Fractional Calculus. Student: Manal AL-Ali Dr. Abdalla Obeidat

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1 Fracional Calculu Suen: Manal AL-Ali Dr. Aballa Obeia

2 Deignaion Deignaion mean inegraion an iffereniaion of arbirary orer, In oher ereion i mean ealing wih oeraor like,, i arbirary real or Comle value. Many efiniion are rooe o fin he fracional erivaive an inegral, he mo common one i Riemann Liouville Definiion.

3 Riemann Liouville Definiion. Riemann Liouville Definiion i given by he following equaion : a D m m ai f... Re... n Re n f u f u u... Re Γ a When a -, eq. i equivalen o Riemann' efiniion, an when a, we have Liouville' efiniion.

4 amle: evaluaion of he fracional erivaive for he funcion f b, b >-. We have for m- m, m N For b, * You can ee ha he fracional erivaive of conan i no ero! Γ Γ Γ b b m m b m m b b u u u m D f I f D { Γ c c D

5 Miag-Leffler Funcion Miag-Leffler funcion of one arameer i enoe by k Γ k k 3 A wo- arameer Miag-Leffler funcion i efine by he erie eanion k Γ k, β, >, β > 4 k β

6 Relaion o ome oher funcion I i follow from he efecion ha :, k k Γ k e, k k Γ k e k, coh k Γk k k, co k Γk, k k Γk inh, k Γk k k in

7 Lalace ranform Lalace ranform of fracional ifferenial oeraor i given by I{ n k k f ; } F [ D f ] k D where n-<<n. 5 Lalace ranform of Miag-Leffler funcion k! a β k β k e, β ± a k,re > a, 6

8 Fracionaliaion of hyical roblem Fracionaliaion of hyical roblem mean bringing he ool of fracional erivaive / fracional inegral ino he heory of he roblem by fracionaliaion of ome aroriae oeraor. Then earch for hyical meaning. The queion now i how we can chooe hoe oeraor???

9 The roblem fracional muliole Cae oin monoole Φ r L L Inermeiae cae Φ r oin iole Φ r Cae he general roblem of of elecroaic oenial of of a aic elecric charge iribuion in in free ace

10 Poion quaion Φ ρ / ε ρ i he volume charge eniy of he ource Cae ρ qδ r L q Then he fracional oeraor will be ρ L. δ r q Cae

11 The inermeiae fracional ource can be efine by he following equaion: ρ r q l L δ ρ ρ 7 By he hel of Riemann Liouville efiniion of fracional oeraor eq.7 can be wrien a } { Γ u y u u q q δ δ δ ρ 8

12 ρ ql δ δ y U Γ 9

13 The calar oenial of hi facional ource i foun o be q ψ, y, L ψ l D 4πε R ql Γ P co θ 4πε R Figure.. The Fracional oenial

14 Fracional roblem uing Miag-Leffler funcion Now we li here ome olve fracional hyical roblem uing erie oluion. an hen rewrie he oluion of hee roblem in erm of Miag-Leffler funcion.

15 The fracional LC-RC circui A.A.Rouan, N.Y.Ayoub, eal uggee a fracional ifferenial equaion ha combine he imle harmonic ocillaion of an LC circui wih icharging of an RC circui. Q Q where - When, equaion goe over o he equaion of RC circui Q Q where /RC

16 an, when, equaion become he LC circui equaion Q Q where /LC 3 Aly lalace ranform o eq.,an ue he following iniial coniion Q Q Q We ge IQ Q 4

17 comare eq. 4 by eq. 6, we ge Q Q, 5 Now when Q Q, Qe Soluion of eq. An when Q Q, i Q in I Q Q co I co Soluion of eq.3

18 Fracional Simle Harmonic Ocillaor The equaion of moion ha cover a ynamic yem i given by he following equaion: m b k 6 Akram. A. Rouan, Nabil.Y.Ayoub an Kheam Khaawinah uggee a moifie equaion of eq. 6 uing fracionalie econ erm a follow: m b k 7

19 Define he aming raio η m b an m k Then ake lalace ranform of eq.7 ] [ ] [ I I I η } ] [ { ] [ ] [ η Ue he iniial coniion ] [ ] [ c ] [,,

20 We ge } { c η 8 Rearranging he equaion, one can wrie i a follow c η η Le η η η c

21 One can rewrie an a follow η c η η 9 Comaring eq.9 an by eq6, we ge,! η,! c η η

22 Now! { η, εη Where an c aken o be, } a Simle harmonic ocillaor co, b Dame ocillaor µ {, η η, }!

23 3 Fracional Domain Wall Moion Weam Al Sharo a emloye he iea of fracionaliaion of he econ erm in he equaion of moion uggee in he roblem of imle harmonic ocillaor o uy omain wall moion for wo cae: f δ f con f The fracional omain wall moion equaion i: η f Ue he ame iniial coniion in S.H.O roblem, an follow he Same roceure, one ge :

24 a Cae: f δ,, } {! η 3 For an f eq.3 reuce o Γ, co!!! 4 For,, } {! µ η η η an c 5

25 Cae : f con f 3,,, } {! f 6 Where: η an η For eq.5 reuce o co co f f For 3,,, } {! f 7 8 η Where: an η

28 4- Concluion We were able o rewrie ome olve roblem from JUST which are ublihe in cienific journal in erm of Meag- Leffler funcion, which i a eman from mo reigiou journal.

29 5. Reference: [] I. Polubny, Fracional Differenial quaion, Volume 98,ACADMIC PRSS, 999. [] R.Hilfer, alicaion of fracional calculu in hyic, WORLD SCINTIFIC,. [3] D.H.Werner an R.Mira, Fronier in lecromagneic, I Pre, chaer. [4] A.Rouan an N.Ayoub, A Fracional LC-RC Circui, An Inernaional Journal for Theory an Alicaion. [5] A.Rouan, N.Ayoub an K.Khaawenah, Fracional Simle Harmonic Ocillaor, Inernaional Journal of Theoreical Phyic..[6] Weam Al Sharo a, Fracional Domain Wall Moion,

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