D β, I k, γ n+k f) I n α β, I β, D β, I k, γ n+k f)γ n+k f) I n α β,[ n I k, γ n+k γ n j I n β, I k, γ n+k f)) ) j ] f) Γβ j + ) I n α β,[ I k, γ n+k f) j n j I n α β,[ ] I k, γ n+k f) 0 I n+k α β, γ n+k f) D α+β, f). Eqution 3.7) is proved using similr rguments. γ n j D β, f)) Γβ j + ) ) j ] Lemm 3.8. Let Reα) > 0, n [Reα) + ] + nd Reβ) > 0. Then D α, t ) β ) Γβ) ) β α, Γβ α) Reβ) > n, D α, t ) β ) Γβ) ) β α, Γβ α) Reβ) > n. 3.8) Proof. D α, t ) β ) Γβ) t Γβ n) Γβ) ) β α Γβ n) Γβ) Γβ n) Γβ) ) β α. Γβ α) Eqution 3.8) cn e proved likewise. t ) n α t 0 ) n α [ t d dt ) β dt t ) n t ) β ] dt t u) n α u β du, u t ) β α Γβ n) Γβ α) differentition inside the integrl) using properties of the et function) Lemm 3.9. D α, ) k 0, D α, ) k) 0, k 0,, 2,, n.

12 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), Prticulrly Proof. D α, t The rest formuls cn e proved similrly. D α, ) 0, D α, ) 0. ) k) I n α,[ t d dt) n t ) k ] ) I n α, [0]) 0. From Definition 3. nd Lemm 3.8, we cn set the following reltion etween generlized puto nd Riemnn derivtives: Theorem 3.0. For nd α > 0 nd 0, we hve nd D α, f) D α, f) D α, γ k f) Γk + α) ) k α, f) Dα, f) k )γ k f) Γk + α) ) k α. At lst, we give the reltion etween the generlized puto frctionl derivtives nd the known ones. Theorem 3.. Let α, Reα) > 0 nd n [Reα)] +. Then lim 0 lim 0 lim lim D α, f) D α, f) D α, f) D α, f) t) n α f n) t)dt D α f). 3.9) t) n α ) n f n) t)dt D α f). 3.20) log log t) n α t d dt )n f)t) dt t D α f). 3.2) log t log ) n α ) n t d dt )n f)t) dt t D α f). 3.22) Proof. The limits in 3.9) nd 3.20) re evluted replcing y 0 directly. While, the limits in 3.20) nd 3.2) re evluted y using the L Hôspitl rule. It should e noted tht the derivtives on the right hnd sides in 3.9) nd 3.20) re respectively the left nd right puto derivtives [6, 20]. While, the derivtives on the right hnd sides in 3.2) nd 3.22) re respectively the left nd right puto-hdmrd derivtives developed in [8] nd [2]. 4. onclusions The fundmentl issue of the frctionl opertors nd their generlized versions is to define them correctly in the right spce of functions. In this pper, we defined the generlized frctionl derivtives of functions in the spces of solutely differentile continuous nd differentile continuous functions. Since puto derivtives descrie etter some physicl prolems involving memory effect, we defined the puto version of the generlized frctionl derivtives. We elieve tht this puto

13 F. Jrd, T. Adeljwd, D. Blenu, J. Nonliner Sci. Appl., 0 207), version of the generlized frctionl derivtive would e useful for reserchers working on modeling rel world phenomen descried y frctionl opertors. Finlly, we noticed tht the limiting cse s 0 leds to the Hdmrd nd puto-hdmrd results y noting tht lim 0 ) ln ) nd lim 0 ) ln ). Also, the cse will result in Riemnn-Liouville s nd puto frctionl derivtives. References [] T. Adeljwd, Dul identities in frctionl difference clculus within Riemnn, Adv. Difference Equ., ), 6 pges. [2] T. Adeljwd, On delt nd nl puto frctionl differences nd dul identities, Discrete Dyn. Nt. Soc., ), 2 pges. [3] T. Adeljwd, D. Blenu, Frctionl differences nd integrtion y prts, J. omput. Anl. Appl., 3 20), [4] T. Adeljwd, D. Blenu, Discrete frctionl differences with nonsingulr discrete Mittg-Leffler kernels, Adv. Difference Equ., ), 8 pges. [5] R. Almeid, A. B. Mlinowsk, T. Odzijewicz, Frctionl differentil equtions with dependence on the puto- Ktugmpol derivtive, J. omput. Nonliner Dyn., 206), pges. [6] F. M. Atıcı, S. Şengül, Modeling with frctionl difference equtions, J. Mth. Anl. Appl., ), 9. [7] M. puto, M. Frizio, A new definition of frctionl derivtive without singulr kernel, Progr. Frct. Differ. Appl., 205), [8] Y. Y. Gmo F. Jrd, D. Blenu, T. Adeljwd, On puto modifiction of the Hdmrd frctionl derivtives, Adv. Difference Equ., ), 2 pges., 3 [9] F. Go, X.-J. Yng, Frctionl Mwell fluid with frctionl derivtive without singulr kernel, Therm. Sci., ), S87 S877. [0]. Goodrich, A.. Peterson, Discrete frctionl clculus, Springer, hm, 205). [] R. Hilfer Ed.), Applictions of frctionl clculus in physics, World Scientific Pulishing o., Inc., River Edge, NJ, 2000). [2] F. Jrd, T. Adeljwd, D. Blenu, puto-type modifiction of the Hdmrd frctionl derivtives, Adv. Difference Equ., ), 8 pges., 3 [3] U. N. Ktugmpol, New pproch to generlized frctionl integrl, Appl. Mth. omput., 28 20), , 2, 3, 3, 3 [4] U. N. Ktugmpol, A new pproch to generlized frctionl derivtives, Bull. Mth. Anl. Appl., 6 204), 5., 2.6 [5] A. A. Kils, Hdmrd-type frctionl clculus, J. Koren Mth. Soc., ), , 2 [6] A. A. Kils, H. M. Srivstv, J. J. Trujillo, Theory nd pplictions of frctionl differentil equtions, North-Hollnd Mthemtics Studies, Elsevier Science B.V., Amsterdm, 2006)., 2, 2, 3 [7] J. Losd, J. J. Nieto, Properties of new frctionl derivtive without singulr kernel, Progr. Frct. Differ. Appl., 205), [8] J. T. Mchdo, V. Kirykov, F. Minrdi, Recent history of frctionl clculus, ommun. Nonliner Sci. Numer. Simul., 6 20), [9] R. L. Mgin, Frctionl clculus in ioengineering, Begell House Pulishers, T, 2006). [20] I. Podluny, Frctionl differentil equtions, An introduction to frctionl derivtives, frctionl differentil equtions, to methods of their solution nd some of their pplictions, Mthemtics in Science nd Engineering, Acdemic Press, Inc., Sn Diego, A, 999)., 3 [2] S. G. Smko, A. A. Kils, O. I. Mrichev, Frctionl integrls nd derivtives, Theory nd pplictions, Edited nd with foreword y S. M. Nikolskiĭ, Trnslted from the 987 Russin originl, Revised y the uthors, Gordon nd Brech Science Pulishers, Yverdon, 993). [22] X.-J. Yng, D. Blenu, H. M. Srivstv, Locl frctionl integrl trnsforms nd their pplictions, Elsevier/Acdemic Press, Amsterdm, 206). [23] X.-J. Yng, F. Go, J. A. Tenreiro Mchdo, D. Blenu, A new frctionl derivtive involving the normlized sinc function without singulr kernel, ArXiv, ), pges.