Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control

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1 Numericl Approximions o Frcionl Problems of he Clculus of Vriions nd Opiml Conrol Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres To cie his version: Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres. Numericl Approximions o Frcionl Problems of he Clculus of Vriions nd Opiml Conrol. Jcky Cresson. Frcionl Clculus in Anlysis, Dynmics nd Opiml Conrol, Nov Science Publishers, pp , 204, Mhemics Reserch Developmens, <hl-02234> HAL Id: hl hps://hl.inri.fr/hl Submied on 3 Mr 205 HAL is muli-disciplinry open ccess rchive for he deposi nd disseminion of scienific reserch documens, wheher hey re published or no. The documens my come from eching nd reserch insiuions in Frnce or brod, or from public or prive reserch ceners. L rchive ouvere pluridisciplinire HAL, es desinée u dépô e à l diffusion de documens scienifiques de niveu recherche, publiés ou non, émnn des éblissemens d enseignemen e de recherche frnçis ou érngers, des lboroires publics ou privés.

2 This is preprin of pper whose finl nd definie form ppered in: Chper V, Frcionl Clculus in Anlysis, Dynmics nd Opiml Conrol Edior: Jcky Cresson), Series: Mhemics Reserch Developmens, Nov Science Publishers, New York, 204. hp:// Chper V rxiv: v2 [mh.oc] Nov 203 NUMERICAL APPROXIMATIONS TO FRACTIONAL PROBLEMS OF THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL Shkoor Pooseh, Ricrdo Almeid nd Delfim F. M. Torres Cener for Reserch nd Developmen in Mhemics nd Applicions CIDMA) Deprmen of Mhemics, Universiy of Aveiro, Aveiro, Porugl Keywords: frcionl clculus of vriions, frcionl opiml conrol, numericl mehods, direc mehods, indirec mehods AMS Subjec Clssificion: 49K05, 49M25, 26A33. Inroducion A frcionl problem of he clculus of vriions nd opiml conrol consiss in he sudy of n opimizion problem in which he objecive funcionl or consrins depend on derivives nd inegrls of rbirry, rel or complex, orders. This is generlizion of he clssicl heory, where derivives nd inegrls cn only pper in ineger orders... Preliminries Ineger order derivives nd inegrls hve unified mening in he lierure. In conrs, here re severl differen pproches nd definiions in frcionl clculus for derivives nd inegrls of rbirry order. The following definiions nd noions will be used hroughou his chper. See [9]. Definiion. Gmm funcion). The Euler inegrl of he second kind Γz) = is clled he gmm funcion. 0 z e d, Rez) > 0, E-mil ddress: spooseh@u.p E-mil ddress: ricrdo.lmeid@u.p E-mil ddress: delfim@u.p

3 2 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres The gmm funcion hs n imporn propery, Γz +) = zγz), nd hence Γz) = z )! for z N, which llows o exend he noion of fcoril o rel numbers. Oher properies of his specil funcion cn be found in [5]. Definiion.2 Mig Leffler funcion). Le α > 0. The funcion E α defined by E α z) = j=0 z j Γαj +), whenever he series converges, is clled he one prmeer Mig Leffler funcion. The wo-prmeer Mig Leffler funcion wih prmeers α, β > 0 is defined by E α,β z) = j=0 z j Γαj +β). ) Definiion.3 Grünwld Lenikov derivive). Le 0 < α < nd α k) be he generlizion of binomil coefficiens o rel numbers. The lef Grünwld Lenikov frcionl derivive is defined s GL D α x) = lim h 0 + h α The righ Grünwld Lenikov derivive is GL Db α x) = lim h 0 + h α ) α ) k x kh). 2) k ) α ) k x+kh). 3) k In he bove menioned definiions, α k) is he generlizion of binomil coefficiens o rel numbers, defined by ) α = k Γα+) Γk +)Γα k +). In his relion, k nd α cn be ny ineger, rel or complex number, excep h α / {, 2, 3,...}. Definiion.4 Riemnn Liouville frcionl inegrl). Le x ) be n inegrble funcion in [,b] nd α > 0. The lef Riemnn Liouville frcionl inegrl of order α is given by I α x) = Γα) τ) α xτ)dτ, [,b]. The righ Riemnn Liouville frcionl inegrl of order α is given by I α b x) = Γα) τ ) α xτ)dτ, [,b].

4 Numericl Approximions o Frcionl Problems... 3 Definiion.5 Riemnn Liouville frcionl derivive). Lex ) be n bsoluely coninuous funcion in [,b], x ) AC[,b], nd 0 α <. The lef Riemnn Liouville frcionl derivive of order α is given by D α x) = d Γ α) d τ) α xτ)dτ, [,b]. The righ Riemnn Liouville frcionl derivive of order α is given by Db α x) = d ) τ ) α xτ)dτ, [,b]. Γ α) d Anoher ype of frcionl derivives, inroduced by Cpuo, is closely reled o he Riemnn Liouville definiions. Definiion.6 Cpuo s frcionl derivive). For funcion x ) AC[, b] wih 0 α < : The lef Cpuo frcionl derivive of order α is given by C Dα x) = Γ α) τ) α ẋτ)dτ, [,b]. The righ Cpuo frcionl derivive of order α is given by C Dα b x) = Γ α) τ ) α ẋτ)dτ, [,b]. Definiion.7 Hdmrd frcionl inegrl). Le x : [, b] R. The lef Hdmrd frcionl inegrl of order α > 0 is defined by I α x) = Γα) ln ) α xτ) dτ, ],b[. τ τ The righ Hdmrd frcionl inegrl of order α > 0 is defined by I α b x) = Γα) ln τ ) α xτ) dτ, ],b[. τ When α = m is n ineger, hese frcionl inegrls re m-fold inegrls: nd I m x) = I m b x) = dτ τ dτ τ τ dτ τm 2 xτ m )... dτ m, τ 2 τ m τ dτ 2 τ 2... τ m xτ m ) τ m dτ m.

5 4 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Definiion.8 Hdmrd frcionl derivive). For α > 0 nd n = [α] +, The lef Hdmrd frcionl derivive of order α is defined by D α x) = d ) n d Γn α) ln ) n α xτ) dτ, ],b[. τ τ The righ Hdmrd frcionl derivive of order α is defined by Db α x) = d ) n b ln τ ) n α xτ) dτ, ],b[. d Γn α) τ When α = m is n ineger, we hve D m x) = d ) m x) nd Db m d x) = d) d m x)..2. Frcionl Clculus of Vriions nd Opiml Conrol Mny generlizions o he clssicl clculus of vriions nd opiml conrol hve been mde o exend he heory o cover frcionl vriionl nd frcionl opiml conrol problems. A simple frcionl vriionl problem, for exmple, consiss in finding funcion x ) h minimizes he funcionl J[x )] = L,x), D α x))d, 4) where D α is he lef Riemnn Liouville frcionl derivive. Typiclly, some boundry condiions re prescribed s x) = x nd/or xb) = x b. Clssicl echniques hve been doped o solve such problems. The Euler Lgrnge equion for Lgrngin of he form L,x), D α x)) hs been derived firsly in [30, 3]. Mny vrins of necessry condiions of opimliy hve been sudied. A generlizion of he problem o include frcionl inegrls, i.e., L = L, I α x), D α x)), he rnsversliy condiions of frcionl vriionl problems nd mny oher specs cn be found in he lierure of recen yers. See [, 4, 6] nd references herein. Furhermore, i hs been shown h vriionl problem wih frcionl derivives cn be reduced o clssicl problem using n pproximion of he Riemnn Liouville frcionl derivives in erms of finie sum, where only derivives of ineger order re presen [6]. On he oher hnd, frcionl opiml conrol problems usully pper in he form of J[x )] = L,x),u))d min { s.. D α x) = f,x),u)) x) = x, xb) = x b, where n opiml conrol u ) ogeher wih n opiml rjecory x ) re required o follow frcionl dynmic nd, he sme ime, opimize n objecive funcionl. Agin, clssicl echniques re generlized o derive necessry opimliy condiions. Euler Lgrnge

6 Numericl Approximions o Frcionl Problems... 5 equions hve been inroduced, e.g., in [2]. A Hmilonin formlism for frcionl opiml conrol problems cn be found in [9] h excly follows he sme procedure of he regulr opiml conrol heory, i.e., hose wih only ineger-order derivives. Due o he growing number of pplicions of frcionl clculus in science nd engineering see, e.g., [, 2, 33, 34]), numericl mehods re being developed o provide ools for solving such problems. Using he Grünwld Lenikov pproch, i is convenien o pproxime he frcionl differeniion operor, D α, by generlized finie differences. In [25] some problems hve been solved by his pproximion. In [3] predicor-correcor mehod is presened h convers n iniil vlue problem ino n equivlen Volerr inegrl equion, while [20] shows he use of numericl mehods o solve such inegrl equions. A good survey on numericl mehods for frcionl differenil equions cn be found in [6]. A numericl scheme o solve frcionl differenil equions hs been inroduced in [7, 8], nd [7], mking n dpion, uses his echnique o solve frcionl opiml conrol problems. The scheme is bsed on n expnsion formul o pproxime he Riemnn Liouville frcionl derivive. The pproximions rnsform frcionl derivives ino finie sums conining only derivives of ineger order. In his chper, we ry o nlyze problems for which n nlyic soluion is vilble. This pproch gives us he biliy of mesuring he ccurcy of ech mehod. To his end, we need o mesure how close we ge o he exc soluions. We use hel 2 -norm nd define he error funcion E[x ), x )] by where x ) is defined on [,b]..3. A Generl Formulion E = x ) x ) 2 = [x) x)] 2 d The ppernce of frcionl erms of differen ypes, derivives nd inegrls, nd he fc h here re severl definiions for such operors, mkes i difficul o presen ypicl problem o represen ll possibiliies. Neverheless, one cn consider he opimizion of funcionls of he form J[x )] = ) 2 L,x),D α x))d 5) h depends on frcionl derivive, D α, in which x = x,x 2,...,x n ), α = α,α 2,...,α n ) nd α i, i =,2,...,n, re rbirry rel posiive numbers. The problem cn be wih or wihou boundry condiions. Mny seings of frcionl vriionl nd opiml conrol problems cn be rnsformed o he opimizion of 5). Consrins h usully pper in he clculus of vriions nd re lwys presen in opiml conrol problems cn be included in he funcionl using Lgrnge mulipliers. More precisely, in presence of dynmic consrins s frcionl differenil equions, we ssume h i is possible o rnsform such equions o vecor frcionl differenil equion of he form D α x) = f,x)).,

7 6 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres In his sge, we inroduce new vrible λ = λ,λ 2,...,λ n ) nd consider he opimizion of J[x )] = [L,x),D α x))+λ)d α x) λ)f,x))]d. When he problem depends on frcionl inegrls, I α, new vrible cn be defined s z) = I α x). Recll h D α I α x = x see, e.g., [9]). The equion D α z) = D α I α x) = x) cn be regrded s n exr consrin o be dded o he originl problem. However, problems conining frcionl inegrls cn be reed direcly o void he complexiy of dding n exr vrible o he originl problem. Ineresed reders re ddressed o [4, 28]. Throughou his chper, by frcionl vriionl problem, we minly consider he following one-vrible problem wih given boundry condiions: J[x )] = L,x),D α x))d min { x) = x, s.. xb) = x b. In his seing D α cn be replced by ny frcionl operor h is vilble in he lierure, sy, Riemnn Liouville, Cpuo, Grünwld Lenikov, Hdmrd nd so forh. The inclusion of consrins is done by Lgrnge mulipliers. The rnsiion from his problem o he generl one, equion 5), is srighforwrd nd is no discussed here..4. Soluion Mehods There re wo min pproches o solve vriionl, including opiml conrol, problems. On he one hnd, here re direc mehods. In brnch of direc mehods, he problem is discreized over mesh on he ineresed ime inervl. Discree vlues of he unknown funcion on mesh poins, finie differences for derivives, nd, finlly, qudrure rule for he inegrl, re used. This procedure reduces he vriionl problem, coninuous dynmic opimizion problem, o sic muli-vrible opimizion. Beer ccurcies re chieved by refining he underlying mesh size. Anoher clss of direc mehods uses funcion pproximion hrough liner combinion of he elemens of cerin bsis, e.g., power series. The problem is hen rnsformed ino he deerminion of he unknown coefficiens. To ge beer resuls in his sense, is he mer of using more deque or higher order funcion pproximions. On he oher hnd, here re indirec mehods h reduce vriionl problem o he soluion of differenil equion by pplying some necessry opimliy condiions. Euler Lgrnge equions nd Ponrygin s mximum principle re used, in his conex, o mke he rnsformion process. Once we solve he resuling differenil equion, n exreml for he originl problem is reched. Therefore, o rech beer resuls using indirec mehods, one hs o employ powerful inegrors. I is worh, however, o menion here h numericl mehods re usully used o solve prcicl problems. These wo mehods hve been generlized o cover frcionl problems, which is he essenil subjec of his chper.

8 Numericl Approximions o Frcionl Problems Expnsion Formuls o Approxime Frcionl Derivives This secion is devoed o presen wo pproximions for he Riemnn Liouville, Cpuo nd Hdmrd derivives h re referred s frcionl operors ferwrds. We inroduce he expnsions of frcionl operors in erms of infinie sums involving only ineger order derivives. These expnsions re hen used o pproxime frcionl operors in problems of he frcionl clculus of vriions nd frcionl opiml conrol. In his wy, one cn rnsform such problems ino clssicl vriionl or opiml conrol problems. Herefer, suible mehod, h cn be found in he clssicl lierure, is employed o find n pproximed soluion for he originl frcionl problem. Here we focus minly on he lef derivives nd he deils of exrcing corresponding expnsions for righ derivives re given whenever i is needed o pply new echniques. 2.. Riemnn Liouville Derivive 2... Approximion by Sum of Ineger Order Derivives Recll he definiion of he lef Riemnn Liouville derivive for α 0,), D α x) = d Γ α) d τ) α xτ)dτ. 6) The following heorem holds for ny funcion x ) h is nlyic in n inervl c,d) [,b]. See [6] for more deiled discussion nd [32] for differen proof. Theorem 2.. Lec,d), < c < d < +, be n open inervl inr, nd[,b] c,d) be such h for ech [,b] he closed bll B b ), wih cener nd rdius b, lies inc, d). If x ) is nlyic inc, d), hen D α x) = ) k αx k) ) k!k α)γ α) )k α. 7) Proof. Since x) is nlyic in c,d), nd B b ) c,d) for ny τ,) wih, b), he Tylor expnsion of xτ) is convergen power series, i.e., nd hen, by 6), xτ) = x τ)) = D α x) = d Γ α) d τ) α ) k x k) ) τ) k, k! ) k x k) ) τ) )dτ. k 8) k!

9 8 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Since τ) k α x k) ) is nlyic, we cn inerchnge inegrion wih summion, so D α x) = = = = Observe h d Γ α) d d Γ α) d Γ α) x) Γ α) ) α + Γ α) ) k x k) ) k! τ) k α dτ ) k x k) ) k!k + α) )k+ α ) k x k+) ) k!k + α) )k+ α + )k x k) ) k! k= ) k k α)k )! + )k k! ) ) ) x k) ) ) k α. ) k α ) ) k k α)k )! + )k k! = k )k +k ) k α ) k k α)k! = )k α k α)k!, since for ny k = 0,,2,... we hve k ) k +k ) k = 0. Therefore, he expnsion formul is reched s required. For numericl purposes, finie number of erms in 7) is used nd one hs D α x) N ) k αx k) ) k!k α)γ α) )k α. 9) Remrk 2.2. Wih he sme ssumpions of Theorem 2., we cn expnd xτ), xτ) = x+τ )) = x k) ) τ ) k, k! where τ, b). Similr clculions resul in he following pproximion for he righ Riemnn Liouville derivive: D α b x) N αx k) ) k!k α)γ α) b )k α. A proof for his expnsion is vilble [32] h uses similr relion for frcionl inegrls. The proof discussed here, however, llows o exrc n error erm for his expnsion esily.

10 Numericl Approximions o Frcionl Problems Approximion Using Momens of Funcion By momens of funcion, we hve no physicl or disribuive sense in mind. The nming comes from he fc h, during expnsion, he erms of he form V p ) := V p x)) = p) τ ) p 2 xτ)dτ, p N, τ, 0) resemble he formuls of cenrl momens cf. [8]). We ssume h V p x )), p N, denoes he p 2)h momen of funcion x ) AC 2 [,b]. The following lemm, h is given here wihou proof, is he key relion o exrc n expnsion formul for Riemnn Liouville derivives. Lemm 2.3 cf. Lemm 2.2 of [2]). Le x ) AC[,b] nd 0 < α <. Then he lef Riemnn Liouville frcionl derivive, D α x ), exiss lmos everywhere in [,b]. Moreover, D α x ) L p [,b] for p < α nd D α x) = [ x) Γ α) ) α + ] τ) α ẋτ)dτ,,b). ) The sme rgumen is vlid for he righ Riemnn Liouville derivive nd [ Db α x) = xb) Γ α) b ) α τ ) α ẋτ)dτ ],,b). Theorem 2.4 cf. [7]). Wih he sme ssumpions of Lemm 2.3, he lef Riemnn Liouville derivive cn be expnded s D α x) = ) α Γ α) x)+bα) ) α ẋ) [ ] Cα,p) ) p α Γp +α) V p ) Γα)Γ α)p )! ) α x), 2) p=2 where V p ) is defined by 0) nd Bα) = + Γ2 α) Cα,p) = p= Γ2 α)γα ) Γp +α), Γα )p! Γp +α). p )! Proof. Inegrion by prs on he righ-hnd-side of ) gives D α x) = x) Γ α) ) α + ẋ) Γ2 α) ) α + Γ2 α) τ) α ẍτ)dτ. 3)

11 0 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Since τ ) ), τ) α = ) α τ ) α. Using he binomil heorem, we hve τ ) α = p=0 Γp +α) Γα )p! ) τ p, in which he infinie series converges. Replcing for τ) α in 3) gives D α x) = x) Γ α) ) α + ẋ) Γ2 α) ) α ) + ) α Γp +α) τ p ẍτ)dτ, >. Γ2 α) Γα )p! p=0 Inerchnging he summion nd inegrion operions is possible, nd yields D α x) = x) Γ α) ) α + ẋ) Γ2 α) ) α + ) α Γ2 α) p=0 Γp +α) Γα )p! ) p τ ) p ẍτ)dτ, >. Decomposing he infinie sum, inegring, nd doing noher inegrion by prs, llow us o wrie D α x) = where = x) Γ α) ) α + ẋ) Γ2 α) ) α + ) α Γ2 α) + ) α Γ2 α) p= γα, p) p! ) p [ ) p ẋ) p x) Γ α) ) α + ẋ) Γ2 α) ) α + ) α Γ2 α) + ) α Γ2 α) p= γα, p) p )! ) p γα,p) = Γp +α). Γα ) τ ) p ẋτ)dτ, Repeing his procedure gin, nd simplifying he resuls, ends he proof. ẍτ)dτ ] τ ) p ẋτ)dτ p= γα, p) ẋ) p! The momens V p ), p = 2,3,..., cn be regrded s he soluions o he following sysem of differenil equions: { Vp ) = p) ) p 2 x) 4) V p ) = 0, p = 2,3,...

12 Numericl Approximions o Frcionl Problems... As before, numericl pproximion is chieved by king only finie number of erms in he series 2). We pproxime he frcionl derivive s D α x) A ) α x)+b ) α ẋ) N Cα,p) ) p α V p ), 5) where A = Aα,N) nd B = Bα,N) re given by Aα,N) = N Γp +α) +, Γ α) Γα)p )! p=2 6) Bα,N) = N Γp +α) +. Γ2 α) Γα )p! 7) Remrk 2.5. This expnsion hs been proposed in [4] nd simplificion hs been mde in [8], which uses he fc h he infinie series Γp +α) p= Γα )p! ends o, nd concludes h Bα) = 0, nd hus D α x) Aα,N) α x) p= p=2 N Cα,p) p α V p ). 8) In prcice, however, we only use finie number of erms in series. Therefore, + N p= p=2 Γp +α) Γα )p! nd we keep here he pproximion in he form of equion 5), [3]. To be more precise, he vlues of Bα,N), for differen choices of N nd α, re given in Tble. I shows h even for lrge N, whenαends o one, Bα,N) cnno be ignored. 0, Tble. Bα, N) for differen vlues of α nd N. N B0., N) B0.3, N) B0.5, N) B0.7, N) B0.9, N) B0.99, N) Remrk 2.6. Similr compuions give rise o n expnsion formul for Db α, he righ Riemnn Liouville frcionl derivive: N Db α x) Ab ) α x) Bb ) α ẋ) Cα,p)b ) p α W p ), p=2

13 2 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres where W p ) = p) b τ) p 2 xτ)dτ. The coefficiens A = Aα,N) nd B = Bα,N) re he sme s 6) nd 7) respecively, nd Cα,p) is s before. Remrk 2.7. As sed before, Cpuo derivives re closely reled o hose of Riemnn Liouville. For ny funcion, x ), nd for α 0,) for which hese wo kind of frcionl derivives, lef nd righ, exis, we hve nd C D α x) = D α x) x) ) α, C Db α x) = Db α xb) x) b ) α. Using hese relions, we cn esily consruc pproximion formuls for he lef nd righ Cpuo frcionl derivives, e.g., C D α x) Aα,N) ) α x)+bα,n) ) α ẋ) N Cα,p) ) p α V p ) x) ) α. p= Exmples To exmine he pproximions provided so fr, we ke some es funcions, nd pply 9) nd 5) o evlue heir frcionl derivives. We compue D αx), wih α = 2, for x) = 4 nd x) = e 2. The exc formuls for he frcionl derivives of polynomils re derived from 0D 0.5 n ) = nd for he exponenil funcion one hs Γn+) Γn+ 0.5) n 0.5, 0D 0.5 e λ ) = 0.5 E, 0.5 λ), where E α,β is he wo prmeer Mig Leffler funcion ). Figure shows he resuls using pproximion 9). As we cn see, he hird pproximions re resonbly ccure for boh cses. Indeed, for x) = 4, he pproximion wih N = 4 coincides wih he exc soluion becuse he derivives of order five nd more vnish. Now we use pproximion 5) o evlue frcionl derivives of he sme es funcions. In his cse, for given funcion x ), we cn compue V p by definiion, equion 0). One cn lso inegre he sysem 4) nlyiclly, if possible, or use ny numericl inegror. I is clerly seen in Figure 2 h one cn ge beer resuls by using lrger vlues of N. Compring Figures nd 2, we find ou h he pproximion 9) shows fser convergence. Observe h boh funcions re nlyic nd i is esy o compue higher-order derivives.

14 Numericl Approximions o Frcionl Problems Anlyic N=, E= N=2, E=0.3 N=3, E= Anlyic N=, E= N=2, E= N=3, E= α 0 D.5 α 0 D ) 0D ) b) 0D 0.5 e 2 ) Figure. Anlyic solid line) versus numericl pproximion 9) Anlyic N=, E= N=2, E=0.482 N=3, E= Anlyic N=, E= N=2, E= N=3, E= α 0 D.5 α 0 D ) 0D ) b) 0D 0.5 e 2 ) Figure 2. Anlyic solid line) versus numericl pproximion 5). Remrk 2.8. A closer look o 9) nd 5) revels h in boh cses he pproximions re no compuble nd b for he lef nd righ frcionl derivives, respecively. A hese poins we ssume h i is possible o exend hem coninuously o he closed inervl [,b]. Following Remrk 2.5, we show here h neglecing he firs derivive in he expnsion 5) cn cuse considerble loss of ccurcy in compuion. Once gin, we compue he frcionl derivives of x) = 4 nd x) = e 2, bu his ime we use he pproximion given by 8). Figure 3 summrizes he resuls. Approximion 5) gives more relisic pproximion using quie smll N,3in his cse Hdmrd Derivives For Hdmrd derivives, he expnsions cn be obined in quie similr wy [27].

15 4 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Anlyic Approxime, B 0, N=3, E= Approxime, B = 0, N=3, E= Anlyic Approxime, B 0, N=3, E= Approxime, B = 0, N=3, E= α 0 D α 0 D ) 0D ) b) 0D 0.5 e 2 ) Figure 3. Comprison of pproximion 5) nd pproximion 8) of [8] Approximion by Sum of Ineger Order Derivives Assume h funcion x ) dmis derivives of ny order, hen expnsion formuls for he Hdmrd frcionl inegrls nd derivives of x, in erms of is ineger-order derivives, re given in [0, Theorem 7]: 0I α x) = S α,k) k x k) ) nd where 0D α x) = Sα,k) k x k) ), Sα,k) = k! k ) k ) k j j α j j= is he Sirling funcion. As pproximions, we runce infinie sums n pproprie order N nd ge he following formuls: N 0I α x) S α,k) k x k) ), nd N 0D α x) Sα,k) k x k) ).

16 Numericl Approximions o Frcionl Problems Approximion Using Momens of Funcion The sme ide of expnding Riemnn Liouville derivives, wih slighly differen echniques, is used o derive expnsion formuls for lef nd righ Hdmrd derivives. The following lemm is bsis for hese new relions. Lemm 2.9. Le α 0,) nd x ) be n bsoluely coninuous funcion on [,b]. Then he Hdmrd frcionl derivives my be expressed by nd D α D α b x) x) = ln α + Γ α) ) Γ α) xb) x) = ln b α Γ α) ) Γ α) ln τ) α ẋτ)dτ 9) ln τ ) αẋτ)dτ. A proof of his lemm, for n rbirry α > 0, cn be found in [8, Theorem 3.2]. Theorem 2.0. Le 0 < < b nd x : [,b] R be n bsoluely coninuous funcion. Then D α x) = Γ α) [ Cα, p) p=2 ln ) α x)+bα) ln ) α ẋ) ln ) α p V p ) Γp+α ) ln α x) Γα)Γ α)p )! ) ] wih Bα) = + Γ2 α) p= Γp+α ), Γα )p! Γp+α ) Cα,p) = Γ α)γ+α)p )!, V p ) = p) ln τ ) p 2 xτ) dτ. τ Proof. We rewrie 9) s D α x) = x) ln α + ln Γ α) ) α τẋτ)dτ Γ α) τ τ) nd hen inegring by prs gives D α x) = x) ln ) α + ẋ) Γ α) Γ2 α) + Γ2 α) ln ) α ln τ) α [ẋτ)+τẍτ)]dτ.

17 6 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Now we use he following expnsion for ln τ) α, using he binomil heorem, ln τ) α = = ln ) α ln τ ln ) α ln ) α Γp +α) Γα )p! p=0 ) ln τ p ) ln p. This implies h D α x) = x) Γ α) p=0 ln ) α + ẋ) Γ2 α) Γp +α) Γα )p! ln ) p ln ) α + ln ) α Γ2 α) ln τ ) p[ẋτ)+τẍτ)]dτ. Exrcing he firs erm of he infinie sum, simplificions nd noher inegrion by prs using u = ln τ p,du ) = p) τ ln τ p ) nd dv = [ẋτ)+τẍτ)]dτ, v = τẋτ) yields D α x) = x) Γ α) p= ln ) α +Bα) ln ) α ẋ) Γp +α) Γα )p )! ln ) p ln ) α Γ2 α) ln τ ) p ẋτ)dτ. A finl sep of exrcing he firs erm in he sum nd inegrion by prs finishes he proof. For prcicl purposes, finie sums up o order N re considered nd he pproximion becomes D α x) Aα,N) ln ) α x)+bα,n) ln ) α ẋ) + N p=2 Cα, p) ln ) α p V p ) 20) wih Aα,N) = Bα,N) = + Γ α) + Γ2 α) N Γp+α ), Γα)p )! N Γp+α ). Γα )p! p=2 p=

18 Numericl Approximions o Frcionl Problems... 7 Remrk 2.. The righ Hdmrd frcionl derivive cn be expnded in he sme wy. This gives he following pproximion: Db α x) Aα,N) ln b α x) Bα,N) ln ) b α ẋ) ) N p=2 Cα, p) ln b α p W p ) ) wih W p ) = p) ln b ) p 2 xτ) dτ. τ τ Exmples In his secion we pply 20) o compue frcionl derivives, of orderα = 2, forx) = 4 nd x) = ln). The exc Hdmrd frcionl derivive is vilble for x) = 4 nd we hve D ) = ln Γ.5). For x) = ln), only n pproximion of he Hdmrd frcionl derivive is found in he lierure: D 0.5 ln) Γ0.5) ln erf3 ln). Γ0.5) The resuls of pplying 20) o evlue frcionl derivives re depiced in Figure Anlyic N=3, E=7.75e Anlyic N=3, E= N=4, E=0.38 N=5, E= ) D 0.5 ln ) b) D ) Figure 4. Anlyic versus numericl pproximion 20).

19 8 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Error Anlysis When we pproxime n infinie series by finie sum, he choice of he order of pproximion is key quesion. Hving n esime knowledge of runcion errors, one cn choose properly up o which order he pproximions should be mde o sui he ccurcy requiremens. In his secion we sudy he errors of he pproximions presened so fr. Seprion of n error erm in 8) concludes in D α x) = d Γ α) d d + Γ α) d τ) α N τ) α ) k x k) ) τ) )dτ k k! k=n+ ) k x k) ) τ) )dτ. k 2) k! The firs erm in 2) gives 9) direcly nd he second erm is he error cused by runcion. The nex sep is o give locl upper bound for his error, E r ). The series k=n+ ) k x k) ) τ) k, τ,),,b), k! M is he reminder of he Tylor expnsion of xτ) nd hus bounded by N+)! τ)n+ in which Then, E r ) M d Γ α)n +)! d M = mx τ [,] xn+) τ). τ) N+ α dτ = M Γ α)n +)! )N+ α. In order o esime runcion error for pproximion 5), he expnsion procedure is crried ou wih seprion of N erms in binomil expnsion s τ ) α = = p=0 N p=0 Γp +α) Γα )p! Γp +α) Γα )p! ) τ p ) τ p +R N τ), 22) where R N τ) = p=n+ Γp +α) Γα )p! ) τ p.

20 Subsiuing 22) ino 3), we ge Numericl Approximions o Frcionl Problems... 9 D α x) = = x) Γ α) ) α + ẋ) Γ2 α) ) α N ) + ) α Γp +α) τ p +R N τ) ẍτ)dτ Γ2 α) Γα )p! p=0 x) Γ α) ) α + ẋ) Γ2 α) ) α N ) + ) α Γp +α) τ p ẍτ)dτ Γ2 α) Γα )p! + ) α Γ2 α) p=0 R N τ)ẍτ)dτ. A his poin, we pply he echniques of [8] o he firs hree erms wih finie sums. Then, we receive 5) wih n exr erm of runcion error: Since 0 τ R N τ) E r ) = ) α Γ2 α) for τ [,], one hs Γp +α) Γα )p! p=n+ e α)2 + α p=n p 2 α = p=n+ R N τ)ẍτ)dτ. ) α p dp = e α)2+ α α)n α. Finlly, ssuming L 2 = mx x 2) τ), we conclude h τ [,] p=n+ E r ) L 2 e α)2 + α Γ2 α) α)n α )2 α. e α)2 + α p 2 α Remrk 2.2. Following similr echniques, one cn exrc n error bound for he pproximions of Hdmrd derivives. When we consider finie sums in 20), he error is bounded by e α)2 + α E r ) L) Γ2 α) α)n α ln ) α ), where L) = mx τ [,] ẋτ)+τẍτ).

21 20 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres 3. Direc Mehods There re wo min clsses of direc mehods in he clssicl clculus of vriions nd opiml conrol. On he one hnd, we specify discreizion scheme by choosing se of mesh poins on he horizon of ineres, sy = 0,,..., n = b for [,b]. Then we use some pproximions for derivives in erms of unknown funcion vlues i nd, using n pproprie qudrure, he problem is rnsformed o finie dimensionl opimizion problem. This mehod is known s Euler s mehod in he lierure [5]. Regrding Figure 5, he solid line is he funcion h we re looking for, neverheless, he mehod gives he polygonl dshed line s n pproxime soluion. x x n ) x x 2 x i x n x 0 ) 0 h 2 i n n Figure 5. Euler s finie differences mehod. On he oher hnd, here is he Riz mehod, h hs n exension o funcionls of severl independen vribles which is clled Knorovich s mehod. We ssume h he dmissible funcions cn be expnded in some kind of series, e.g., power or Fourier s series, of he form x) = k φ k ). Using finie number of erms in he sum s n pproximion, nd some sor of qudrure gin, he originl problem cn be rnsformed o n equivlen opimizion problem for k,k = 0,,...,n. In he presence of frcionl operors, he sme ides re pplied o discreize problem. Mny works cn be found in he lierure h use differen ypes of bsis funcions o esblish Riz-like mehods for frcionl clculus of vriions nd opiml conrol. 3.. Euler-like Mehods The Euler mehod in he clssicl heory of he clculus of vriions uses finie differences pproximions for derivives nd is referred lso s he mehod of finie differences. The

22 Numericl Approximions o Frcionl Problems... 2 bsic ide of his mehod is h insed of considering he vlues of funcionl J[x )] = L, x), ẋ))d wih boundry condiions x) = x nd xb) = x b, on rbirry dmissible curves, we only rck he vlues n n + grid poins, i, i = 0,...,n, of he ineresed ime inervl [29]. The funcionl J[x )] is hen rnsformed ino funcion Ψx ),x 2 ),...,x n )) of he vlues of unknown funcion on mesh poins. Assuming h = i i,x i ) = x i nd ẋ i x i x i h, one hs n J[x )] Ψx,x 2,...,x n ) = h L i,x i, x ) i x i, h i= x 0 = x, x n = x b. The desired vlues of x i,i =,...,n, re he exremum of he muli-vrible funcion Ψ which is he soluion o he sysem Ψ x i = 0, i =,...,n. The fc h only wo erms in he sum,i )h ndih, depend onx i, mkes i rher esy o find he exremum of Ψ solving sysem of lgebric equions. For ech n, we obin polygonl line which is n pproxime soluion of he originl problem. I hs been shown h pssing o he limi s h 0, he liner sysem corresponding o finding he exremum of Ψ is equivlen o he Euler Lgrnge equion of he problem Finie Differences for Frcionl Derivives In clssicl heory, given derivive of cerin order, x n), here is finie difference pproximion of he form x n) ) = lim h 0 + h n n ) n ) k x kh), k where n k) is he binomil coefficien nd ) n nn )n 2) n k +) =, n,k N. k k! The Grünwld Lenikov definiion of frcionl derivive is generlizion of his formul o derivives of rbirry order. The series in 2) nd 3), he Grünwld Lenikov definiions, converge bsoluely nd uniformly if x ) is bounded. The infinie sums, bckwrd differences for he lef nd forwrd differences for he righ derivive in he Grünwld Lenikov definiions for frcionl derivives, revels h he rbirry order derivive of funcion ime depends on ll vlues of h funcion in,] nd [, ), for lef nd righ derivives respecively. This is due o he non-locl propery of frcionl derivives.

23 22 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Remrk 3.. Equions 2) nd 3) need o be consisen in closed ime inervls nd we need he vlues of x) ouside he inervl [,b]. To overcome his difficuly, we cn ke { x x) [,b], ) = 0 / [,b]. Then we ssume GL Dα x) = GL Dα x ) nd GL Dα bx) = GL Dα b x ) for [,b]. This definiion coincides wih Riemnn Liouville nd Cpuo derivives. The ler is believed o be more pplicble in prcicl fields such s engineering nd physics. Proposiion 3.2 See [25]). Le 0 < α < n, n N nd x ) C n [,b]. Suppose lso hx n) ) is inegrble on[,b]. Then, for everyα, he Riemnn Liouville derivive exiss nd coincides wih he Grünwld Lenikov derivive nd he following holds: n D α x) = i=0 = GL D α x). x i) ) ) i α Γ+i α) + Γn α) τ) n α x n) τ)dτ Remrk 3.3. For numericl purposes we need finie series in 2). Given grid on [,b] s = 0,,..., n = b, where i = 0 + ih for some h > 0, we pproxime he lef Riemnn Liouville derivive s D α x i ) h α i ωk α )x i kh), 23) where ωk α) = )k α) k = Γk α) Γ α)γk+). Similrly, one cn pproxime he righ Riemnn Liouville derivive by Db α x i) n i h α ωk α )x i +kh). 24) Remrk 3.4. The Grünwld Lenikov pproximion of Riemnn Liouville is firs order pproximion [25], i.e., D α x i ) = h α i ωk α )x i kh)+oh). Remrk 3.5. I hs been shown h he implici Euler mehod soluion o cerin frcionl pril differenil equion bsed on he Grünwld Lenikov pproximion o he frcionl derivive, is unsble [23]. Therefore, discreizing frcionl derivives, shifed Grünwld Lenikov derivives re used nd, despie he sligh difference, hey exhibi sble performnce, les for cerin cses. The shifed Grünwld Lenikov derivive is defined by sgl D α x i ) h α i ωk α )x i k )h).

24 Numericl Approximions o Frcionl Problems Oher finie difference pproximions cn be found in he lierure. We refer here o he Diehelm bckwrd finie difference formul for Cpuo s frcionl derivive, wih 0 < α < 2 nd α, which is n pproximion of order Oh 2 α ) [6]: C D α x i ) h α Γ2 α) i j=0 i,j α i j) x k h k i j x k) ), k! where, if i = 0, i,j = j +) α 2j α +j ) α, if 0 < j < i, α)i α i α +i ) α, if j = i Euler-like Direc Mehod for Frcionl Vriionl Problems As menioned erlier, we consider simple version of frcionl vriionl problems where he frcionl erm hs Riemnn Liouville form on finie ime inervl[, b]. The boundry condiions re given nd we pproxime he problem using he Grünwld Lenikov pproximion given by 23). In his conex, we discreize he funcionl in 4) using simple qudrure rule on he mesh poins, = 0,,,..., n = b, wih h = b n. The gol is o find he vluesx,x 2,...,x n of he unknown funcionx ) poins i,i =,...,n. The vlues of x 0 nd x n re given. Applying he qudrure rule gives J[x )] = n i= i i L i,x i, D α x i )d n hl i,x i, D α x i ) i= nd by pproximing he frcionl derivives mesh poins using 23) we hve J[x )] n hl i,x i, h α i= ) i ωk α )x i k. 25) Herefer he procedure is he sme s in he clssicl cse. The righ-hnd-side of 25) cn be regrded s funcion Ψ of n unknowns x = x,x 2,...,x n ), Ψx) = n hl i,x i, h α i= ) i ωk α )x i k. 26) To find n exremum for Ψ, one hs o solve he following sysem of lgebric equions: Ψ x i = 0, i =,...,n. 27) Unlike he clssicl cse, ll erms, sring from he ih erm in 26), depend on x i nd we hve Ψ = h L x i x i,x i, D α x n i i)+h h α ωα k ) L D αx i+k,x i+k, D α x i+k). 28)

25 24 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Equing he righ-hnd-side of 28) wih zero, one hs L x i,x i, D α x i )+ n i h α ωk α ) L D αx i+k,x i+k, D α x i+k ) = 0. Pssing o he limi, nd considering he pproximion formul for he righ Riemnn Liouville derivive, equion 24), i is srighforwrd o verify h: Theorem 3.6. The Euler-like mehod for frcionl vriionl problem of he form 4) is equivlen o he frcionl Euler Lgrnge equion s he mesh size, h, ends o zero. L x + Db α L D αx = 0, Proof. Consider minimizer x,...,x n ) of Ψ, vriion funcion η C[,b] wih η) = ηb) = 0 nd define η i = η i ), for i = 0,...,n. We remrk h η 0 = η n = 0 nd h x + ǫη,...,x n + ǫη n ) is vriion of x,...,x n ), wih ǫ < r, for some fixed r > 0. Therefore, since x,...,x n ) is minimizer for Ψ, proceeding wih Tylor s expnsion, we deduce h where 0 Ψx +ǫη,...,x n +ǫη n ) Ψx,...,x n ) [ ] n L = ǫ h x [i]η i + L D α [i] i h α ωk α )η i k +Oǫ), i= [i] = i,x i, h α ) i ωk α )x i k. Since ǫ kes ny vlue, i follows h [ n L h x [i]η i + L D α [i] h α i= ] i ωk α )η i k = 0. 29) On he oher hnd, since η 0 = 0, reordering he erms of he sum, i follows immediely h n L i n D α [i] ωk α )η n i i k = η i ωk α ) L D α [i+k]. i= Subsiuing his relion ino equion 29), we obin [ ] n L η i h x [i]+ n i h α ωk α ) L D α [i+k] = 0. i= i= Since η i is rbirry, for i =,...,n, we deduce h L x [i]+ n i h α ωk α ) L D α [i+k] = 0, for i =,...,n.

26 Numericl Approximions o Frcionl Problems Le us sudy he cse when n goes o infiniy. Le ],b[ nd i {,...,n} such h i < i. Firs observe h, in such cse, we lso hve i nd n i. In fc, le i {,...,n} be such h So,i < )/h+, which implies h Then +i )h < +ih. n i > n b b. lim n,i i =. Assume h here exiss funcion x C[, b] sisfying ǫ > 0 N n N : x i x i ) < ǫ, i =,...,n. Asxis uniformly coninuous, we hve ǫ > 0 N n N : x i x) < ǫ, i =,...,n. By he coninuiy ssumpion of x, we deduce h lim n,i n i h α ωk α ) L D α [i+k] = Db α L D α,x), D α x)). For n sufficienly lrge nd herefore i lso sufficienly lrge), In conclusion, lim n,i L x,x), D α x))+ D α b L L [i] = x x,x), D α x)). L D α,x), D α x)) = 0. 30) Using he coninuiy condiion, we prove h he frcionl Euler Lgrnge equion 30) holds for ll vlues on he closed inervl b Exmples Now we pply he Euler-like direc mehod o some es problems for which he exc soluions re known. Alhough we propose problems for he inervl [0, ], moving o rbirry inervls is only mer of more compuions. To mesure he errors reled o pproximions, differen norms cn be used. Since direc mehod seeks for he funcion vlues cerin poins, we use he mximum norm o deermine how close we cn ge o he exc vlue h poin. Assume h he exc vlue of he funcion x ), he poin i, is x i ) nd i is pproximed by x i. The error is defined s E = mx{ x i ) x i, i =,,n }.

27 26 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Exmple 3.7. Our gol here is o minimize qudric Lgrngin on [0,] wih fixed boundry condiions. Consider he following minimizion problem: { J[x )] = ) 2d 0 0D 0.5 x) 2 Γ2.5).5 min 3) x0) = 0, x) =. Since he Lgrngin is lwys posiive, problem 3) ins is minimum when 0D 0.5 x) 2 Γ2.5).5 = 0 nd hs he obvious soluion of he form x) = 2 becuse 0 D = 2 Γ2.5).5. To begin wih, we pproxime he frcionl derivive by 0D 0.5 x i ) h i ) ω 0.5 k xi kh) for fixedh > 0. The funcionl is now rnsformed ino ) 2 i ) J[x )] = ω 0.5 h 0.5 k xi k 2 Γ2.5).5 d. i= Finlly, we pproxime he inegrl by recngulr rule nd end wih he discree problem ) 2 n i ) Ψx) = h ω 0.5 h 0.5 k xi k 2 Γ2.5).5 i. Since he Lgrngin in his exmple is qudric, sysem 27) hs liner form nd herefore is esy o solve. Oher problems my end wih sysem of nonliner equions. Simple clculions led o he sysem Ax = b, 32) in which A = n i=0 A2 i n 2 n i= A ia i i=0 A n 2 ia i+ i= A2 i n 3 i=0 A n 3 ia i+2 i= A ia i i=0 A ia i+n 2 i=0 A ia i+n 3 where A i = ) i h.5 0.5) i nd b = b,b 2,,b n ) wih b i = n i 2h 2 A k Γ2.5).5 k+i A n ia 0 n i n n 2 i=n 2 A ia i n 2) i=n 3 A ia i n 3) n 3 i=n 4 A ia i n 4) i=0 A2 i ) A k A k+i. Since sysem 32) is liner, i is esily solved for differen vlues of n. As indiced in Figure 6, by incresing he vlue of n we ge beer soluions. Le us now move o noher exmple for which he soluion is obined by he frcionl Euler Lgrnge equion.,

28 Numericl Approximions o Frcionl Problems Anlyic soluion Approximion: n = 5, Error= 0.03 Approximion: n = 0, Error= 0.02 Approximion: n = 30, Error= x) Figure 6. Anlyic nd pproxime soluions of Exmple 3.7. Exmple 3.8. Consider he following minimizion problem: { J[x )] = 0 0D 0.5 x) ẋ 2 ) ) d min x0) = 0, x) =. 33) In his cse he only wy o ge soluion is by use of Euler Lgrnge equions. The Lgrngin depends no only on he frcionl derivive, bu lso on he firs order derivive of he funcion. The Euler Lgrnge equion for his seing becomes L x + D α b L D α d d ) L = 0, ẋ nd by direc compuions necessry condiion for x ) o be minimizer of 33) is D α +2ẍ) = 0 or ẍ) = 2Γ α) ) α. Subjec o he given boundry condiions, he bove second order ordinry differenil equion hs he soluion x) = 2Γ3 α) )2 α + ) + 2Γ3 α) 2Γ3 α). 34) Discreizing problem 33) wih he sme ssumpions of Exmple 3.7 ends in liner

29 28 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Anlyic soluion Approximion: n = 5, Error= Approximion: n = 0, Error= Approximion: n = 30, Error= x) Figure 7. Anlyic nd pproxime soluions of Exmple 3.8. sysem of he form x x 2 x 3. x n = b b 2 b 3. b n, 35) where nd b i = h 2 n i b n = h 2 ) k h k ), i =,2,...,n 2, )) 0.5 ) k h 0.5 +x n. k Sysem 35) is liner nd cn be solved for nyno rech he desired ccurcy. The nlyic soluion ogeher wih some pproximed soluions re shown in Figure 7. Boh exmples bove end wih liner sysems nd heir solvbiliy is simply dependn o he mrix of coefficiens. Now we ry his mehod on more compliced problem, ye nlyiclly solvble, wih n oscilling soluion.

30 Numericl Approximions o Frcionl Problems Exmple 3.9. Consider he problem of minimizing 0 Ld subjec o he boundry condiions x0) = 0 nd x) =, where he Lgrngin L is given by L = 0D 0.5 x) 6Γ6) Γ5.5) Γ4) Γ3.5) ) 4 Γ.5) 0.5. This exmple hs n obvious soluion oo. Since L is posiive, he minimizer is Noe h D α ) ν = Γν+) Γν+α) ν α. x) = The ppernce of fourh power in he Lgrngin, resuls in nonliner sysem s we pply he Euler-like direc mehod o his problem. For j =,,n we hve where n i=j ) ω 0.5 i j h 0.5 i ) ω 0.5 k xi k φ i ) φ) = 6Γ6) Γ5.5) Γ4) Γ3.5) Γ.5) 0.5. ) 3 = 0, 36) Sysem 36) is solved for differen vlues of n nd he resuls re depiced in Figure x) Anlyic Approximion: n = 5, E=.48e+000 Approximion: n = 20, E= 3.0e 00 Approximion: n = 90, E= 6.8e Figure 8. Anlyic nd pproxime soluions of Exmple 3.9.

31 30 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres 4. Indirec Mehods As in he clssicl cse, indirec mehods in frcionl sense provide he necessry condiions of opimliy using he firs vriion. Frcionl Euler Lgrnge equions re now well-known nd well-sudied subjec in frcionl clculus. For simple problem of he form 4), following [], necessry condiion implies h he soluion mus sisfy frcionl boundry vlue differenil equion. Theorem 4. cf. []). Le x ) hve coninuous lef Riemnn Liouville derivive of order α nd J be funcionl of he form J[x )] = L,x), D α x))d 37) subjec o he boundry condiions x) = x nd xb) = x b. Then necessry condiion for J o hve n exremum for funcion x ) is h x ) sisfies he following Euler- Lgrnge equion: { L x + D α b L D αx = 0, x) = x, xb) = x b, which is clled he frcionl Euler Lgrnge equion. Proof. Assume h x ) is he desired funcion nd le x) = x )+ǫη) be fmily of curves h sisfy boundry condiions, i.e., η) = ηb) = 0. Since D α is liner operor, for ny x ), he funcionl becomes J[x )] = L,x )+ǫη), D α x )+ǫ D α η))d, which is funcion of ǫ,j[ǫ]. Since J ssumes is exremum ǫ = 0, one hs dj dǫ ǫ=0 = 0, i.e., [ L x η + L ] D α D α x η d = 0. Using he frcionl inegrion by prs of he form g) D α f)d = f) Db α g)d on he second erm nd pplying he fundmenl heorem of he clculus of vriions complees he proof. Remrk 4.2. Mny vrins of his heorem cn be found in he lierure. Differen ypes of frcionl erms hve been embedded in he Lgrngin nd pproprie versions of Euler Lgrnge equions hve been derived using proper inegrion by prs formuls. See [, 3, 6, 22, 24] for deils. 38)

32 Numericl Approximions o Frcionl Problems... 3 For frcionl opiml conrol problems, so-clled Hmilonin sysem is consruced using Lgrnge mulipliers. For exmple, cf. [9], ssume h we re required o minimize funcionl of he form J[x ),u )] = L, x), u))d such h x) = x, xb) = x b nd D α x) = f,x),u)). Similr o he clssicl mehods, one cn inroduce Hmilonin H = L,x),u)) +λ)f,x),u)), where λ) is considered s Lgrnge muliplier. In his cse we define he ugmened funcionl s J[x ),u ),λ )] = [H,x),u),λ)) λ) D α x)]d. Opimizing he ler funcionl resuls in he following necessry opimliy condiions: D α x) = H λ Db α H λ) = x 39) H u = 0. Togeher wih he prescribed boundry condiions, his mkes wo poin frcionl boundry vlue problem. These rgumens revel h, like he clssicl cse, frcionl vriionl problems end wih frcionl boundry vlue problems. To rech n opiml soluion, one needs o del wih frcionl differenil equion or sysem of frcionl differenil equions. The clssicl heory of differenil equions is furnished wih severl soluion mehods, heoreicl nd numericl. Neverheless, solving frcionl differenil equion is rher ough sk [2]. To benefi hose mehods, especilly ll solvers h re vilble o solve n ineger order differenil equion numericlly, we cn eiher pproxime frcionl vriionl problem by n equivlen ineger-order one or pproxime he necessry opimliy condiions 38) nd 39). The res of his secion discusses wo ypes of pproximions h re used o rnsform frcionl problem o one in which only ineger order derivives re presen; i.e., we pproxime he originl problem by subsiuing frcionl erm by is corresponding expnsion formuls. This is minly done by cse sudies on cerin exmples. The exmples re chosen so h eiher hey hve rivil soluion or i is possible o ge n nlyic soluion using frcionl Euler Lgrnge equions. By subsiuing he pproximions 9) or 5) for he frcionl derivive in 37), he problem is rnsformed o J[x )] = = L,x), N ) ) k αx k) ) k!k α)γ α) )k α d L ),x),ẋ),...,x N) ) d

33 32 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres or J[x )] = = Ax) L,x), ) α + Bẋ) N ) α L,x),ẋ),V 2 ),...,V N ))d { Vp ) = p) ) p 2 x) V p ) = 0, p = 2,3,... Cα,p)V p ) d ) p+α p=2 The former problem is clssicl vriionl problem conining higher order derivives. The ler is muli-vrible problem, subjec o some ordinry differenil equion consrin. Togeher wih he boundry condiions, boh bove problems belong o clsses of well sudied vriionl problems. To ccomplish deiled sudy, s es problems, we consider here Exmple 3.8, { J[x )] = 0 0D 0.5 x) ẋ 2 ) ) d min 40) x0) = 0, x) =, nd he following exmple. Exmple 4.3. Given α 0, ), consider he funcionl J[x )] = 0 D α x) )2 d 4) o be minimized subjec o he boundry condiions x0) = 0 ndx) = Γα+). Since he inegrnd in 4) is non-negive, he funcionl ins is minimum when D α x) =, i.e., for x) = α Γα+). We illusre he use of he wo differen expnsions seprely. 4.. Expnsion o Ineger Orders Using pproximion 9) for he frcionl derivive in 40), we ge he pproximed problem [ N ] min J[x )] = Cn,α) n α x n) ) ẋ 2 ) d 42) 0 n=0 x0) = 0, x) =, which is clssicl higher-order problem of he clculus of vriions h depends on derivives up o order N. The corresponding necessry opimliy condiion is wellknown resul. Theorem 4.4 cf., e.g., [2]). Suppose h x ) C 2N [,b] minimizes L,x),x ) ),x 2) ),...,x N) ))d

34 wih given boundry condiions Numericl Approximions o Frcionl Problems x) = 0, xb) = b 0, x ) ) =, x ) b) = b,. x N ) ) = N, x N ) b) = b N. Then x ) sisfies he Euler Lgrnge equion L x d L )+ d2 L d x ) d 2 x 2) ) + ) N dn d N ) L x N) = 0. 43) In generl 43) is n ODE of order 2N, depending on he order N of he pproximion we choose, nd he mehod leves 2N 2 prmeers unknown. In our exmple, however, he Lgrngin in 42) is liner wih respec o ll derivives of order higher hn wo. The resuling Euler Lgrnge equion is he second order ODE h hs he soluion where N ) n Cn,α) dn d nn α ) d d [ 2ẋ)] = 0 n=0 x) = M α,n) 2 α +M 2 α,n), [ N ] M α,n) = ) n Γn+ α)cn,α), 2Γ3 α) n=0 [ ] N M 2 α,n) = + ) n Γn+ α)cn,α). 2Γ3 α) n=0 Figure 9 shows he nlyic soluion ogeher wih severl pproximions. I revels h by incresing N, pproxime soluions do no converge o he nlyic one. The reson is he fc h he soluion 34) o Exmple 3.8 is no n nlyic funcion. We conclude h 9) my no be good choice o pproxime frcionl vriionl problems. In conrs, s we shll see, he pproximion 5) leds o good resuls. To solve Exmple 3.8 using 9) s n pproximion for he frcionl derivive, he problem becomes N 2 min J[x )] = Cn,α) n α x ) ) n) d, 0 n=0 x0) = 0, x) = Γα+). The Euler Lgrnge equion 43) gives 2N order ODE. For N 2 his pproch is inpproprie since he wo given boundry condiions x0) = 0 ndx) = Γα+) re no enough o deermine he 2N consns of inegrion.

35 34 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres Anlyic N= N=3 N= x) Figure 9. Anlyic versus pproxime soluions o Exmple 3.8 using pproximion 9) wihα = Expnsion hrough he Momens of Funcion If we use 5) o pproxime he opimizion problem 40), wih A = Aα,N), B = Bα,N) nd C p = Cα,p), we hve N J[x )] = A α x)+b α ẋ) C p p α V p ) ẋ 2 ) d, 0 V p ) = p) p 2 x), p = 2,...,N, V p 0) = 0, p = 2,...,N, x0) = 0, x) =. Problem 44) is consrined wih se of ordinry differenil equions nd is nurl o look o i s n opiml conrol problem [26]. For h we inroduce he conrol vrible u) = ẋ). Then, using he Lgrnge mulipliers λ,λ 2,...,λ N, nd he Hmilonin sysem, one cn reduce 44) o he sudy of he wo poin boundry vlue problem wih boundry condiions { x0) = 0, V p 0) = 0, p = 2,...,N, p=2 ẋ) = 2 B α 2 λ ), V p ) = p) p 2 x), p = 2,...,N, λ ) = A α N p=2 p)p 2 λ p ), λ p ) = C p p α), p = 2,...,N, { x) =, λ p ) = 0, p = 2,...,N, where x0) = 0 nd x) = re given. We hve V p 0) = 0, p = 2,...,N, due o 4) nd λ p ) = 0, p = 2,...,N, becuse V p is free finl ime for p = 2,...,N [26]. In 44) 45)

36 Numericl Approximions o Frcionl Problems Anlyic N=2 N=5 N=0 N=6 0.6 x) Figure 0. Anlyic versus pproxime soluions o Exmple 3.8 using pproximion 5) wihα = 0.5. generl, he Hmilonin sysem is nonliner, hrd o solve, wo poin boundry vlue problem h needs specil numericl mehods. In his cse, however, 45) is non-coupled sysem of ordinry differenil equions nd is esily solved o give N x) = Mα,N) 2 α Cα, p) N Cα, p) 2p2 p α) p + Mα,N)+, 2p2 p α) where Mα,N) = p=2 Bα,N) Aα,N) N 22 α) α p=2 p=2 Cα,p) p). α)2 p α) Figure 0 shows he grph of x ) for differen vlues of N. Le us now pproxime Exmple 4.3 using 5). The resuling minimizion problem hs he following form: 2 N min J[x )] = A α x)+b α ẋ) C p p α V p ) d, 0 V p ) = p) p 2 x), p = 2,...,N, V p 0) = 0, p = 2,...,N, x0) = 0, x) = Γα+). Following he clssicl opiml conrol pproch of Ponrygin [26], his ime wih N u) = A α x)+b α ẋ) C p p α V p ), p=2 p=2 46)

37 36 Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres.4.2 Anlyic: J=0 Approximion: N=2, J= x) Figure. Anlyic versus pproxime soluion o Exmple 4.3 using pproximion 5) wihα = 0.5. we conclude h he soluion o 46) sisfies he sysem of differenil equions ẋ) = AB x)+ N p=2 B C p p V p )+ 2 B 2 2α 2 λ )+B α, V p ) = p) p 2 x), p = 2,...,N, λ ) = AB λ N p=2 p)p 2 λ p ), λ p ) = B C p p λ, p = 2,...,N, 47) where A = Aα,N), B = Bα,N) nd C p = Cα,p) re defined ccording o Secion 2..2, subjec o he boundry condiions { x0) = 0, V p 0) = 0, p = 2,...,N, { x) = Γα+), λ p ) = 0, p = 2,...,N. 48) The soluion o sysem 47) 48), wih N = 2, is shown in Figure. 5. Conclusion The relm of numericl mehods in scienific fields is vsly growing due o he very fs progresses in compuionl sciences nd echnologies. Neverheless, he inrinsic complexiy of frcionl clculus, cused prilly by non-locl properies of frcionl derivives nd inegrls, mkes i rher difficul o find efficien numericl mehods in his field. I seems enough o menion here h, up o he ime of his mnuscrip, nd o he bes of our knowledge, here is no rouine vilble for solving frcionl differenil equion s Runge Ku for ordinry ones. Despie his fc, however, he lierure exhibis growing ineres nd improving chievemens in numericl mehods for frcionl clculus in generl nd frcionl vriionl problems specificlly.