Research Article An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

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1 Hindwi Pulishing Corporion The Scienific World Journl Volume 23, Aricle ID 95437, pges hp://dx.doi.org/.55/23/95437 Reserch Aricle An Expnsion Formul wih Higher-Order Derivives for Frcionl Operors of Vrile Order 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 Correspondence should e ddressed o Delfim F. M. Torres; delfim@u.p Received 27 Augus 23; Acceped 9 Sepemer 23 Acdemic Ediors: A. Angn, S. C. O. ouchie, nd A. Secer Copyrigh 23 R. Almeid nd D. F. M. Torres. This is n open ccess ricle disriued under he Creive Commons Ariuion License, which permis unresriced use, disriuion, nd reproducion in ny medium, provided he originl wor is properly cied. We oin pproximion formuls for frcionl inegrls nd derivives of Riemnn-Liouville nd Mrchud ypes wih vrile frcionl order. The pproximions involve ineger-order derivives only. An esimion for he error is given. The efficiency of he pproximion mehod is illusred wih exmples. As pplicions, we show how he oined resuls re useful o solve differenil equions, nd prolems of he clculus of vriions h depend on frcionl derivives of Mrchud ype.. Inroducion Frcionl clculus is nurl exension of he inegerorder clculus y considering derivives nd inegrls of rirry rel or complex order α K, wih K R or K C. The sujec ws orn from fmous correspondence eween L Hopil nd Leiniz in 695 nd hen developed y mny fmous mhemicins, lie Euler, Lplce, Ael, Liouville, nd Riemnn, jus o menion few nmes. Recenly, frcionl clculus hs rced he enion of vs numer of reserchers, no only in mhemics, u lso in physics nd in engineering, nd hs proven o eer descrie cerin complex phenomen in nure [, 2]. Since he order α of he inegrls nd derivives my e ny vlue, noher ineresing exension is o consider he order no s consn during he process u s vrile α) h depends on ime. This provides n exension of he clssicl frcionl clculus nd i ws inroduced y Smo nd Ross in 993 [3]seelso[4]).Thevrileorder frcionl clculus is nowdys recognized s useful ool, wih successful pplicions in mechnics, in he modeling of liner nd nonliner viscoelsiciy oscillors, nd in oher phenomen where he order of he derivive vries wih ime. For more on he sujec, nd is pplicions, we menion [5 ]. For numericl pproch see, for exmple, [2 4]. Resuls on differenil equions nd he clculus of vriions wih frcionl operors of vrile order cn e found in [5, 6] nd references herein. In his pper we show how frcionl derivives nd inegrls of vrile order cn e pproximed y clssicl ineger-order operors. The ouline of he pper is he following. In Secion 2 we presen he necessry definiions, nmely, he frcionl operors of Riemnn-Liouville nd Mrchud of vrile order. Some properies of he operors re lso given. The min core of he pper is Secion 3, where we prove he expnsion formuls for he considered frcionl operors, wih he size of he expnsion eing he derivive of order n.insecion 4 we show he ccurcy of our mehod wih some exmples nd how he pproximions cn e pplied in differen siuions o solve prolems involving vrile order frcionl operors. 2. Frcionl Clculus of Vrile Order In he following, he order of he frcionl operors is given y funcion α C [, ], ], [); x ) is ssumed o ensure convergence for ech of he involved inegrls. For complee nd rigorous sudy of frcionl clculus we refer o [7].

2 2 The Scienific World Journl Definiion. Le x ) e funcion wih domin [, ].Then, for [,], i) he lef Riemnn-Liouville frcionl inegrl of order α ) is given y I α) x ) Γ α )) τ) α) x τ) dτ, ) Exmple 3 see [3]). Le x e he power funcion x) ) γ.then,forγ>,wehve I α) Γγ) x ) Γγα) ) )γα), D α) Γγ) x ) Γγ α) ) )γ α), ii) he righ Riemnn-Liouville frcionl inegrl of order α ) is given y I α) x ) Γ α )) τ ) α) x τ) dτ, 2) D α) x ) Γγ) Γγ α) ) )γ α) α ) ) Γγ) Γγ α) 2) )γ α) [ln ) ψγ α) 2) 8) iii) he lef Riemnn-Liouville frcionl derivive of order α ) is given y D α) x ) d Γ α)) d τ) α) x τ) dτ, 3) iv) he righ Riemnn-Liouville frcionl derivive of order α ) is given y D α) x ) d Γ α)) d τ ) α) x τ) dτ, 4) v) he lef Mrchud frcionl derivive of order α ) is given y D α) x ) x ) Γ α)) ) α) α ) Γ α)) x ) xτ) τ) α) dτ, 5) vi) he righ Mrchud frcionl derivive of order α ) is given y D α) x ) x ) Γ α)) ) α) α ) Γ α)) x ) xτ) τ ) α) dτ. 6) ψ α))], where ψ is he Psi funcion, h is, he derivive of he logrihm of he Gmm funcion: ψ ) d d ln Γ )) Γ ) Γ ). 9) From Exmple 3 we see h D α) x) D α) x).also, hesymmeryonpowerfuncionsisvioled,whenweconsider I α) x) nd D α) x), uholdsfor I α) x) nd D α) x). Ler we explin his eer, when we deduce he expnsion formul for he Mrchud frcionl derivive. In conrs wih he consn frcionl order cse, he lw of exponens fils for frcionl inegrls of vrile order. However, we form holds see [3]): if β) β, ],[, hen I α) Iβ x) Iα)β x). 3. Expnsion Formuls wih Higher-Order Derivives The min resuls of he pper provide pproximions of he frcionl derivives of given funcion x y sums involving only ineger derivives of x. The pproximions use he generlizion of he inomil coefficien formul o rel numers: α ) ) α ) ) ) Γ α ) ) Γ α ))!. ) Theorem 4. Fix n nd n,ndlex ) C n [, ], R). Define he lef) momen of x of order y Remr 2. I follows from Definiion h D α) x ) d d I α) x ), D α) x ) d d I α) x ). 7) Then, V ) n) τ ) n x τ) dτ. ) D α) x ) S ) S 2 ) E, ) E 2, ) 2)

3 The Scienific World Journl 3 wih S ) ) α) [ where A α ),) B α ),) n A α ),) ) x ) ) n Γ α)) [ [ pn B α ),) ) n V )], Γ nα)) Γ α )) Γ α)) n)!, S 2 ) x ) α) ) Γ α)) ) α) ln ) [ α) α)) 2 ln ) 3) Γp nα)) Γ α ) ) p n)! ], ] α ) ) ),...,n, α ) ) ) pp) ] α ) ) Γ α)) ) α) n [ln ) n p α ) n ) ) n n )n V ) n n α ) n ) ) n ppp n) )n p V p )]. 4) The error of he pproximion D α) x) S ) S 2 ) is given y E, ) E 2, ), wheree, ) nd E 2, ) re ounded y E, ) L n ) exp n α)) 2 n α)) Γ n α)) n α)) n α) ) n α), 5) E 2, ) L ) α) ) )2 α) exp α 2 ) α)) Γ 2 α)) α) 6) wih L j ) mx τ [,] xj) τ), j {, n }. 7) Proof. Sring wih equliy D α) x ) d Γ α)) d τ) α) x τ) dτ, 8) doing he chnge of vrile τu over he inegrl, nd hen differeniing i, we ge D α) x ) wih S ) S 2 ) d Γ α)) d u ) α) x u) du Γ α)) [ x ) ) α) d d [u ) α) x u)]du] Γ α)) x ) [ [ α ) )u ) α) ) α) S ) S 2 ) ln u ) x u) u ) α) x ) u)]du] 9) Γ α)) [ x ) ) τ) α) x ) τ) dτ], α) 2) α ) ) Γ α)) τ) α) ln τ) x τ) dτ. 2)

4 4 The Scienific World Journl The equivlence eween 2) nd3) followsfromhe compuions of [8]. To show he equivlence eween 2) nd 4) we sr in he smewy s done in [9], o ge α ) ) S 2 ) Γ α)) [x ) u) α) ln u) du x ) τ) τ u) α) ln u) du) dτ] α ) ) [x ) ) α) Γ α)) ln ) [ α) α)) 2 ] x ) τ) τ ) α) u ) α) [ln ) ln u )] du) dτ]. 22) ow, pplying Tylor s expnsion over u ) α) 23) nd we deduce h S 2 ) α ) ) Γ α)) ln u ), 24) [x) ) α) ln ) [ α) α)) 2 ] E 2, ) x ) τ τ) ) α) ln ) τ α ) ) ) u ) ) ) α) u ) ) p du α ) ) ) u ) p p du) dτ] p ) α ) ) [x ) ) α) Γ α)) E 2, ) ln ) [ α) α)) 2 ] x ) τ) ) α) ln ) α ) ) ) ) τ u ) du) dτ x ) τ) ) α) α ) p ) p p ) ) ) τ u ) p du) dτ] α) ) ) α) Γ α)) ln ) [x) ) [ α) α)) 2 ] ln ) α ) ) ) ) ) x ) τ)τ ) dτ) α ) ) ) ) p ) p p) p x ) τ)τ ) p dτ)]e 2, ). 25) Inegring y prs, we conclude wih he wo following equliies: x ) τ)τ ) dτ x ) ) V n ), x ) τ)τ ) p dτ x ) ) p V pn ). 26)

5 The Scienific World Journl 5 The deducion of relion 4) fors 2 ) follows now from direc clculions. Finlly, we prove he upper ound formul for he error. The ound 5) for he error E, ) ime follows esily from [8]. Wih respec o sum S 2,heerror is ounded y E 2, ) α ) ) ) α) Γ α)) ln ) α ) ) ) x ) τ) τ ) ) Define he quniies I 2 ) p α ) ) ) pp) x ) τ) τ )p ) p dτ) dτ). I ) α) )2) d, α) pp)p2) Inequliy 6) follows from relion nd he upper ounds dp d. 27) 28) ) α ) exp α2 ) α)) 29) α) I ) < d 2 α) α)), α) I 2 ) < for I nd I 2. 2 α) p 2 dp d α)) 2 α) 3) Similrly s done in Theorem 4 for he lef Riemnn- Liouville frcionl derivive, n pproximion formul cn e deduced for he righ Riemnn-Liouville frcionl derivive. Theorem 5. Fix n nd n,ndlex ) C n [, ], R). Define he righ) momen of x of order y W ) n) τ) n x τ) dτ. 3) Then, wih D α) x ) S ) S 2 ) E, ) E 2, ) 32) S ) ) α) [ where A α ),) n A α ),) ) x ) ) n ) Γ α)) [ [ B α ),) pn S 2 ) x ) α) ) Γ α)) ) α) B α ),) ) n W )], 33) Γp nα)) Γ α ) ) p n)! ], ],...,n, ) n Γ nα)) Γ α )) Γ α)) n)!, ln ) [ α) α)) 2 ln ) α ) ) ) α ) ) ) pp) ] p α ) ) Γ α)) ) α) n [ln ) n α ) n ) ) n n )n W ) n n α ) n ) ) n pp n) )n p W p )]. p 34)

6 6 The Scienific World Journl The error of he pproximion D α) x) S ) S 2 ) is given y E, ) E 2, ), wheree, ) nd E 2, ) re ounded y E, ) L n ) exp n α)) 2 n α)) Γ n α)) n α)) n α) ) n α), E 2, ) L ) α) ) )2 α) exp α 2 ) α)) Γ 2 α)) α) wih [ ln ) ] 35) L j ) mx τ [,] xj) τ), j {, n }. 36) Using he echniques presened in [2], similr formuls s he ones given y Theorems 4 nd 5 cneprovedforhe lef nd righ Riemnn-Liouville frcionl inegrls of order α ). For exmple, for he lef frcionl inegrl one hs he following resul. Theorem 6. Fix n nd n,ndlex ) C n [, ], R).Then, x ) ) α) [ A α ),) ) x ) ) I α) where A α ),) E ), n n Γ α)) [ [ B α ),) pn B α ),) ) n V )] 37) Γp n α)) Γ α ) ) p n )! ], ] Γ n α)) Γ α )) Γ α)) n)!, V ) n) τ ) n x τ) dτ,,...,n, n,...,. 38) AoundforheerrorE ) is given y E ) L n ) exp nα)) 2 nα)) Γ nα)) nα)) nα) 39) ) nα). We now focus our enion on he lef Mrchud frcionl derivive D α) x). Spliingheinegrl5), we deduce h D α) α ) x ) Γ α)) x τ) dτ. 4) α) τ) Inegring y prs, D α) x ) Γ α)) [ x ) ) τ) α) x ) τ) dτ], α) 4) which is represenion for he lef Riemnn-Liouville frcionl derivive when he order is consn, h is, when α) α [7, Lemm2.2].Forhisreson,he Mrchud frcionl derivive is more suile s he inverse operion for he Riemnn-Liouville frcionl inegrl. Wih 4) ndtheorem 4 inmind,iisnodifficulooinhe corresponding formul for D α) x). Theorem 7. Fix n nd n,ndlex ) C n [, ], R).Then, D α) x ) S ) E, ), 42) where S ) nd E, ) re s in Theorem 4. Similrly, hving ino considerion h D α) x ) Γ α)) [ x ) ) τ ) α) x ) τ) dτ], α) 43) he following resul holds. Theorem 8. Fix n nd n,ndlex ) C n [, ], R).Then, D α) x ) S ) E, ), 44) where S ) nd E, ) re s in Theorem Exmples For illusrive purposes, we consider he lef Riemnn- Liouville frcionl inegrl nd he lef Riemnn-Liouville nd Mrchud frcionl derivives of order α) )/4. Similr resuls s he ones presened here re esily oined for he oher frcionl operors nd for oher funcions α ). All compuions were done using he Compuer Alger Sysem Mple.

7 The Scienific World Journl 7 I α) x) Exc Figure : Exc 45) nd numericl pproximions of he lef Riemnn-Liouville inegrl I α) x) wih x) 4 nd α) )/4 oined from Theorem 6 wih n2nd {3,5}.The error 48)isE.269 for 3nd E.292 for 5. D α) x) Exc Figure 2: Exc 46) nd numericl pproximions of he lef Riemnn-Liouville derivive D α) x) wih x) 4 nd α) )/4 oined from Theorem 4 wih n2nd {3,5}.The error 48)isE.3294 for 3nd E.3976 for Tes Funcion. We es he ccurcy of our pproximions wih n exmple. Exmple. Le x e he funcion x) 4 wih [, ]. Then, for α) )/4,ifollowsfromExmple 3 h D α) x ) I α) 24 x ) Γ 2)/4) 24 Γ 9 ) /4) 5 )/4 [ln ) ψ 7)/4, 45) 6 Γ 23 ) /4) 9 )/4 23 )ψ )], 46) D α) 24 x ) Γ 9 ) /4) 5 )/4. 47) In Figures, 2, nd3one cn compre he exc expressions of he frcionl operors of vrile order 45), 46), nd 47), respecively, wih he pproximions oined from our resuls of Secion 3 wih n2nd {3,5}.TheerrorE is mesured using he norm Ef,g)) f ) g)) 2 d. 48) 4.2. Frcionl Differenil Equions of Vrile Order. Consider he following frcionl differenil equion of vrile order: D α) x ) x) Γ 7 )/4) 3 )/4, 49) x ), wih α) )/4. Iisesyochechx) is soluion o 49). We exemplify how Theorem 7 my e pplied in order o pproxime he soluion of such ype D α) x) Exc Figure 3: Exc 47) nd numericl pproximions of he lef Mrchud derivive D α) x) wih x) 4 nd α) )/4 oined from Theorem 7 wih n2nd {3,5}.Theerror48) is E.499 for 3nd E.477 for 5. of prolems. The min ide is o replce ll he frcionl operors h pper in he differenil equion y finie sum up o order, involving ineger derivives only, nd, y doing so, o oin new sysem of sndrd ordinry differenil equions h is n pproximion of he iniil frcionl vrile order prolem. As he size of increses, he soluion of he new sysem converges o he soluion of he iniil frcionl sysem. The procedure for 49) ishe following. Firs, we replce D α) x) y D α) x ) Aα ),) α) x ) Bα ),) α) x ) ) 2 C α ),) α) V ), 5)

8 8 The Scienific World Journl where A α ),) Γ α)) [ Γp α)) Γ α )) p )! ], B α ),) C α ),) p2 Γ 2 α)) [ Γp α)) Γ α ) ) p! ], p Γ α)) Γ α )) Γ α)) )!, 5) x 3 ) nd V ) is he soluion of he sysem V ) ) ) 2 x ), V ), 2,3,...,. 52) Thus, we ge he pproximed sysem of ordinry differenil equions [A α ),) α) ]x) Bα ),) α) x ) ) 2 C α ),) α) V ) Γ 7 ) /4) 3 )/4, V ) ) ) 2 x ), 2,3,...,, x ), V ), 2,3,...,. 53) ow we pply ny sndrd echnique o solve he sysem of ordinry differenil equions 53). We used he commnd dsolve of Mple. In Figure 4 we find he grph of he pproximion x 3 ) o he soluion of prolem 49), oined y solving 53) wih 3. Tle gives some numericl vlues of such pproximion, illusring numericlly he fc h he pproximion x 3 ) is lredy very close o he exc soluion x) of 49). In fc he plo of x 3 ) in Figure 4 is visully indisinguishle from he plo of x) Frcionl Vriionl Clculus of Vrile Order. We now exemplify how he expnsions oined in Secion3 re useful o pproxime soluions of frcionl prolems of he clculus of vriions [2]. The frcionl vriionl clculus of vrile order is recen sujec under srong curren developmen [5, 6, 22, 23]. So fr, only nlyicl mehods o solve frcionl prolems of he clculus of vriions of vrile order hve een developed in he lierure, which consis in he soluion of frcionl Euler-Lgrnge differenil equions of vrile order [5, 6, 22, 23]. In mos cses, however, o solve nlyiclly such frcionl differenil equions is exremely hrd or even impossile, so numericl/pproximing mehods re needed. Our resuls Figure 4: Approximion x 3 ) o he exc soluion x) of he frcionl differenil equion 49), oined from he pplicion of Theorem 7, h is, oined y solving 53)wih3. provide wo pproches o his issue. The firs ws lredy illusred in Secion 4.2 nd consiss in pproximing he necessry opimliy condiions proved in [5, 6, 22, 23], which re nohing else hn frcionl differenil equions of vrile order. The second pproch is now considered. Similr o Secion 4.2, he min ide here is o replce he frcionl operors of vrile order h pper in he formulion of he vriionl prolem y he corresponding expnsion of Secion 3, which involves only ineger-order derivives.bydoingi,wereduceheoriginlprolemo clssicl opiml conrol prolem, whose exremls re found y pplying he celered Ponrygin mximum principle [24]. We illusre his mehod wih concree exmple. Consider he funcionl J x) [ D α) 2 x) Γ7 )/4) 3 )/4 ] d, 54) wih frcionl order α) )/4,sujecoheoundry condiions x ), x). 55) Since Jx) for ny dmissile funcion x nd ing x), which sisfies he given oundry condiions 55), gives Jx),weconcludehxgives he glol minimum o he frcionl prolem of he clculus of vriions h consiss in minimizing funcionl 54)sujecoheoundrycondi- ions 55). The numericl procedure is now explined. Since we hve wo oundry condiions, we replce D α) x) y he expnsion given in Theorem 7 wih nnd vrile size 2.Thepproximionecomes D α) x ) Aα ),) α) x ) Bα ),) α) x ) ) 2 C α ),) α) V ). 56)

9 The Scienific World Journl 9 Tle : Some numericl vlues of he soluion x 3 ) of 53) wih3, very close o he vlues of he soluion x) of he frcionl differenil equion of vrile order 49) x 3 ) Using 56), we pproxime he iniil prolem 54)-55) y he following one: o minimize J x) [A α ),) α) x ) sujec o Bα ),) α) x ) ) 2 C α ),) α) V ) Γ7 )/4) 3 )/4 ] d V ) ) ) 2 x ), V ), x ), x), 2 2,...,, 57) 58) where α) )/4. This dynmic opimizion prolem hs sysem of ordinry differenil equions s consrin, so i is nurl o solve i s n opiml conrol prolem. For h, define he conrol u y u ) Aα ),) α) x ) Bα ),) α) x ) ) 2 We hen oin he conrol sysem C α ),) α) V ). x ) ) B α) u ) AB x ) 2 where, for simplificion, B C V ) : f, x ),u),v)), AAα ),), BBα ),), C Cα ),), 59) 6) 6) V ) V 2 ),...,V )). In conclusion, we wish o minimize he funcionl 2 J x, u, V) [u) Γ7 )/4) 3 )/4 ] d 62) sujec o he firs-order dynmic consrins x ) ) f, x, u, V), V ) ) ) 2 x ), 2,...,, nd he oundry condiions V ), x ), x ), 2,...,. In his cse, he Hmilonin is given y H, x, u, V, λ) [u 2 Γ7 )/4) 3 )/4 ] λ f, x, u, V) 2 λ ) 2 x 63) 64) 65) wih he djoin vecor λλ,λ 2,...,λ ) [24]. Following he clssicl opiml conrol pproch of Ponrygin e l. [24], we hve he following necessry opimliy condiions: V ) p H u, x) H λ, H λ p, λ ) H x, λ ) p H V p. 66) Th is, we need o solve he sysem of differenil equions x ) ) B Γ 7 ) /4) 2 B 2 2α) 2 λ ) AB x ) 2 B C V ), V ) ) ) 2 x ), 2,...,, λ ) ) AB λ ) 2 λ ), 2 λ ) ) B C λ, 2,...,, 67)

10 The Scienific World Journl Tle 2: Some numericl vlues of he soluion x 2 ) of 67)-68) wih2, close o he vlues of he glol minimizer x) of he frcionl vriionl prolem of vrile order 54)-55) x 2 ) x 2 ) Figure 5: Approximion x 2 ) o he exc soluion x) of he frcionl prolem of he clculus of vriions 54)-55), oined from he pplicion of Theorem 7 ndheclssiclponrygin mximum principle, h is, oined y solving 67)-68)wih 2. sujec o he oundry condiions x ), V ), x ), λ ), 2,...,, 2,...,. 68) Figure 5 plos he numericl pproximion x 2 ) o he glol minimizer x) of he vrile order frcionl prolem of he clculus of vriions 54)-55), oined y solving 67)-68) wih 2.Thepproximion x 2 ) is lredy visully indisinguishle from he exc soluion x), nd we do no increse he vlue of. The effeciveness of our pproch is lso illusred in Tle 2, where some numericl vlues of he pproximion x 2 ) re given. Acnowledgmens This wor is suppored y FEDER Funds hrough COMPETE Operionl Progrmme Fcors of Compeiiveness Progrm Opercionl Fcores de Compeiividde ) nd y Poruguese Funds hrough he Cener for Reserch nd Developmen in Mhemics nd Applicions Universiy of Aveiro) nd he Poruguese Foundion for Science nd Technology FCT Fundção pr Ciênci e Tecnologi ), wihin Projec PEs-C/MAT/UI46/2 wih COMPETE no. FCOMP--24-FEDER Delfim F.M. Torres ws lso suppored y EU Funding under he 7h Frmewor Progrmme FP7-PEOPLE-2-IT, Grn Agreemen no SADCO. References [] M. Dlir nd M. Bshour, Applicions of frcionl clculus, Applied Mhemicl Sciences, vol.4,no.2 24,pp.2 32, 2. [2] J.A.TenreiroMchdo,M.F.Silv,R.S.Brosel., Some pplicions of frcionl clculus in engineering, Mhemicl Prolems in Engineering, vol.2,aricleid6398,34 pges, 2. [3]S.G.SmondB.Ross, Inegrionnddiffereniiono vrilefrcionlorder, Inegrl Trnsforms nd Specil Funcions,vol.,no.4,pp.277 3,993. [4] S. G. Smo, Frcionl inegrion nd differeniion of vrile order, Anlysis Mhemic,vol.2,no.3,pp , 995. [5] A. Almeid nd S. Smo, Frcionl nd hypersingulr operors in vrile exponen spces on meric mesure spces, Medierrnen Journl of Mhemics,vol.6,no.2,pp , 29. [6] C. F. M. Coimr, Mechnics wih vrile-order differenil operors, Annlen der Physi Leipzig), vol. 2, no. -2, pp , 23. [7]C.M.Soon,C.F.M.Coimr,ndM.H.Koyshi, The vrile viscoelsiciy oscillor, Annlen der Physi Leipzig), vol. 4, no. 6, pp , 25. [8] G. Diz nd C. F. M. Coimr, onliner dynmics nd conrol of vrile order oscillor wih pplicion o he vn der Pol equion, onliner Dynmics,vol.56,no.-2,pp.45 57, 29. [9] C.F.LorenzondT.T.Hrley, Vrileordernddisriued order frcionl operors, onliner Dynmics,vol.29,no. 4, pp , 22. [] L.E.S.RmirezndC.F.M.Coimr, Onheselecionnd mening of vrile order operors for dynmic modeling, Inernionl Journl of Differenil Equions,vol.2,Aricle ID8467,6pges,2. [] L. E. S. Rmirez nd C. F. M. Coimr, On he vrile order dynmics of he nonliner we cused y sedimening pricle, Physic D, vol. 24, no. 3, pp. 8, 2. [2] S. M, Y. Xu, nd W. Yue, umericl soluions of vrileorder frcionl finncil sysem, Journl of Applied Mhemics, vol. 22, Aricle ID 47942, 4 pges, 22. [3] D. Vlério nd J. Sá D Cos, Vrile-order frcionl derivives nd heir numericl pproximions, Signl Processing, vol.9,no.3,pp ,2. [4] P.Zhung,F.Liu,V.Anh,ndI.Turner, umericlmehodsfor he vrile-order frcionl dvecion-diffusion equion wih nonlinersourceerm, SIAM Journl on umericl Anlysis, vol.47,no.3,pp.76 78,29. [5] T. Odzijewicz, A. B. Mlinows, nd D. F. M. Torres, Frcionl vriionl clculus of vrile order, in Advnces in

11 The Scienific World Journl Hrmonic Anlysis nd Operor Theory, he Sefn Smo Anniversry Volume, A.Almeid,L.Csro,ndF.O.Spec, Eds.,vol.229ofOperor Theory: Advnces nd Applicions, pp.29 3,Springer,23. [6] T. Odzijewicz, A. B. Mlinows, nd D. F. M. Torres, A generlized frcionl clculus of vriions, Conrol nd Cyerneics, vol. 42, no. 2, pp , 23. [7] S. G. Smo, A. A. Kils, nd O. I. Mrichev, Frcionl Inegrls nd Derivives,GordonndBrech,Yverdon,Swizerlnd, 993. [8] S. Pooseh, R. Almeid, nd D. F. M. Torres, umericl pproximions of frcionl derivives wih pplicions, Asin Journl of Conrol,vol.5,no.3,pp ,23. [9] T. M. Ancovic, M. Jnev, S. Pilipovic, nd D. Zoric, An expnsion formul for frcionl derivives of vrile order, Cenrl Europen Journl of Physics,23. [2] S. Pooseh, R. Almeid, nd D. F. M. Torres, Approximion of frcionl inegrls y mens of derivives, Compuers & Mhemics wih Applicions, vol.64,no.,pp.39 3, 22. [2] A. B. Mlinows nd D. F. M. Torres, Inroducion o he Frcionl Clculus of Vriions, Imperil College Press, London, UK, 22. [22] T. Odzijewicz, A. B. Mlinows, nd D. F. M. Torres, Vrile order frcionl vriionl clculus for doule inegrls, in Proceedings of he 5s IEEE Conference on Decision nd Conrol, no , pp , Mui, Hwii, USA, Decemer 22, Aricle [23] T. Odzijewicz, A. B. Mlinows, nd D. F. M. Torres, oeher s heorem for frcionl vriionl prolems of vrile order, CenrlEuropenJournlofPhysics,vol.,no.6,pp.69 7, 23. [24] L. S. Ponrygin, V. G. Bolynsii, R. V. Gmrelidze, nd E. F. Mishcheno, The Mhemicl Theory of Opiml Processes, Inerscience, John Wiley & Sons, ew Yor, Y, USA, 962, edied y L. W. eusd.

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