Aeric Jourl o Couiol d Alied Mheics 4, 4(6): 87-9 DOI: 593/jjc446 Frciol Fourier Series wih Alicios Abu Hd I, Khlil R * Uiversiy o Jord, Jord Absrc I his er, we iroduce coorble rciol Fourier series We use such series o solve ceri ril rciol diereil equios Keywords Frciol ourier series, Coorble rciol derivive Iroducio Fourier series is oe o he os ior ools i lied scieces For exle oe c solve ril diereil equios usi Fourier series Furher oe c id he su o ceri uericl series usi Fourier series Frciol ril diereil equios ered o hve y licios i hysics d eieeri There re y deiiios o rciol derivive Oe o he os rece oes is he coorl rciol derivive [5] Recely [], rciol Tylor ower series ws iroduced, d beuiul heory ws lyed here However, o work is doe o rciol Fourier series, houh here is soe work o rciol ourier rsor The i o his er is o iroduce coorble rciol Fourier series As licio we solve soe rciol ril diereil equios usi rciol Fourier series Foe ore licios o coorble rciol derivive we reeree o [-4] Bsics o Coorble Frciol Derivive The subjec o rciol derivive is s old s clculus I 5 d 695, L Hoil sked i he exressio hs y 5 dx ei Sice he, y reserchers hve bee ryi o eerlize he coce o he usul derivive o rciol derivives These dys, y deiiios or hec rciol derivive re vilble Mos o hese deiiios use ierl or The os oulr deiiios re: (i) Rie - Liouville Deiiio: I is osiive * Corresodi uhor: roshdi@juedujo (Khlil R) Published olie h://jourlsubor/jc Coyrih 4 Scieiic & Acdeic Publishi All Rihs Reserved ieer d [, ) ive by, he h derivive o is d ( x) Γ( ) d ( x) (ii) Cuo Deiiio For [, ) D dx derivive o is +, he ( ) ( x) D ( ) dx Γ( ) x + Now, ll deiiios re eed o sisy he usul roeries o he sdrd derivive The oly roery iheried by ll deiiios o rciol derivive is he lieriy roery However, he ollowi re he se- bcks o oe deiiio or oher: (i) The Rie-Liouville derivive does o sisy D ( D or he Cuo derivive), i is o url uber (ii) All rciol derivives do o sisy he kow roduc rule: D ( ) D ( ) + D ( ) (iii) All rciol derivives do o sisy he kow quoie rule: D D D / (iv) All rciol derivives do o sisy he chi rule: ( D ) ο (v) All rciol derivives do o sisy:
88 Abu Hd I e l: Frciol Fourier Series wih Alicios + i eerl D D D (vi) Cuo deiiio ssues h he ucio is diereible T λ, or ll cos ucios (vii) λ I ew deiiio clled coorble rciol derivive ws iroduced Deiiio I > he we deie T [ ] ([ ] ) [ ] ( + ε ) ( ) li, ε ε where [ ] is he ceili o We cll T he rciol derivive o o order We shll wrie ( ) T ( ) The ew deiiio sisies: T ( b ) T bt ( ) T ( λ ), or ll cos ucios or + +, or ll b, λ Furher, or (,] d, be -diereible oi, wih The 3 T ( ) T ( ) + T ( ) T ( ) T ( ) 4 T We lis here he rciol derivives o ceri ucios, or he urose o cori he resuls o he ew deiiio wih he usul deiiio o he derivive: T ( ) 3 4 T si cos T cos si T e e O lei i hese derivives, we e he corresodi ordiry derivives Oe should oice h ucio could be -diereible oi bu o diereible, or exle, ke T The Hece T Bu T ( ) does o exis This is o he cse or he kow clssicl rciol derivives 3 Frciol Fourier Series Le <, d ϕ :[, ) R be deied by d [ [ ϕ :, ) R be y ucio Le :, ) R be deied by ϕ is clled -eriodicl For exle, i cos, he cos Deiiio 3 A ucio wih eriod i or ll [, ) ( ϕ ) ϕ + As exle, cos eriod is -eriodic wih Deiiio 3 Two ucios, h re clled -orhool o [,b ] i b hd Exles 3 cos d cos, Proo Pu x d The dx d -orhool o Furher, whe, x, d whe Hece ( ) re, x coscos cos( x)cos( x) dx d
Aeric Jourl o Couiol d Alied Mheics 4, 4(6): 87-9 89 I eerl, usi he ide i exle??? oe c esily rove: Theore 3, ) (i) cos d cos, or ll, ) (ii) si d si, or ll, ) (iii) si d cos, or ll, re orhool o re orhool o re orhool o Now le us deies he Fourier coeicies o -eriodic ucio wih eriod Deiiio 33 Le : [, ) R be ive eicewise coiuous -eriodic wih eriod : The we deie: (i) The cosie -Fourier coeicies o s d ()cos( ),,,3 (ii) The sie -Fourier coeicies o s d b ()si( ),,,3 For exle, he cosie -Fourier coeicies o he ucio cos is:, d or ll, where, Now, we ive he deiiio o he rciol Fourier series: Deiiio 34 Le : [, ) R be ive eicewise coiuous ucio which is -eriodicl wih eriod : The he -rciol Fourier series o ssocied wih he iervl [, ] is S + cos( ) + si( ) where d b re s i???? Le us hve soe exles Exle 3 Le i () i < wih o he iervl [, ]: The, d cos( ) d cos( ) d cos( ), d d + ( )cos( ), d Usi che o vribles: θ, we e dθ, θ i,, θ i, d θ i () Hece, he ierl becoes θcos( θ) dθ + ( θ )cos θdθ Siilrly b θsi( θ) dθ + ( θ )si θdθ So, 4 + + S 4 ( )( ) ( )si +
9 Abu Hd I e l: Frciol Fourier Series wih Alicios d i Exle 33 Le () i < < i The b ( ) d cos ( ) d si Hece S( ) si ( ) Oe c esily rove he ollowi clssicl resul Theore 3 The rciol Fourier series o iece wise coiuous - eriodicl ucio coveres oiwise o he vere lii o he ucio ech oi o discoiuiy, d o he ucio ech oi o coiuiy 4 Alicios I his secio we will use rciol Fourier series o solve soe rciol ril diereil equios Nely, we will solve he equio (, ) u( x, ) u x (4) <, < (4) u o, u L,, u x,, u x, x, < x< L (43) d Soluio We will use serio o vribles echique u x, P x Q Subsiue i he equio So le o e Fro which we e ( ) ( ) P x Q P x Q ( ) P( x) ( ) Q P x Q Sice x d re ideede vribles, he we e ( ) P( x) ( ) Q P x Q λ, cos o be deeried Hece d ( ) λ P x P x (44) ( ) λ Q Q (45) Codiios (43) suess h we work wih equio (44) irs There re hree ossibiliies or λ: (i) λ The equio (44) becoes P ( ) ( x), d ro he roery () o coorble rciol P x c Codiio (43) shows derivive, we e h c : (ii) λ > The equio (44) becoes P x λp x, d ro orul (4) o he coorble rciol derivive, we e λ P x ce Codiio (43) shows h c : (iii) λ < The equio (44) becoes µ P x + P x Usi oruls () d (3) we e x x P( x) ccos µ csi µ + (46) Codiio (43) ilies h c So si x c P x Aoher use o codiio (43) µ L ives si µ Hece So µ, wih, L (47) si x P x c L (48) Now, we o bck o equio (45) o e µ Q Q Usi orul (4) we e
Aeric Jourl o Couiol d Alied Mheics 4, 4(6): 87-9 9 µ µ + Q x e e Codiio (43) ilies h - Hece Q x Cobii (48) d (4) o e (49) sih µ (4) x u( x, ) b si sih (4) L L Now, usi he codiio u( x ),, o e x x x b si sih L L Usi he - Fourier series o - x, we id b REFERENCES [] Abdeljwd, T O coorble rciol clculus To er [] Abu-Hd, M, d Khlil, R Coorble Frciol He Diereil Equio Ieriol Jourl o Pure d Alied Mheics, 94 (4) 5-7 [3] Abu-Hd, M, d Khlil, R Abel S Forul Ad Wroski For Coorble Frciol Diereil Equios IJ Diereil Equios d Alicios, 3 (4) 77-83 [4] Abu-Hd, M, d Khlil, R Leedre rciol diereil equio d Leeder rciol olyoils IJ o Alied Mheicl Reserch, 3 (3) (4) 4-9 [5] Khlil, R, Al Hori, M, Youse A d Sbbheh, M, A ew Deiio o Frciol Derivive, J Cou Al Mh 64 657, 4