A modified method for solving Delay differential equations of fractional order

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1 IOSR Joural of Mahemais (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 3 Ver. VII (May. - Ju. 6), PP 5- A modified mehod for solvig Delay differeial equaios of fraioal order Osama Hameed mohammed a,quaiba Wadi b a,b Al-Nahrai Uiversiy, College of Siee, Deparme of Mahemais. dr_usama79@yahoo.om, qu_aiba99@yahoo.om Absra: I his paper, geeralized Ha fuios operaioal mariesare proposed ombied wih he mehod of seps o solve liear oliear delay differeial equaios of fraioal order. We over he delay differeial equaiosof fraioal order o o-delay differeial equaiosof fraioal ordero a give iervalby apply he mehod of seps, he apply he operaioal maries for geeralized Ha fuio o he obaied o-delay differeial equaiosof fraioal order o rasform liear oliear o-delay differeial equaiosof fraioal order io a sysem of algebrai equaios he fid he soluio.wo illusraive examples will be preseed o show he auray effiieyof he proposed mehod. Keywords: Fraioal alulus, Delay differeial equaios, Fraioal delay differeial equaios, Ha fuios, Mehod of seps. I. Iroduio Fraioal alulus (ha is, alulus of iegrals derivaives of ay arbirary real or omplex order) is a old mahemaial problem, maily developed as a pure mahemais problem for early hree euries []-[3]. hough havig a log hisory, i was o used i physis egieerig for a log period. However, i he las few deades, fraioal alulus bega o ara ireasig aeio of sieiss from he viewpoi of appliaio [3]-[6]. Fraioal alulus fraioal differeial equaios have foud appliaios i several differe disiplies [7]-[8]. Over he years, may mahemaiias, usig heir ow oaio approah, have foud various defiiios ha fi he idea of a o-ieger order iegral or derivaive. he mos famous of hese defiiios ha have bee popularized i he world of fraioal alulus are Riema-Liouville Grüwald-Leikov defiiio. Also, Capuo, [9] reformulaed he more "lassi" defiiio of he Riema-Liouville fraioal derivaive i order o use ieger order iiial odiios o solve his fraioal order differeial equaios. Delay differeial equaios (DDEs) are a ype of differeial equaio i whih he derivaive of he ukow fuio a a erai ime is give i erms of he values of he fuio a previous imes. Delay differeial equaios play a impora role i he researh field of various applied siees suh as orol heory, elerial eworks, populaio dyamis, evirome siee, biology life siee []. Fraioal delay differeial equaios are a very ree opi. Alhough i seems aural o model erai proesses sysems i egieerig oher siees wih his kid of equaios, oly i he las few years has he aeio of he sieifi ommuiy bee devoed o hem [], [], [3].his paper is orgaized as follows: I seio we reall he defiiios of fraioal derivaives fraioal iegraio, i seio 3, a review of geeralized Hafuios heir properies is desribed. I seio 4, he operaioal maries of iegraio for geeralized Ha fuios is derived. I seio 5, he proposed mehodis desribed. I seio 6, some illusraive examples are preseed. Fially, a olusio is draw i seio 7. II. Fraioal iegral differeial operaors I his seio, we review basi defiiios of fraioaldiffereiaio fraioal iegraio [3]. Defiiio.: he Riema Liouville fraioal order iegral operaori α of order α, of a fuio u() L [a, b] is give by: - I α u() = a b. Γ(α) s α u s ds, α >, u, α =, () Defiiio.:he Riema-Liouville fraioal order derivaives operaor D α of order α, of a fuio u() L [a, b] is give by: - D α u = ( d Γ( α) d ) s α u s ds () for a b, α R + is ieger. DOI:.979/ Page

2 A modified mehod for solvig Delay differeial equaios of fraioal order he Riema Liouville derivaives have erai disadvaages whe ryig o model real-world pheomea wih fraioal α differeial equaios. herefore, we shall irodue a modified fraioal differeial operaor D whih is proposed by Capuo [9]. Defiiio.3: he Capuo fraioal derivaive of a fuio u() L [a, b] is give by: D α u = Γ( α) s α ( d d ) u s ds (3) for a b, α R + is ieger. For f C m a, b, α, β, < α, α + β m, ν, he fraioal iegral derivaives saisfy he followig:. I α I β u = I β I α u = I α+β u. k k=.. (I α D α u) = u() u k ( + ) 3. (I α v ) = Γ(v+) Γ(v+α+) α+v. III. Geeralized Ha fuios heir properies [4] he radiioal Ha fuios are oiuous fuios, also alled riagle, e or riagular fuios are defied o he ierval [,].he geeralized Ha fuios are exesio of radiioal Ha fuios o he fiie ierval [, A]. he ierval [, A]is divided io equidisa subiervals, i, i + of equal leghs = A is a arbirary posiive ieger. he geeralized Ha fuio s family of firs ( + ) Ha fuios are usually defied o [, A] as [5]:, < φ = (4), oerwise (i ), i < φ i = i+, i < i +. i =,,, (5), oerwise (A ), A A φ = (6), oerwise Aordig o he defiiio of Ha fuios:, i = j φ i j = (7), i j φ i φ j =, i j. i= φ i () =. 3. Fuio Approximaio A arbirary fuio g L [a, b] is approximaed i veor form as: DOI:.979/ Page

3 g = A modified mehod for solvig Delay differeial equaios of fraioal order i= f i φ i () =G + G + [g, g, g,, g ] Φ + () =Φ + () G + (8) Φ + () [φ (), φ (),φ (),,φ ()] () he impora aspe of usig geeralized Ha fuios i heapproximaio of fuio g, lies i he fa ha he oeffiies g i i he Eq. (9), are give by g i = g i, i =,,,,. (9) IV. Operaioal Maries of he Iegraio for Geeralized Ha Fuios he ieger order fraioal order operaioal maries of iegraio for geeralized Ha fuios is give i he subseios (4.) (4.) respeively. 4.. Ieger OrderOperaioal Marix of Iegraio of he Geeralized Ha Fuios Sie φ i τ dτ L [, A], Eq. () is used o approximae i i he erms of he geeralized Ha basis fuios as φ i τ dτ b j = ij φ j, i =,,,, () Usig Eq. (), we alulae he oeffiies a ij as i b ij = φ i τ dτ, j =,,,, () he oeffiies b ij will form a ( + ) ( + ) marix P + wih (i +, j + ) ery asb ij, fori =,,,,, j =,,,,.Usig he values of b ij s from Eq. (), we obai he marix P + as: P + =( ) + (+) (3) he marix P + is alled he ieger order Ha fuios operaioal marix of iegraio. I plays a pivoal role i deermiaio of g τ dτ for a arbirary g L [, A]. wih he help of Eq. () Eq. (), we have φ + τ dτ=p + Φ + (). 4..Fraioal orderoperaioal Marix of Iegraio of he Geeralized Ha Fuios he fraioal iegraio of geeralized Ha fuio i Eq. () a be approximaed as (I α α Φ + )()=P + Φ + ((. α P + = α Γ(α+) γ γ ζ γ ζ ζ + (+) γ k = k α α k + + (k ) α+, k =,,,. ζ k = (k + ) α+ k α+ + (k ) α+, k =,,,. For more deails,oe a see[5]. V. he Approah I his seio, we shall approximae soluio of he followig fraioal order delay differeial equaios: D α u = F, u, u φ, < α, > (4) DOI:.979/ Page

4 α D u = ψ, τ, u (i) = u (i) A modified mehod for solvig Delay differeial equaios of fraioal order DOI:.979/ Page (5), i =,,,, (6) iscapuo fraioal derivaive of order α, F is a oliear operaor, is he idepede variable, u() is he (i) ukow fuio, φ is he delay fuio ψ is give fuios u are give osas. Firs we over he fraioal order delay differeial equaio o fraioal order o-delay differeial equaio by applyig he mehod of seps [3], as D α u = F, u, ψ φ, < α, > (7) u (i) (i) = u, i =,,,, (8) Now i order o solve Eq.'s (7)- (8) by usig he operaioal maries of geeralized Ha fuios, we approximae D α u() u i erms of geeralized Ha fuios as follows ( D α u)()=c + Φ + () (9) Ad upo operaig I α o he boh sides of equaio (9) leads o Hee u()=c + P + Φ + () + k= u k () Φ + () [φ (), φ (),φ (),,φ ()], C + () [,,,, ]. F, u, u φ Subsiuig Eq.'s (9) () io Eq. (7) gives + k k= = F(, C + P + Ψ + () + u (k) + k, ψ(φ )) () C + Φ + ()=F(, C + P + Φ + () + k= u (k), ψ(φ )) () Also, by subsiuig Eq.'s () (9) io Eq. (8), we ge u (i) i = C + Φ + ()=u, i =,,,, (3) From Eq. ()-(3), we a obai he oeffiies C +. he usig Eq.(), we a ge he oupu respose u. VI. Illusraive Examples I his seio, we shall solve liear oliear delay differeial equaios of fraioal order by usig he approah give i seio 5, ompare he resuls ha we have bee obaied wih he exisig mehods he exa soluio. we refer u a o represe he soluio by geeralized Ha fuios, u o represe he soluio by Chebyshev waveles mehod u exa o represe he exa soluio. Example (): Cosider he delay differeial equaios of fraioal order wih oliear delay fuio D α u () =-u, < α, < (4) u = si (), (5) u = (6) he exa soluio, whe α =, is u = si (). Soluio: Firs we over he delay differeial equaio of fraioal order o o-delay differeial equaio of fraioal order by applyig he mehod of seps, as D α u () =-si. < α. < (7) u = (8) Now we approximae D α u() i Eq. (7), i erms of geeralized Ha fuios as follows ( D α u)()=c + Φ + () (9) Hee u()=c + P + Φ + () (3) Also wriig he erm si Where + k i Eq. (7) i erms of geeralized Ha fuios leads o si = G +Φ + () (3) G + [g, g, g,, g ],

5 A modified mehod for solvig Delay differeial equaios of fraioal order g i = si i, i =,,,,. Subsiuig Eq.'s (9) (3) io Eq. (7), we have C + Φ + ()=G + Φ + () (3) whih implies ha C + =G + (33) Solvig Eq. (33), we a obai he oeffiies C +. he usig Eq.(3), oe a ge he oupu resposeu. For = 8, i seems from able () ha he resuls obaied from he proposed mehod whe α = provides beer resuls as ompared wih he exisig mehods suh as Chebyshev wavele mehod he exa soluio. (able )ompariso of he approximae soluio of example () usig he proposed mehod Chebyshev wavele mehod whe α = he exa soluio. u h α = u ha α = u exa α = Followig Figure () represe he approximae soluio of example () usig he proposed mehod for differe values of α he exa soluio whe α =. Fig. : he approximae soluio of example () by usig he proposed mehod a differe values of α he exa soluios a α =. Example (): Cosider he delay differeial equaio of fraioal order D u u =, < α, < (34) u = +, (35) u = (36) he exa soluio is u = ( ) k(k ) k= k. Soluio: Firs we over he delay differeial equaio of frraioal order oo-delay differeial equaio of fraioal order by applyig he mehod of seps, as: D α u = +, < α, < (37) u = (38) Now we approximae D α u() i Eq. (37), i erms of geeralized Ha fuios as follows ( D α u)()=c + Φ + () (39) Hee DOI:.979/ Page

6 A modified mehod for solvig Delay differeial equaios of fraioal order u()=c + P + Φ + () + (4) Also wriig he erm + i Eq. (37) i erms of geeralized Ha fuios leads o + = G +Φ + () (4) G + [g, g, g,, g ], g i = + i, i =,,,,. Subsiuig Eq.'s (39) (4) io Eq. (37), we have whih implies ha C + Φ + ()=G + Φ + () (4) C + =G + he usig Eq.(4), oe a ge he oupu respose u. For = 8, i seems from able () ha he resuls obaied from he proposed mehod whe α = provides beer resuls as ompared wih he exisig mehods suh as Chebyshev wavele mehod he exa soluio. (able )ompariso of he approximae soluio of example () usig he proposed mehod Chebyshev wavele mehod whe α = he exa soluio. u h α = u ha α = u exa α = (43) Followig Figure ()represe he approximae soluio of example () usig he proposed mehod for differe values of α he exa soluio whe α =. Fig. : he approximae soluio of example () by usig he proposed mehod a differe values of α he exa soluios a α =. VII. Colusio I his paper, weprese he ieger fraioal orders of iegraio for he geeralized ha fuios operaioal maries ombied hem wih he mehod of seps o solve liear oliear delay differeial equaios of fraioal order umerially. he obaied resuls are ompared wih he exa soluios wih he soluios obaied by some oher umerial mehods suh as Chebyshev wavele mehod. he resuls obaied from he proposed mehodare more aurae beer ha he resuls obaied from Chebyshev wavele mehod are i good agreeme wih he exa soluio. DOI:.979/ Page

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