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1 Segi Rhm, Mohmd Rfi (2014) Iegrl rform mehod for olvig frciol dymic equio o ime cle. Abrc d Applied Alyi, 2014 (2014). pp ISSN Acce from he Uiveriy of Noighm repoiory: hp://epri.oighm.c.uk/27710/1/iegrl%20rform%20mehod%20for %20frciol%20dymic%20equio.pdf Copyrigh d reue: The Noighm epri ervice mke hi work by reercher of he Uiveriy of Noighm vilble ope cce uder he followig codiio. Thi ricle i mde vilble uder he Creive Commo Aribuio licece d my be reued ccordig o he codiio of he licece. For more deil ee: hp://creivecommo.org/licee/by/2.5/ A oe o verio: The verio preeed here my differ from he publihed verio or from he verio of record. If you wih o cie hi iem you re dvied o coul he publiher verio. Plee ee he repoiory url bove for deil o cceig he publihed verio d oe h cce my require ubcripio. For more iformio, plee coc epri@oighm.c.uk

2 Abrc d Applied Alyi, Aricle ID , 10 pge hp://dx.doi.org/ /2014/ Reerch Aricle Iegrl Trform Mehod for Solvig Frciol Dymic Equio o Time Scle Mohmd Rfi Segi Rhm School of Applied Mhemic, The Uiveriy of Noighm Mlyi Cmpu, Jl Brog, Semeyih, Selgor, Mlyi Correpodece hould be ddreed o Mohmd Rfi Segi Rhm; mohd.rfi@oighm.edu.my Received 23 April 2014; Acceped 6 Augu 2014; Publihed 31 Augu 2014 Acdemic Edior: H Elyeb Copyrigh 2014 Mohmd Rfi Segi Rhm. Thi i ope cce ricle diribued uder he Creive Commo Aribuio Licee, which permi urericed ue, diribuio, d reproducio i y medium, provided he origil work i properly cied. We iroduce bl ype Lplce rform d Sumudu rform o geerl ime cle. We iveige he properie d he pplicbiliy of hee iegrl rform d heir efficiecy i olvig frciol dymic equio o ime cle. 1. Iroducio I i kow h he mehod coeced o he employme of iegrl rform re very ueful i mhemicl lyi. Thoe mehod re uccefully pplied o olve differeil d iegrl equio, o udy pecil fucio, d o compue iegrl. Oe of he more widely ued iegrl rform i he Lplce rform defied by he followig formul: L F(z) f () e z d, z C. (1) 0 The fucio F of complex vrible i clled he Lplce rform of he fucio f. Wugl[1] iroducedew iegrl rform clled Sumudu rform defied by he followig formul: S {f} (u) 1 u f (u) e /u d, u ( τ 1,τ 2 ), (2) 0 d pplied o he oluio of ordiry differeil equio i he corol egieerig problem (ee lo [2]). I ppered like he modificio of he Lplce rform. The Sumudu rform rivl he Lplce rform i problem olvig. I mi dvge i he fc h i my be ued o olve problem wihou reorig o ew frequecy domi, becue i preerve cle d ui properie. The heory of ime cle clculu w iiied by Hilger [3](eelo[4]). Thi heory i ool h uifie he heorie ofcoiuouddicreeimeyem.iiubjecofrece udie i my differe field i which dymic proce c be decribed wih coiuou d dicree model. For he deiled iformio o heory of ime cle clculu, we refer o [5, 6]. The del Lplce rform o rbirry ime cle (T) i iroduced by Boher d Peero i [7](eelo [8]) by he followig formul: L {x}(z) : x () e z (σ (), 0 )Δ, z D {x}, (3) 0 where D{x} coi of ll complex umber z Cfor which he improper iegrl exi d for which 1+μ()z0for ll T.Iimilrfhio,Agwel.i[9]iroducehe Sumudu rform o rbirry ime cle T,byhefollowig formul: S {x}(z) : 1 z x () e σ (1/z) (, 0)Δ, (4) 0 for z D{x}, whered{x} coi of ll complex umber z Cforwhich he improper iegrl exi d for which 1+μ()/z 0 for ll T. NoehifT R (for rel lyi), (3) (1) d(4) (2) 0 0.Iheceof T Z (for dicree lyi), we hve Z {x}(z+1) L {x}(z), (5) z+1 where Z{x}(z) 0 x()z i he clicl Z-rform, which will be ued o olve higher order lier forwrd

3 2 Abrc d Applied Alyi differece equio (ee [7]). Similrly, formul (3)clo be exeded o oher priculr dicree eig uch T q Z, q > 1 (which h impor pplicio i quum heory), T hz (i h-clculu) (ee [10]), d lo T T 0 (q,h) (i (q, h)-clculu) (ee [11]). Likewie, he del Sumudu rform o ime cle o oly c be pplied o ordiry differeil equio whe T R d o forwrd differece equio whe T hz(h>0)bu lo c be pplied for q-differece equio whe T q Z d o differe ype of ime cle like T hzd T T ; for he pce of he hrmoic umber, ee [9]. Coiuou frciol clculu i field of mhemic udyhgrowouofherdiioldefiiioofhe clculu iegrl d derivive operor. Frciol differeiio h plyed impor role i vriou re rgig from mechic o imge proceig. Their fudmel reul hve bee urveyed, for exmple, i he moogrph [12, 13]. O he oher hd, dicree frciol clculu i very ew re for ciei. Foudio of hi heory were formuled i pioeerig work by Agrwl [14] ddíz d Oler [15, 16], where bic pproch, defiiio, d properie of he heory of frciol um d differece were repored (ee lo [17, 18]). Recely, erie of pper coiuig hi reerch h ppered (ee e.g., [19 26] d he referece cied herei). The exeio of bic oio of frciol clculu o oher dicree eig w performed i [27, 28]. I hee pper, he uhor ofe preferred he power fucio oio bed o he ime cle heory, which eily expoe imilriie mog he reul i q-clculu, h-clculu, (q, h)- clculu, d he coiuou ce. However, hi oio w employed oly formlly, ice here w o geerl ime cle defiiio of he power fucio d herefore he chieved reul could o be geerlized o oher ime cle. O hi ccou, ome ide regrdig fudmel properie which hould be me by power fucio o ime cle were oulied i [29]. I [30], he uhor iroduced frciol derivive d iegrl o ime cle vi he geerlized Lplce rform. However, hi pproch uffer by ome echicl difficulie, coeced o he ivere Lplce rform (ee [8]). Recely, i [31, 32](eelo[33]), he uhor idepedely uggeed xiomic defiiio of power fucio o rbirry ime cle. The im of hi pper i o iroduce he bl ype Lplce rform d Sumudu rform, heir properie, d pplicbiliy d i efficiecy i olvig frciol dymic equio o rbirry ime cle. Of coure, i i poible o coider lo he del ype Lplce d Sumudu rform (3) d (4), repecively; however, he bl verio eem be more uible for frciol clculu oulied, for exmple, i [27, 28, 34]. Thi pper i orgized follow. I Secio 2, we recll bic of he ime cle heory d he foudio of frciol clculu o ime cle. Secio 3 i devoed o bl Lplce rform, i properie, covoluio heorem, d exmple of oluio of frciol dymic equio o ime cle i erm of Mig-Leffler fucio. Filly, i Secio4, we iroduce bl Sumudu rform d i properie o rbirry ime cle. A cloe reliohip bewee bl Sumudu rform d bl Lplce rform d everl impor reul were obied. Thi ecio eded up wih olvig ome frciol dymic equio wih bl Sumudu rform mehod. 2. Prelimirie AimecleT i rbirry oempy cloed ube of he rel umber R. The mo well-kow exmple re T R, Z, dq Z : {q : Z} {0},whereq > 1.LeT hve righ-cered miimum m d defie T κ : T {m}; oherwie, e T κ T. If, b T wih <b, we deoe by [, b] T he cloed iervl [, b] T. The bckwrd jump operor ρ:t T i defied by ρ () : up { T :<}, (6) d he bckwrd griie fucio ] : T κ [0,) i defied by ]() : ρ(). For deil d dvceme o ime cle, ee he moogrph [3, 5, 7, 35 37]. For f : T R d T κ, he bl-derivive (briefly, he -derivive) [21] off, deoed by f (), i he umber (provided i exi) wih he propery h, give y ε>0, here exi eighborhood U of uch h f(ρ()) f() f () (ρ () ) ε ρ () U. (7) For T R, f () f ()) i he uul derivive; for T Z, he -derivive i he bckwrd differece operor, f () f() f( 1) f(). Afuciof:T C i lef-dee coiuou or ldcoiuou provided i i coiuou lef-dee poi i T d i righ-ided limi exi (fiie) righ-dee poi i T. IfT R, hef i ld-coiuou if d oly if f i coiuou. The e of ld-coiuou fucio f:t C will be deoed by C ld (T, C) d he e of fucio f:t C h re -differeible d whoe derivive re ld-coiuou i deoed by C 1 ld (T, C). I i kow from [5]hiff C ld (T, C), he here exi fuciofuch h F () f(). I hi ce, we defie he Cuchy iegrl by b f () F (b) F(), for, b T. (8) Le [, b] T d f C ld (T, C).IfT R,he b b f () f () d, (9) where he righ-hd ide iegrl i he Riem iegrl from clculu d if T Z,he b b f () f (). (10) +1

4 Abrc d Applied Alyi 3 For f, g C ld (T, C) d, b T, he iegrio by pr formul i give by b f(ρ())g () [f ()()] b b f () g (). (11) Afuciof C ld (T, C) i clled ]-regreive if 1 ]f 0o T κ d poiively ]-regreiveifiirelvlued d 1 ]f > 0 o T κ.theeof]-regreive fucio d he e of poiively ]-regreive fucio re deoed by R ] (T, C) d R + ] (T, C), repecively,dr ] (T, C) i defied imilrly. For impliciy, we deoe by c R ] (T, C) he e of complex ]-regreive co d, imilrly, we defie he e c R + ] (T, C) d cr ] (T, C). Le f C ld (T, C). The he bl expoeil fucio e f (, ) idefiedobeheuiqueoluioofhefollowig iiil vlue problem: x fx o T κ, x () 1 for ome fixed T.Leh>0;e (12) C h : {z C :z 1 h }, Z h : {z C : π h < Im (z) π (13) h }, d C 0 : Z 0 : C. Forh > 0,heHilgerrelprd imgiry pr of complex umber re give by R h (z) : 1 (1 1 hz ), (14) h I h (z) : 1 Arg (1 hz), (15) h repecively, where Arg deoe he priciple rgume fucio; h i, Arg : C ( π,π] R,dleR 0 (z) : R(z) d I 0 (z) : I(z). For y fixed complex umber z, hehilger rel pr R h (z) i odecreig fucio of h [0,)(ee [38]). For h 0, we defie he ]-cylider rformio ξ h : C h Z b by ξ h (z) : { z, h 0, { 1 { h Log (1 hz), h > 0 (16) for z C h. The, he bl expoeil fucio c lo be wrie i he followig form: e f (, ) : exp { ξ ](τ) (f (τ)) τ} for, T. (17) I i kow h he bl expoeil fucio e f (, ) i ricly poiive o T, providedf R + ] (T, C) (ee Theorem 3.18 [6]). For f, g R ] (T, C), he]-circul plu d he ]- circle miu re defied by f ] g:f+g ]fg, f ] g: f g 1 ]g, (18) repecively. For furher deil o bl expoeil fucio, we refer o [5]. We recll he oio of Tylor moomil iroduced i [39](eelo[7]). Thee moomil ĥ : T T C, N 0, re defied recurively follow: d, give ĥ for N 0,wehve ĥ 0 (, ) 1, T (19) ĥ +1 (, ) ĥ (τ, ) τ, T. (20) Exmple 1. For he ce T R,wehve ĥ (, ) ( )k, k!, R. (21) For he ce T Z,wehve ĥ (, ) ( )()! j0 ( j),, Z, (22)! where () ( + 1)( + 2) ( + 1). For he ime cle T q Z for ome q>1,wehve q k ĥ (, ),, k qz. (23) j0 qj Lemm 2 (bl Cuchy formul [37]). Le Z +,, b T, d le f:t C be -iegrble o T : [, b] T.If T, he f () ĥ (, ρ (τ)) f (τ) τ. (24) The formul (24) i corer oe i he iroducio of he bl frciol iegrl α f() for α>0.however, i require reoble d url exeio of dicree yem of moomil (ĥ, N 0 ) o coiuou yem (ĥα,α R + ).However,heclculioofĥ for > 1 i difficul k which eem o be werble oly i ome priculr ce (ee Exmple 1). Recely, [31, 32] idepedely uggeed quie imilr xiomic defiiio of ime cle power fucio. I [40], he uhor hve coidered he power fucio d eeil of frciol clculu o ioled ime cle. The defiiio below follow from [31]. Defiiio 3. Le, T d α, β > 1. Theimecle power fucio ĥα(, ) re defied fmily of oegive fucio ifyig (i) ĥα(, ρ(τ))ĥβ(τ, ) τ ĥα+β+1(, ) for ; (ii) ĥ0(, ) 1 for ; (iii) ĥα(, ) 0 for α (0, 1). Furher, we hve he followig.

5 4 Abrc d Applied Alyi Defiiio 4. Le α 0, β>0,d, b T. Theforf C ld ([, b] T, C) oe defie he followig. (i) The frciol iegrl of order α>0wih he lower limi ( α f) () : ĥ α 1 (, ρ (τ)) f (τ) τ (25) d for α0oe pu ( 0 f)() f(). (ii) The Riem-Liouville frciol derivive of order β>0wih lower limi ( β f) () : [ (m β) f] m (), [σ (),b] T, (26) where m[β]+1. (iii) The Cpuo frciol derivive C γ f() (γ > 0) o [σ(), b] T i defied vi he Riem-Liouville frciol derivive by where m[γ]+1. C γ f () : ( (m γ) f m ) (), (27) 3. Nbl Lplce Trform Noe h below we ume h z c R ] ;he( ] z) c R ] d herefore e ] z(, 0 ) i well defied o T. Fromowowe ume h T i ubouded bove. The followig heorem i cocerig he ympoic ure of he bl expoeil fucio. To hi ed, we defie he miiml griie fucio ] : T [0,) by ] () : if τ [,) T ] (τ) for T, (28) d for h 0d λ R, we defie C h (λ) : {z C h : R h (z) >λ}. (29) Theorem 5 (decy of he bl expoeil fucio). Le T, λ c R + ] ([, ) T, C). (30) The, for y z C ] ()(λ),wehvehefollowigproperie: (i) e λ ] z(, ) e λ ] R ] ) for ll [,) ()(z)(, T, (ii) lim e λ ] R ] (, ) 0, ()(z) (iii) lim e λ ] z(, ) 0. Proof. The proof i imilr o Theorem 3.4 of [38]. Defiiio 6 (expoeil order). Le T. A fucio f C ld (T, C) h expoeil order α o [, ) T,if (i) α c R + ] ([, ) T, C), (ii) here exi K>0,uchh f() K e α (, ) for ll [,) T. Lemm 7. Le T d f C ld ([, ) T, C) be fucio of expoeil order α.the, lim f () e ] z (, ) 0, (31) where z C ] ()(α). Proof. I follow h f () e ] z (, ) K e α (, ) e z (, ) K e α z (, ) (32) for ll [, ) T d ome K>0.ByTheorem 5(iii) d leig i (32), we ge (31). Thi complee he proof. Defiiio 8. Le f C ld (T, C) be fucio. The, he - Lplce rform L {f}( ) bou he poi T of he fucio f i defied by L : e ] z (ρ (),)f() for z D ] {f}, (33) where D ] {f} coi of ll complex umber z R ] (T, C) for which he improper iegrl exi. Theorem 9. Le f C ld ([, ) T, C) be of expoeil order α.the,he -Lplce rform L {f}( ) exi o C ] ()(α) d coverge boluely. Proof. The proof i imilr o Theorem 5.1 i [38]. Theorem 10 (lieriy of he rform). Le f 1,f 2 C ld ([, ) T, C) be of expoeil order α 1,α 2,repecively. The, for y c 1,c 2 R,wehve L {c 1 f 1 +c 2 f 2 } (z) c 1 L {f 1 } (z) +c 2 L {f 2 } (z) (34) for ll z C ] ()(mx{α 1,α 2 }). Proof. The proof follow from he lieriy propery of he - iegrl (ee Theorem 8.47(i) i [5]). Theorem 11 (rform of derivive). Le f C ld ([, ) T, C) be fucio of expoeil order α.the,oeh L {f } (z) z F (z) f(), (35) for ll z C ] ()(α),where F deoe L {f}. Proof. By uig iegrio by pr formul (11), we ge L {f } (z) e ] z (ρ (),)f () [ e ] z(, )f()] +z e ] z (ρ (),)f() f() +z F (z), for ll z C ] ()(α).thicompleeheproof. By iducio, we hve he followig reul. (36)

6 Abrc d Applied Alyi 5 Corollry 12. Le f C ld ([, ) T, C) be fucio of expoeil order α.the for y N,oe h L {f } (z) z F (z) z k 1 f k () (37) for ll z C ] ()(α). Defiiio 13 (ee [41]). For give f:[ 0,) T oluio of he hifig problem C, he u (, ρ ()) u (, ),, T, 0, (38) u(, 0 )f(), T, 0 i deoed by f diclledhehif(ordely)off. I hi ecio, we will ume h he problem (38)h uiqueoluio f for give iiil fucio f d h he fucio f, g, d he complex umber z re uch h he operio fulfilled re vlid. Defiiio 14 (ee [41]). For give fucio f, g : T C, heir covoluio f gi defied by f () g() f(,ρ(τ))g(τ) τ,, T, (39) where f i he hif of f iroduced i Defiiio 8. We e he followig reul wihou proof, ice he proofofhemreimilrohoei[6]. Theorem 15. The covoluio i ociive; h i, (f g) hf (g h). (40) Theorem 16. If f i -differeible, he (f g) f g+f() g (41) d if g i bl-differeible, he (f g) f g +fg(). (42) Corollry 17. The followig formul hold: f(, ρ (τ)) τ f (τ) τ. (43) Theorem 18 (covoluio heorem). Suppoe f, g : T R re loclly -iegrble fucio o T d heir covoluio f gi defied by (39).The, L {f g} (z) L L {g} (z), z D ] {f} D ] {g}. (44) Le Φ α () ĥα 1(, ).The,bymeofcovoluio,he bl operor i Defiiio4 c be reed ( α f) () Φ α () f(), ( β f) () (Φ m β () f()), m [β]+1, ( C γ f) () Φ m γ () f m, m [γ]+1. (45) I i kow [5, 7]h,forllk N 0 d z D ], 0 L {ĥk (, 0 )} (z) 1. (46) zk+1 I geerl, we hve Theorem 19. For α>0d z D ], hold. L {ĥα (, )} (z) 1 z α+1 (47) Proof. Fir, we wrie Defiiio 3(i) i covoluio form; h i, ĥ α (, ρ (τ)) ĥβ (τ, ) τ ĥα (, ) ĥβ (, ). (48) The, obviouly ĥ α (, ) ĥβ (, ) ĥα+β+1 (, ). (49) We how h (47) ifie he Lplce rform of (49). Le β 0. Tkig Lplce rform o he lef-hd ide followed by pplyig covoluio heorem (39)yield L {ĥα (, ) ĥ0 (, )} ( 1 z α+1 )(1 z ) 1. (50) zα+2 Bu, from he righ ide of (50), we hve 1 z α+2 1 z (α+1)+1 L {ĥα+1 (, )} (z). (51) Hece he reul follow from (50)d(51). Thi complee he proof. From (45), kowig L {Φ α } (z) L {ĥα 1 (, )} (z) 1, (52) zα we hve (by kig ) he followig reul. Theorem 20. For α>0, L { α f} (z) z α F (z). (53) For he Riem-Liouville frciol derivive derivive (26), we hve he followig reul. Theorem 21. For β>0d m[β]+1, L { β f} (z) z β m 1 F (z) Proof. Wrie (26) z m k 1 [ (m β) f] k (). (54) ( β f) () g m (), where g () ( (m β) f) (). (55)

7 6 Abrc d Applied Alyi From (53), we hve G (z) z (m β) F (z). (56) Thu, by (37) d(56), we hve L { β f} (z) L {g m } (z) z m m 1 G (z) z β m 1 F (z) z m k 1 g k () z m k 1 [ (m β) f] k (). (57) The Lplce rform (54)i equivle o he followig oe: L { β f} (z) z β F (z) l j1 z j 1 ( β j f) (), l 1<β l. (58) The bl Lplce rform of Cpuo frciol derivive of order α i give follow. Theorem 22. For α>0d m[α]+1, L { C α f} (z) z α m 1 F (z) Proof. Wrie (27) z α k 1 f k (). (59) ( C α f) () ( (m α) g) (), where g () f m (). (60) By followig (53)d(37), we ge L { C α f} (z) L { (m α) g} (z) z (m α) G (z) z (m α) [z m m 1 F (z) z α m 1 F (z) z α k 1 f k (). z m k 1 f k ()] (61) Now, le u coider he geerlized Mig-Leffler fucio o ime cle (ee [28, 42]). Defiiio 23. Le α, β, λ R d, T. The ime cle Mig-Leffler fucio, E,λ α,β (), i defied by he followig erie expio: α,β () λ k ĥ αk+β 1 (, ). (62) E,λ I he followig heorem, we give he Lplce rform of geerlized Mig-Leffler fucio o ime cle. Theorem 24. For α, β, λ R d, T,iholdh provided λ/z α <1. L z β α,β ()} (z) 1 λz α, (63) Proof. By uig Theorem 10 d he relio (47), we obi L α,β ()} (z) L { λ k ĥ αk+β 1 (, )} λ k L {ĥαk+β 1 (, )} z 1 kα+β z {1 + λ β z λ k z β 1 λz α, α + λ2 + } z2α λ z α <1. (64) Exmple 25. Coider he followig iiil vlue problem: ( α y) () λy() f(), [σ (),b] T, (65) ( α j y) () b j, (b j R;j1,2,...,[α]). (66) By kig Lplce rform of boh ide of (65) duig (58), we ge z α L {y} (z) j1 L, L {y} (z) j1 z j 1 ( α j y) () λ L {y} (z) z α 1 λz α z j 1 ( α j z α y) () + 1 λz α L j1 b j L α,α j+1 ()} (z) + L α,α ()} (z) L j1 b j L α,α j+1 ()} (z) + L α,α () f()}. (67)

8 Abrc d Applied Alyi 7 Thu, we hve y () j1 j1 b j E,λ α,α j+1 b j E,λ α,α j+1 () + () +E,λ α,α () f() E ρ(τ),λ α,α () f (τ) τ. (68) The bove exmple coicide wih he ce T R (ee [43]). Now, we coider he Cuchy problem for dymic equio wih he bl ype Cpuo frciol derivive. Exmple 26. Coider he followig iiil vlue problem: ( C α y) () λy() f(), [σ (),b] T, (69) y j () c j, (c j R;j0,1,2,...,). (70) By kig Lplce rform of boh ide of (69) duig (59), we ge z α L {y} (z) j0 L, L {y} (z) j0 z α j 1 y j () λ L {y} (z) z α 1 λz α c j z α j 1 z α + 1 λz α L j0 c j L α,j+1 ()} (z) + L α,α ()} (z) L j0 Thu, we hve y () c j L α,j+1 ()} (z) + L α,α () f()}. j0 j0 c j E,λ α,j+1 c j E,λ α,j+1 () + () +E,λ α,α () f() E ρ(τ),λ α,α () f (τ) τ. (71) (72) The l exmple clerly coicide wih he rel couer pr; ee [43]. 4. Nbl Sumudu Trform I [9],heuhoriroduceddudiedhe(del)Sumudu rform o ime cle. My impor reul were produced d pplied o dymic equio o ime cle. I hi ecio, we will coider he bl Sumudu rform. Mo of he reul were coed from [9, 44, 45]wihouproof iceheirproofreimilr. Defiiio 27. Le f C ld (T, C) be fucio. The, he -Sumudu rform S {f}( ) bou he he poi T of he fucio f i defied by S : 1 (73) z e ] (1/z) (ρ (),)f() for z D {f}, where D{f} coi of ll complex umber z R ] (T, C) for which he improper iegrl exi. Le u defie he e C h (λ) : {z C h : R h ( 1 ) > λ}. (74) z We oice h, followig Lemm 7,iff C ld ([, ) T, C) i fucio of expoeil order α,he lim f () e ] (1/z) (, ) 0, (75) where z C h (α).hece,wehvehefollowig. Theorem 28. Le f C ld ([, ) T, C) be of expoeil order α. The,he -Sumudu rform S {f}( ) exi o C ] ()(α) d coverge boluely. I he pecil ce T N {, + 1, + 2,...}, R fixed (ee [44]), we hve S 1 ( z 1 k 1 z z ) f (+k) (76) k1 for ech z C \ {0, 1} for which he erie coverge. The followig heorem e he cloe reliohip bewee bl Sumudu rform d bl Lplce rform. Theorem 29. Le f C ld ([, ) T, C) be fucio. The S 1 z L {f} ( 1 z ) 1 z F( 1 ). (77) z The followig heorem c be eily verified uig iducio. Theorem 30. Le f C ld ([, ) T, C) be of expoeil order α.the, S {f } (z) 1 z S where z C ] ()(α). 1 f z k k (), (78) The followig heorem pree he bl-sumudu rformio of covoluio of wo fucio o ime cle.

9 8 Abrc d Applied Alyi Theorem 31. Le f, g C ld ([, ) T, C).The S {f g} (z) z[ 0 S 0 S {g} (z)]. (79) Proof. The proof i direc coequece of relio (77)d Theorem 18. Now, we coider he -Sumudu rform o ime cle frciol clculu. We begi wih -Sumudu rform of power fucio o T. Theorem 32. Le T.Forα> 1,oeh S {ĥα (, )} (z) z α. (80) Proof. Uig Theorem 28 d he reul (41), we hve S {ĥα (, )} (z) 1 z L {ĥα (, ) f} ( 1 z ) Thi complee he proof. 1 z ( 1 (1/z) α+1 ) z α. (81) I priculr, ĥ0(, ) 1 d hece he -Sumudu rform of f() 1 i give follow. Corollry 33. The -Sumudu rform of f(x) 1 i give by S {1}(z) S {ĥ0 (, )} (z) z 0 1. (82) I he followig heorem, we give he Sumudu rform of geerlized Mig-Leffler fucio o ime cle. Theorem 34. For α, β, λ R d, T,iholdh S α,β ()} (z) zβ 1 1 λz α, (83) provided λz α <1. Proof. Uig he relio (77)d he reul(63), we ge S α,β ()} (z) 1 z L α,β ()}(1 z ) 1 z ( (1/z) β 1 λ(1/z) α ) zβ 1 1 λz α, λzα <1. (84) The bl Sumudu rform of frciol iegrl d frciol derivive re follow. Theorem 35. (i) For α>0, S { α f} (z) z α S. (85) (ii) For β>0d m[β]+1, S { β f} (z) 1 m 1 1 z β S z [ m k (m β) f] (iii) For γ>0d m[γ]+1, S { C γ f} (z) 1 m 1 z γ S 1 k f zγ k (). (86) k (). (87) Proof. The proof o ech pr follow immediely fer pplyig (77) d he repecive Lplce rform (53), (54), d (59). A i he ce of Lplce rform (ee relio (58)), he -Sumudu rformi Theorem 35(ii) i equivle o S { β f} (z) 1 1 z β S z j ( β j f) (), l j1 l 1<β l. (88) I he followig exmple we will illure he ue of he -Sumudu rform by pplyig i o olve iiil vlue problem. Exmple 36. Coider he followig iiil vlue problem: ( α y) () λy() f(), T, 0<α 1, (89) ( α 1 y) () y 0, y 0 R. (90) We begi by kig he -Sumudu rform of boh ide of (89). By uig Theorem 35(ii) for 0<α 1,wege z α S {y} (z) z 1 (1 α) y () λs {y} (z) S. Hece, S {y} (z) z 1 z α λ α 1 1 y () + z α λ S ( zα 1 Thu, we hve zα 1 1 λz α )y 0 +z( 1 λz α ) S y 0 S α,α 1 ()} (z) +z{ S α,α 1 ()} (z) S } y 0 S α,α 1 ()} (z) + S { α,α 1 ()} (z) {f}(z)}. (91) (92) y () y 0 E,λ α,α 1 () +E,λ α,α 1 () f(). (93)

10 Abrc d Applied Alyi 9 Exmple 37. Coider he followig Cpuo ype iiil vlue problem: ( C α y) () λy() f(), ( T,<α<, N,λ R), (94) y k () b k,,1,2,...,. (95) By kig he -Sumudu rform of boh ide of (94)d uig Theorem 35(iii), we ge 1 z α S {y} (z) S, S {y} (z) 1 y zα k k () λs {y} (z) z α b 1 λz α k z + z α α k 1 λz α S b k z k 1 λz α + b k S α,k+1 z α 1 λz α S ()} (z) +z( S α,α ()} (z) S ) Thu, we hve b k S α,k+1 ()} (z) + S α,α () f()}. (96) y () b k E,λ α,k+1 () +E,λ α,α () f(). (97) I priculr, whe 0<α<1, heiiilvlueproblem C α y () λy() f(), T, y () b 0, y 0 R h oluio of he followig form: (98) y () b 0 E,λ α,1 () +E,λ α,α () f(). (99) Exmple 38. Coider he followig Cpuo ype iiil vlue problem: ( C α y) () f(), T, 0<α 1, (100) y () y 0, y 0 R. (101) By kig he -Sumudu rform of boh ide of (100)d uig Theorem 35(iii) for 0<α 1,we ge z α S {y} (z) z α y () S. (102) Hece, S {y} (z) y 0 +z α S Thu, we hve y 0 S {ĥ0 (, )} (z) +z S {ĥα 1 (, )} (z) S. y () y 0 + ĥα 1 (, ) f() y 0 + α f (). (103) (104) Remrk 39. Followig Theorem 29 d he exmple o olvig frciol dymic equio, oe c coclude h () if he oluio of frciol dymic equio exi by -Sumudud rform, he he oluio exi by -Lplce rform, d vie ver; (b) if he oluio of frciol dymic equio exi by -Sumudud rform, he he oluio exi by Sumudu d Lplce rform (here T R). Coflic of Iere The uhor declre h here i o coflic of iere regrdig he publicio of hi pper. Ackowledgme The uhor i very greful o he referee for heir helpful uggeio. Referece [1] G. K. Wugl, Sumudu rform: ew iegrl rform o olve differeil equio d corol egieerig problem, Ieriol Jourl of Mhemicl Educio i Sciece d Techology,vol.24,o.1,pp.35 43,1993. [2] Q. D. Kbeh d F. B. M. Belgcem, Applicio of he Sumudu rform o frciol differeil equio, Nolier Sudie,vol.18,o.1,pp ,2011. [3] S. Hilger, Alyi o meure chi uified pproch o coiuou d dicree clculu, Reul i Mhemic, vol. 18,o.1-2,pp.18 56,1990. [4] S. Hilger, Specil fucio, Lplce d Fourier rform o meure chi, Dymic Syem d Applicio, vol.8,o. 3-4, pp , [5] M. Boher d A. Peero, Dymic Equio o Time Scle: A Iroducio Wih Applicio, Birkhäuer, Boo, M, USA, [6] M. Boher d A. Peero, Ed., Advce i Dymic Equio o Time Scle, Birkhäuer, Boo, M, USA, [7] M. Boher d A. Peero, Lplce rform d Z- rform: uificio d exeio, Mehod d Applicio of Alyi,vol.9,o.1,pp ,2002. [8]J.M.Dvi,I.A.Grvge,B.J.Jcko,R.J.MrkII,d A. A. Rmo, The Lplce rform o ime cle reviied, Jourl of Mhemicl Alyi d Applicio,vol.332,o. 2, pp , 2007.

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12 Advce i Operio Reerch Advce i Deciio Sciece Jourl of Applied Mhemic Algebr Jourl of Probbiliy d Siic The Scieific World Jourl Ieriol Jourl of Differeil Equio Submi your mucrip Ieriol Jourl of Advce i Combioric Mhemicl Phyic Jourl of Complex Alyi Ieriol Jourl of Mhemic d Mhemicl Sciece Mhemicl Problem i Egieerig Jourl of Mhemic Dicree Mhemic Jourl of Dicree Dymic i Nure d Sociey Jourl of Fucio Spce Abrc d Applied Alyi Ieriol Jourl of Jourl of Sochic Alyi Opimizio