Local Fractional Kernel Transform in Fractal Space and Its Applications

Transcription

1 From he SelecedWorks of Xio-J Yg 22 Locl Frciol Kerel Trsform i Frcl Spce d Is Applicios Yg Xioj Aville : hps://works.epress.com/yg_ioj/3/

2 Advces i Compuiol Mhemics d is Applicios 86 Vol. No Copyrigh World Sciece Pulisher Uied Ses Locl Frciol Kerel Trsform i Frcl Spce d Is Applicios Xio-J Yg Deprme of Mhemics d Mechics Chi Uiversiy of Miig d Techology Xuzhou Cmpus Xuzhou Jigsu 228 P. R. Chi Emil: dygioj@63.com Asrc I he prese pper we poi ou he locl frciol kerel rsform sed o locl frciol clculus (LFC) d is pplicios o he Yg-Fourier rsform he Yg-Lplce rsform he locl frciol Z rsform he locl frciol Sieljes rsform he locl frciol volerr/ Fredholm iegrl equios he locl frciol volerr/ Fredholm iegro-differeil equios he locl frciol vriiol ierio lgorihms he locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer he modified locl frciol vriiol ierio lgorihms d he modified locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer. Keywords Frcl spce; Locl frciol clculus; Locl frciol kerel rsform; No-differeile fcios; Locl frciol volerr / Fredholm iegrl/iegro-differeil equios; Locl frciol vriiol ierio lgorihms. Iroducio Frequely i mhemicl physics we ecoer pirs of fcios reled y epressio of he followig form: g f K d (.) he fcio g is clled he iegrl rsform of f d he fcio K is he kerel of he fcio f which is o frcl chrcerisic fcio. However some fcios re frcl curves which re every coiuous u o differeile. As resul we c employ he fcios should e differeile o descrie he iegrl rsform i frcl spce. Recely he heory of locl frciol iegrls d derivives [-] ws delig wih frcl fcios d ws successfully pplied i frcl elsiciy [3] frcl sigls [2 28] frcl wve equio [7] Yg-Fourier rsform [8 2 22] Yg-Lplce rsform[ ] discree Yg-Fourier rsform [ 23] fs Yg- Fourier rsform [26] locl frciol Sieljes rsform i frcl spce [2] locl frciol shor ime rsforms [56] d locl frciol wvele rsform [56]. Bsed o he vriiol ierio mehod [3] he vriiol ierio lgorihm c ws cosruced i [3 4]. The Modified vriiol ierio mehod is repored [4-8]. Yilmz d Ic [9] cosruced vriiol ierio lgorihm wih uiliry prmeer. I his pper we sudy he locl frciol kerel rsform sed o he heory of locl frciol iegrls d give is pplicios o locl frciol iegrl rsforms i frcl spce he locl frciol volerr/ Fredholm iegrl equios he locl frciol volerr/ Fredholm iegro-differeil equios he locl frciol vriiol ierio lgorihms (wih uiliry prmeer) d he modified locl frciol vriiol ierio lgorihms (wih uiliry prmeer). The pper is orgized s follows. I secio 2 he elemery resuls re preseed. The locl frciol kerel rsform is ivesiged i secio 3. Is pplicios re show i secio 4. The coclusios re i secio Prelimiries 2. Locl frciol coiuiy of fcios Defiiio If here eiss [5678] f f (2.) wih for d ow f is clled locl frciol coiuous deoe y lim f f. The f is clled locl frciol coiuous o he iervl deoed y f C. (2.2) Defiiio 2 A fcio f is clled odiffereile fcio of epoe which sisfies Hölder fcio of epoe he for y X such h [5678] f f y C y. (2.3)

3 Xio-J Yg e l. ACMA Vol. No. 2 pp Defiiio 3 A fcio f is clled o e coiuous of order or shorly coiuous whe we hve h[567] f f o. (2.4) Remrk. Compred wih (2.4) (2.) is sdrd defiiio of locl frciol coiuiy. Here (2.3) is ified locl frciol coiuiy. 2.2 Locl frciol derivive Defiiio 4 Le f C derivive of [5678] f of order. Locl frciol is give d f f f f lim d f f f f. For y here eiss f D f deoed y f D. Locl frciol derivive of high order is derived s k imes (2.5) k f D... D f (2.6) d locl frciol pril derivive of high order k imes f Locl frciol iegrls Defiiio 4 Le f C iegrl of f.. Locl frciol f of order i he iervl is give [ ] I f lim f jn d j j f j j j j m 2 j... j... N N iervl d d (2.7) j j is priio of he. For coveiece we ssume h I f if I f I f if. For y deoed y Remrk 2. If I we ge I f f I. we hve h f C 2.4 Specil Fcios i Frcl Spce. Defiiio 5 The Mig-Leffler fcio i frcl spce is defied y[5628] k E : k k d. ( 2.8) Defiiio 6 The sie fcio i frcl spce is y he epressio [5628] (2k) k si : d. k 2k (2.9) Defiiio 7 The cosie fcio i frcl spce is give[5628] 2k k cos : d. 2 k k The followig rules hold [5628]: cos si (2.) E E y E y ; (2.) E E y E y ; (2.2) E i E i y E i y ; (2.3) E i i ; (2.4) si cos E i E i ; (2.5) 2i E i E i. (2.6) 2.5 Frciol-order comple umer Defiiio 8 Frciol-order comple umer is defied y [5628] 2 I i y y (2.7) is cojuge of comple umer shows h I i y d he frciol modulus is derived s (2.8) 2 2 I I I I I y. (2.9) Defiiio 9 Comple Mig-Leffler fcio i frcl spce is defied y[5628]

4 Xio-J Yg e l. ACMA Vol. No. 2 pp E z k z : (2.2) k k C (comple umer se) d. The for z followig rules re vlid: E z E z2 E z z ; (2.2) E z E z E z z ; (2.22) E i z E i z E i z z. (2.23) Whe z i he comple Mig-Leffler fcio is compued y[5628] wih d cos si E i i (2.24) 2k k cos : 2 k k k si : k 2k for d we hve h (2k) E i E i y E i y (2.25) d E i E i y E i y. (2.26) 2.6 Geerlized lier operors To egi wih we give he defiiio of geerlized lier operor [5 6]. Le X d Y e geerlized lier spces over field F d le T : X Y. If T y T T y y X F (2.27) ;. We syt is geerlized lier operor or geerlized lier rsformio from X io Y. Also we wrie T X T : X. (2.28) The locl frciol differeil operor D is geerlized lier operor [5 6]: D f f g lim. (2.29) The locl frciol iegrl operor I is geerlized lier operor [5 6]: I f f d. (2.3) The locl frciol lier fciol is geerlized lier operor [5 6]: I f f d. (2.3) Defiiio The locl frciol kerel rsform is defied y (3.) g f K m d oh f d K m coiuous fcio f. The fcio K m sisfy (2.2) d is frcl kerel of locl frciol g is locl frciol kerel rsform. Properies of locl frciol kerel rsform re vlid: () (2) f f K m d 2 f K m d 2 f K m d ; f Km d f K m d. I is remrked h he ove re derived from he heory of he geerlized lier operor. 4. Applicios 4. Applicios o locl frciol iegrl rsforms The locl frciol iegrl rsform provides comprehesive d widely ccessile cco of he sujec coverig oh heory d pplicios. Here we discuss pplicios o locl frciol iegrl rsforms i frcl spce. 4.. The Yg-Fourier rsform i frcl spce Suppose h f C F F f f Fourier rsform deed y wrie i he form [ ] F f f F he Yg- E i f d is (4.) he ler coverges. Ad of course sufficie codiio for covergece is 3. Locl frciol kerel rsform i frcl spce

5 Xio-J Yg e l. ACMA Vol. No. 2 pp If F f f F i he form [4 7 2] f F f F 2 f E i d f d K. is iversio formul is wrie : F E i f d. As is show he Yg-Fourier rsform is sed o he comple Mig-Leffler kerel E i The Yg-Lplce rsform i frcl spce Suppose h f C rsform deed y he form [92225] L f L s s f L s L f f s he Yg-Lplce is give i E s f d (4.2) he ler coverges d s. Ad of course sufficie codiio for covergece is f d K. L s L f f s Suppose h is iverse is give y he epressio [92225] f d L s : L f s 2 i i L s E s f s ds s i Re s. I is remrked h he defiiio is esy eough for egieer o dersd i i he form. Here we regrd s lim s s d lim Res. Hece i is coveie o dersd hese forms. I is show h he Yg-Lplce rsform is sed o he comple Mig-Leffler kerel E s The locl frciol Z-rsform i frcl spce y[29] The locl frciol Z-rsform of Z f : F z f z Is iversio is f is give (4.3) Z F z : f F z z dz C 2 i As direc resul he locl frciol Z-rsform is sed o he comple kerel z The locl frciol Sieljes rsform i frcl spce The locl frciol Sieljes rsform of fcio f ( ) o is [2] f ( ) S { f ( )} f ( z) ( d) ( ) (4.4) ( z) z is comple vrile i he cu ple rg z. I is show h he locl frciol Sieljes rsform is sed o he comple kerel z. 4.2 Applicios o specil locl frciol iegrl equios Here we cosruc he locl frciol volerr/ Fredholm iegrl equios sed o he locl frciol iegrl rsform Locl frciol volerr iegrl equio- I Locl frciol volerr iegrl equio- I is cosruced i he form f k d (4.5) Locl frciol volerr iegrl equio- I I Locl frciol volerr iegrl equio- I I is cosruced i he form g k d (4.6) is frcl prmeer Locl frciol Fredholm iegrl equio- I Locl frciol Fredholm iegrl equio- I is cosruced i he form

6 Xio-J Yg e l. ACMA Vol. No. 2 pp f k d (4.7) Locl frciol Fredholm iegrl equio- I I Locl frciol Fredholm iegrl equio- I I is cosruced i he form g k d (4.8) k is clled s he locl frciol kerel fcio d oh d re wo coss. 4.3 Applicios o specil locl frciol olier iegro-differeil equios Here we cosruc he locl frciol volerr/ Fredholm iegro-differeil equios derived from he locl frciol iegrl rsform Locl frciol olier volerr iegrodiffereil equio Locl frciol olier volerr iegro-differeil equio c e wrie i he form 2 y y y E d y (4.9) wih he iiil codiios y d y Locl frciol olier Fredholm iegrodiffereil equio Locl frciol olier Fredholm iegrodiffereil equio c e wrie s 2 y y E y d wih he iiil codiios y d y. (4.) I is remrked h he ove olier locl frciol equio c e wrie s L u N u (4.) L d N re lier d olier locl frciol operors respecively. 4.4 Applicios o locl frciol vriiol ierio lgorihms Here we cosruc he locl frciol vriiol ierio lgorihms vi he locl frciol iegrl rsform d he ide is sed o he mehodology i [2 3]. Cosider he followig geerl olier locl frciol equio which is rewrie s L u N u Where L d N re lier d olier locl frciol operors respecively. Here we my wrie he followig lgorihms: 4.4. Locl frciol vriiol ierio lgorihm-i Locl frciol vriiol ierio lgorihm-i c e wrie s L s N s ds. (4.2) Here we c cosruc correcio fciol s follows L s sds (4.3) u is cosidered s resriced locl frciol vriio. i.e.. For more deils see[3]. u Locl frciol vriiol ierio lgorihm-i I Locl frciol vriiol ierio lgorihm-i I c e wrie s u sds ; (4.4) There is correcio fciol s follows u sds. (4.5) Locl frciol vriiol ierio lgorihm-i I I Locl frciol vriiol ierio lgorihm-i I I c e wrie s 2 s sds (4.6) is he ideified frcl Lgrge muliplier. There is correcio fciol which c e wrie s 2 s sds. (4.7) 4.5 Applicios o locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer Here we cosruc he locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer sed o he locl frciol iegrl rsform d he ide is sed o he mehodology i [3 4].

7 Xio-J Yg e l. ACMA Vol. No. 2 pp Locl frciol vriiol ierio lgorihm-i wih uiliry frcl prmeer Locl frciol vriiol ierio lgorihm-i wih uiliry frcl prmeer is srucured i he form h L s N s ds. (4.8) Here we c cosruc correcio fciol s follows h L s s ds (4.9) u is cosidered s resriced locl frciol vriio. i.e.. For more deils see[3]. u Locl frciol vriiol ierio lgorihm- I I wih uiliry frcl prmeer Locl frciol vriiol ierio lgorihm-i I wih uiliry frcl prmeer is srucured i he form h u sds. (4.2) Here correcio fciol c e cosruced s follows h u sds. (4.2) Locl frciol vriiol ierio lgorihm- I I I wih uiliry frcl prmeer Locl frciol vriiol ierio lgorihm-i I I wih uiliry frcl prmeer is srucured i he form h 2 s sds (4.22) h is uiliry frcl prmeer. Here here is correcio fciol which c e cosruced s follows h 2 s sds. (4.23) 4.6 Applicios o modified locl frciol vriiol ierio lgorihms Cosider he followig geerl olier locl frciol equio L un u g (4.24) L is lier differeil operor N is olier operor d g is give lyicl fcio. We c cosruc modified locl frciol vriiol ierio lgorihms d he ide is sed o he mehodology i [5-8] Modified locl frciol vriiol ierio lgorihm-i A modified locl frciol vriiol ierio lgorihm-i is srucured i he form L s N s gs ds ; (4.25) Here we c cosruc correcio fciol s follows L s s gs ds (4.26) u is cosidered s resriced locl frciol vriio. i.e.. For more deils see[3]. u Modified locl frciol vriiol ierio lgorihm-i I A modified locl frciol vriiol ierio lgorihm-i I is srucured i he form u N u s g s ds ; (4.27) Here we c cosruc correcio fciol s follows: u N u s g s ds. (4.28) Modified locl frciol vriiol ierio lgorihm-i I I A modified locl frciol vriiol ierio lgorihm-i I I is srucured i he form 2 N s N sds (4.29) is he ideified frcl Lgrge muliplier. Hece correcio fciol is cosruced s follows: 2 s sds. u (4.3) is cosidered s resriced locl frciol vriio ie. u 4.7 Applicios o modified locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer

8 Xio-J Yg e l. ACMA Vol. No. 2 pp Here we ivesige he locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer sed o he locl frciol iegrl rsform d he ide is sed o he mehodology i [9] Modified locl frciol vriiol ierio lgorihm-i wih uiliry frcl prmeer A modified locl frciol vriiol ierio lgorihm-i wih uiliry frcl prmeer is srucured i he form h L s N s gs ds. (4.3) There is correcio fciol which is cosruced s follows: h L s N s gs ds u (4.32) is cosidered s resriced locl frciol vriio. i.e.. For more deils see[3]. u Modified locl frciol vriiol ierio lgorihm-i I wih uiliry frcl prmeer A modified locl frciol vriiol ierio lgorihm-i I wih uiliry frcl prmeer is srucured i he form h u N u s g s ds ; (4.33) There eiss correcio fciol which is cosruced s follows: h u N u s g s ds. (4.34) Modified locl frciol vriiol ierio lgorihm-i I I wih uiliry frcl prmeer A modified locl frciol vriiol ierio lgorihm-i I I wih uiliry frcl prmeer is srucured i he form h 2 s sds h (4.35) is he ideified frcl Lgrge muliplier is uiliry frcl prmeer. Here correcio fciol is cosruced s follows: h 2 s sds (4.36) is he ideified frcl Lgrge muliplier h is uiliry frcl prmeer d u is cosidered s resriced locl frciol vriio i.e.. u I is remrked h he clssicl vriio is i cse of locl frciol vriio whe frcl dimesio is equl o. 5. Coclusios I his pper we sudy he elemes. The we proposed he locl frciol kerel rsform i frcl spce sed o he locl frciol clculus (LFC) d discuss is pplicios. I is show h pplicios o he locl frciol iegrl rsforms (he Yg-Fourier rsform he Yg-Lplce rsform he locl frciol Z rsform d he locl frciol Sieljes rsform) re preseed sed o i. Filly he locl frciol volerr/ Fredholm iegrl equios he locl frciol volerr/fredholm iegro-differeil equios he locl frciol vriiol ierio lgorihms (I I I I I I) locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer (I I I I I I) he modified locl frciol vriiol ierio lgorihms (I I I I I I) d he modified locl frciol vriiol ierio lgorihms wih uiliry frcl prmeer (I I I I I I) re ivesiged. Refereces [] K.M. Kolwkr A.D.Ggl Frciol differeiiliy of o differeile fcios d dimesios Chos 6 (4)(996) [2] F. B.Add J.Cresso Aou o-differeile fcios J. Mh. Al. Appl. 263(2) [3] A.Crpieri P.Corei A frciol clculus pproch o he descripio of sress d sri loclizio i frcl medi Chos Solios d Frcls 3(22) [4] Y. Che Y.Y K. Zhg O he locl frciol derivive J. Mh. Al. Appl. 362(2)7-33. [5] X.J. Yg Locl Frciol Iegrl Trsforms Prog. i Noli. Sci. 4(2)-225. [6] X.J. Yg Locl Frciol Fciol Alysis d Is Applicios Asi Acdemic pulisher Limied Hog Kog 2. [7] X.J. Yg Applicios of locl frciol clculus o egieerig i frcl ime-spce: Locl frciol differeil equios wih locl frciol derivive ArXiv:6.3v [mh-ph]. [8] W.P. Zhog F. Go X.M. She Applicios of Yg-Fourier Trsform o Locl Frciol Equios wih Locl Frciol Derivive d Locl Frciol Iegrl Advced Merils Reserch 46 (22) [9] W. P. Zhog F. Go Applicio of he Yg-Lplce rsforms o soluio o olier frciol wve equio wih frciol derivive. I: Proc. of he 2 3rd Ieriol Coferece o Compuer Techology d Developme ASME 2 pp [] X.J. Yg A New Viewpoi o he Discree Approimio Discree Yg-Fourier Trsforms of Discree-ime Frcl Sigl. 7.26v[mh-ph]. [] S.M. Guo L. Q. Mei Y. Li Y.F. S The improved frciol su-equio mehod d is pplicios o he spceime frciol differeil equios i fluid mechics Phys. Le. A. 376(4) (2)47-4. [2] J. H. He G.C. Wu F. Ausi The vriiol ierio mehod which should e followed Nolier Sci. Le. A (2) -3.

9 Xio-J Yg e l. ACMA Vol. No. 2 pp [3] J. H. He Noes o he opiml vriiol ierio mehod Appl. Mh. Le. doi:.6/j.ml [4] A. M. Jssim A modified vriiol ierio mehod for Schrödiger d Lplce prolems. I. J. Coemp. Mh. Scieces 7 (3)(22) [5] T.A. Assy M.A. El-Twil H. El-Zoheiry Towrd modified vriiol ierio mehod (MVIM) Jourl of Compuiol d Applied Mhemics 27() (27) [6] T.A. Assy M.A. El-Twil H. El-Zoheiry Modified vriiol ierio mehod for Boussiesq equio Compuers d Mhemics wih Applicios 54 (27) [7] M. A. Noor Modified Vriiol Ierio Mehod for Gours d Lplce Prolems World Applied Scieces Jourl 4 (4) (28) [8] M. A. Noor S. T. Mohyud-Di A modified vriiol ierio mehod for odry lyer prolem i oded domi Ieriol Jourl of Nolier Sciece 7 (4)(29) [9] E. Yilmz M. Ic Numericl simulio of he squeezig flow ewee wo ifiie ples y mes of he modified vriiol ierio mehod wih uiliry prmeer Nolier Sci. Le. A (2) [2] Yg X.J Lio M.K d Che J.W. A ovel pproch o processig frcl sigls usig he Yg-Fourier rsforms. Procedi Egrg. 29 (22) [2] G.S. Che The locl frciol Sieljes rsform i frcl spce Advces i Iellige Trsporio Sysems ()(22)29-3. [22] X.J. Yg Locl frciol pril differeil equios wih frcl odry prolems Advces i Compuiol Mhemics d is Applicios () (22) [23] X.J. Yg The discree Yg-Fourier rsforms i frcl spce Advces i Elecricl Egieerig Sysems (2) (22) [24] X.J. Yg Epressio of geerlized Newo ierio mehod vi geerlized locl frciol Tylor series Advces i Compuer Sciece d is Applicios (2) (22) [25] X.J. Yg A shor iroducio o Yg-Lplce Trsforms i frcl spce Advces i Iformio Techology d Mgeme (2) (22) [26] X.J. Yg Fs Yg-Fourier rsforms i frcl spce Advces i Iellige Trsporio Sysems ()(22) [27] X.J. Yg Locl frciol Fourier lysis Advces i Mechicl Egieerig d is Applicios()(22) 2-6. [28] X.J. Yg Geerlized Smplig Theorem for Frcl Sigls Advces i Digil Mulimedi (2) (22) [29] Y. Guo Locl frciol Z rsform i frcl spce 22 Fiished. [3] X.J. Yg Reserch o Frcl Mhemics d Some Applicios i Mechics M.S. hesis Chi Uiversiy of Miig d Techology 29.