Numerical Solution of Fractional Differential Equations by Using the Jacobi Polynomials
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1 J. Basc. Appl. Sc. Res., ( ,, TexRoad Publcao ISSN Joural of Basc ad Appled Scefc Research Nuercal Soluo of Fracoal Dffereal Equaos by Usg he Jacob Polyoals Seyedahad Behesh, Hassa Khosrava-Arab, Ia Zare 3,,3 Khoe Brach, Islac Azad Uversy, Khoe, Ira ABSTRACT I hs paper, we prese a ew accurae ad sple approach for he uercal soluo of al value probles wh Capuo ype fracoal dervave of order >. The ehod s based upo expadg fracoal dervave of uow soluo er of Jacob polyoals. The properes of Jacob polyoal are he ulzed o reduce he fracoal al value proble o he soluo of algebrac equaos. Through several uercal exaples, he accuracy ad valdy of he ehod are repored. Keywords: Fracoal Calculus, Capuo fracoal dervaves, Fracoal al value proble, Jacob polyoals, Jacob-Gauss quadraure.. INTRODUCTION I hs paper, we cosder he fracoal dffereal equao ( y ( f (, y(, >, subec o he followg al codos ( p ( p y ( y, p,,[ ], ( where f s a arbrary fuco, y p (, p,,[ ] s he p -h dervave of y ad y p, p,,[ ] ( he specal al codos. Furherore y ( s he Capuo ype fracoal dervave of order, defed by y ( ([ ] d f ( ( [ ] ([ ] d ( y( d. For ore deals o fracoal dervave coceps ad aoher defos for he fracoal dervave, see [7, 3]. Fracoal al value probles (FIVPs arse soe applcao areas. See for sace he fracoal oscllao equao [9, 3], he lear ad olear fracoal Bagley-Torv equao [], he Basse equao [, ], he fracoal Lorez syse [3, 9], he fracoal dyacal syses [3, 3], ad ec. Several ehods have recely bee proposed o solve he FIVPs. I [], Ourac e al. gave a aalycal ehod for fracoal dffereal equao, Galeoe [5] used he Adas ulsep ehods for FIVPs, I [4], Aroglu e al. used dffereal rasfor ehod for FIVPs. As he oher uercal approach o solve he fracoal ordary ad paral dffereal equaos, we refer o [, 4, 6, 8, 7, 3, ]. I hs paper, by esablshg a relaoshp bewee Jacob polyoals ad fracoal dervaves, a ufed ehod for soluo of FIVPs s preseed. The Jacob polyoals have bee used exesvely aheacal aalyss ad uercal soluo of dffereal equaos (cf. [6]. Our proposed ehods are based o he approxao of uow soluo by usg a -er Jacob polyoals expaso. The properes of Jacob polyoal ad Jacob-Gauss quadraure are ulzed o reduce he FIVP o a syse of algebrac equaos. The a advaage of he preseed ehod s ha gas effce resuls eve wh usg a sall uber of dscrezao paraeer.. SOME PRELIMINARIES I hs seco, we brefly revew he Jacob polyoal ad Jacob-Gauss quadraure rule [6, 4, 6]. are *Correspodg Auhor: Seyedahad Behesh, Khoe Brach, Islac Azad Uversy, Khoe, Ira. Eal: sabehesh@auhoe.ac.r,tel:
2 Behesh e al.,. Jacob Polyoals The well-ow Jacob polyoals (, P,,, are gve explcly by ( ( ( a x P (, (!(!( ( b a where ( a a( a ( a, ( a. ( These polyoals ce expressed equvalely by oher explc forula or by he Rodrgues forula [6, 4, 6]. I pracce, we ca use he recurrece Boe's relao o geerae Jacob polyoals, P (, P ( ( a ( a b x, (, (, P ( ( a x b P ( P (,,,, wh ( a b ( a b a, ( ( a b ( a b ( a b b, ( ( a b ( a ( a( ( a b. ( ( a b ( a (, These expressos show ha P ( are aalyc fucos of paraeers a ad b. The classcal Jacob polyoals correspod o he paraeers a, b >. For hese paraeers, he Jacob polyoals are orhogoal o he caocal erval [,] wh respec o he wegh fuco ( x (..e.,, f, ab ( a ( b P P ( ( dx, f. (3!( a b ( a b (, As a resul, all he zeros of P ( are sple ad belog o he erval (,.. Jacob-Gauss Pos ad Quadraures For a gve posve eger, we deoe he Jacob-Gauss pos, wh paraeers a ad b, by { }, (, whch s he se of roos of P ( x. The Jacob-Gauss quadraure rule, wh paraeers a ad b, s based o Jacob-Gauss pos { }, ad ca be used o approxae he egral of a fuco over he rage [,] wh wegh x ( as ( f ( ( ( dx f (, (4 where,,, are Jacob-Gauss quadraure weghs. The Jacob-Gauss pos ad weghs ay be deered by accurae ad sable ehods [6]. Also s well ow ha Jacob-Gauss quadraure has degree of exacess,.e. s exac wheever f ( s a polyoal of degree equal or less ha. 3. The Prese Mehod A frs, we prese he followg heore ha has a ey rule our ehod. Theore 3. For > ad < x <, we have 4895
3 J. Basc. Appl. Sc. Res., ( , where d x P dx ( g,,,. ( (, x ( gp (, x (, x Proof 3. By subsug Eq.(, we have (, x ( ( x x P (,!(!( ow by ag he Rea-Louvlle fracoal dervave of order of boh sde, we ge d (, x ( ( d (. x P x dx!(!( dx d dx ( ( By og ha x x, >,,,, we coclude d (, x ( ( ( x x P ( dx!(!( ( ( (, x P (. ( For sae of splcy, we cosder he case < <. Furherore, he case does o see o be of aor praccal eres. However he preseed ehod s sply exeded for he case. Ideed, we cosder he followg fracoal al value proble ( y ( f x, y(, < x <, (5 y ( a, (6 y ( b, (7 where he secod al codo (7 s for > oly. For, we approxae y ( as y( ~ (, x y( a c x P (, (8 ad for <, we approxae y ( by ~ (, x y( y( a bx c x P (. (9 Noe ha he approxao (8 sasfes al codo (6 ad he approxao (9 sasfes al codos (6 ad (7. For suarzg he boh cases, we se ~ * (, x y ( a b x c x P (, ( where, * b b, ( I vew of Theore 3., we ca approxae y ( as, <. 4896
4 Behesh e al., ( ~ ( (, x y ( y ( c gp (. ( By subsug ( ad ( fracoal ODE (5, we ge (, x * (, x c gp ( f x, a b x c x P (. (, x x Now by ulplyg boh sde of above equaos by P ( ( ad egrag he he erval [,], we ge c g (, P x ( ( P, x x ( ( dx * (, x (, x x f x, a b x c x P ( P ( ( dx. By applyg rasforao x (, ad by usg orhogoaly propery (3, fally we ge g c ( * (, (, f, c P ( P ( ( d. Now, order o oba hgh order accuracy, he egral er above equao s approxaed by usg Jacob Gauss quadraure (4 ad we ge g c ( (, (, * (, f ˆ,, a b ˆ,,,,, c (, (, (, (, (, ˆ P ( P (, (,, (, where ˆ,,,,. The above equaos for a (olear syse of algebrac equaos. By solvg, we ca fd uow coeffces c,,, ad he approxao ( s followed. 4. NUMERICAL ILLUSTRATION Ths seco s devoed o he uercal experes. We pleeed he proposed ehod for uercal soluo of FIVPs wh Malab a persoal copuer. Meawhle, we use Malab roues provded by Gausch [6] our pleeao o geerae Jacob-Gauss odes ad weghs. Table. lss soe FIVPs, whch are used as es exaples our sulaos ad repors. I hs able, E a, b s Mag-Leffler fuco[3], E ( :, a, >., ( a b Ths fuco plays he sae role dffereal equaos of fracoal order whch he expoeal fuco e plays ordary dffereal equaos; fac, E ( e,,. A frs, we apply our ehod o Ex., wh.3,,, ad,4,6 ad 3. The resulg 4897
5 J. Basc. Appl. Sc. Res., ( , soluos ad exac soluo are ploed Fg. 4.a. We see ha, he preseed ehod s provdes accurae resuls eve wh usg 6. Also he error of obaed soluo wh 3 s ploed Fg. 4.b. To explore he depedece of errors o he paraeer, we use he followg oao ~ e Max y( y (,,,,, ad we plo e for Ex. (.8 ad 6, Ex. (. ad ad Ex. 3 (.3,, c ad Fg. 5. Ths fgure shows ha he preseed ehod coverges qucly. To ae a coparso, we cosder he four uercal ehods [,, 8, 5] whch are vesgaed []. I Tables. ad 3, he obaed resuls of hese ehods ad our ehod, for Ex. wh.5,.5 ad 6.4 are repored. I fal, we apply our ehod o Ex. 4, wh 3 ad.,.4,,.8,. The resuls are ploed Fg. 6. We eo ha whe ad he exac soluos are y( ( e x cos( ad x y( x e respecvely. These exac soluos are hghlghed hs fgure. Also Table. 4, he obaed values of ~ y ( x x,,3,4,5 wh,5, are preseed. Through hs Table, he accuracy ad covergece rae of preseed ehod for Ex. 4 are show. Fgure : (lef Coparso bewee exac soluo ad obaed soluos by he prese ehod for Ex. wh,4,6. (rgh Plo of error y( ~ y ( for 3. Fgure : Error e as a fuco of dscrezao paraeer for Exaples
6 Behesh e al., Fgure 3: Coparso of obaed soluo of Ex. 4 by he prese ehod wh 3 for varous values of Ex. [, 7] Ex. [, 9] Ex. 3 [8, 5] Ex. 4 [9, 3] ODE Ial Co. Table : Soe Fracoal Ial Values Probles ( y (.x y(, < < Exa y (, ' y ( Sol. y(.x E,( x y( E,( x ODE Ial Co. Sol. ODE Exa (5 4 ( 43 8 y x 3 x (9 ( ( ( x x y, < < 4 ' y (, y ( y 4 8 ( x 3x cy ( y (, 9 4 ( y( Ial Co. Exa Sol. E ( x / c ODE Ial Co. Sol. Exa y ( ( y xe ' y (, y ( No Exac Soluo x x x x, < <, < < 4899
7 J. Basc. Appl. Sc. Res., ( , Table : Coparso of errors bewee soe uercal ehods ad our ehod a dffere x for Ex. wh.5 Mehods (wh h.65( 4 Preseed Mehod x Ref.[] Ref.[] ef.[8] Ref.[5].8-5.e e-5-3.9e-5.6e e e e e-5 -.6e e e-4.36e e-5-8.9e-6 -.6e e e-4.e e e e-6.7e e e e-6-5.e-6-6.7e-6.7e-4 -.7e e e-6-4.8e-6-4.9e-6.8e-4.33e e e e-6-4.7e-6.57e e-4.e e-6-7.e-6-7.5e-6.35e-4-3.5e e-5 Table 3: Coparso of errors bewee soe uercal ehods ad our ehod a dffere x for Ex. wh.5 Mehods (wh h.( 64 Preseed Mehod x Ref.[] Ref.[] Ref.[8] Ref.[3] e-4.63e e-5.6e e-5-8.e e-4.6e-5.84e e-6 8.4e e e e-6.9e e-6 -.9e e e e-6 -.6e-6-4.6e-6.5e-5.59e e-4-5.5e e e-6 -.8e-5.8e e e-6-6.9e-6-8.3e-7.8e-5-4.6e e-4 -.6e-6-4.4e-6.86e e-6.e e e e e-6.43e-4.83e-8 Table 4: The resulg value of he preseed ehod o Ex. 4 for ~ y ( x x,,3,4,5 wh,5,. x 5 x e e e x e e e x e e e x e e e x e e e 5. CONCLUSION I he prese wor, a uercal procedure has bee developed for obag he soluo of fracoal al value probles. I hs ehod he Jacob polyoals were used o reduce he fracoal al value proble o a syse of algebrac equaos. The ehod s characerzed by splcy, effcecy ad s also readly pleeed. By soe uercal exaples, we aalyzed he accuracy ad valdy of he preseed ehod hrough sulaos. We beleve ha he preseed ehod hs wor ce exeded o solve he uler fracoal dffereal equaos. REFERENCES [] O P. Agrawal, Bloc-by-bloc ehod for uercal soluo of fracoal dffereal equaos, : Proceedgs of IFAC4, Frs IFAC Worshop o Fracoal Dffereao ad Is Applcaos. Bordeaux, Frace, July 9ï ½- (4, 9ï ½-. [] O P Agrawal ad Paa Kuar, Coparso of fve uercal schees for fracoal dffereal equaos, Advaces fracoal calculus, Sprger, Dordrech, 7, pp [3] R. Nazar C.P. L A.K. Aloar, M.S.M. Noora, Hooopy aalyss ehod for solvg fracoal lorez syse, Cou. Nolear Sc. Nuer. Sul, Volue 5, Issue 7, July, [4] Ayac Aroglu ad Ibrah Ozol, Soluo of fracoal dffereal equaos by usg dffereal rasfor ehod, Chaos Solos Fracals 34, 7, o. 5,
8 Behesh e al., [5] T. M. Aaacovc ad B. Saovc, O a uercal schee for solvg dffereal quaos for fracoal order, Mech. Res. Co. 35, 8, o. 7, [6] Ka B. Daa ad B. M. Moha, Orhogoal fucos syses ad corol, Advaced Seres Elecrcal ad Copuer Egeerg, vol. 9, World Scefc Publshg Co. Ic., Rver Edge, NJ, 995. [7] Mehd Dehgha, Jall Maafa, ad Abbas Saadaad, Solvg olear fracoal paral dffereal equaos usg he hooopy aalyss ehod, Nuer. Mehods Paral Dffereal Equaos. 6,, o., [8] Ka Dehel, A vesgao of soe oclasscal ehods for he uercal approxao of Capuo-ype fracoal dervaves, Nuer. Algorhs. 47, 8, o. 4, [9] Ka Dehel, A provee of a oclasscal uercal ehod for he copuao of fracoal dervaves, J. Vbr. Acous. 3, 9, o., 4-. [] Ka Dehel, Nevlle J. Ford, ad Ala D. Freed, A predcor-correcor approach for he uercal soluo of fracoal dffereal equaos, Nolear Dya. 9,, o. -4, 3-. [] Joh T. Edwards, Nevlle J. Ford, ad A. Charles Spso, The uercal soluo of lear ul-er fracoal dffereal equaos; syses of equaos, J. Copu. Appl. Mah. 48,, o., [] A. E. M. El-Mesry, A. M. A. El-Sayed, ad H. A. A. El-Saa, Nuercal ehods for ul-er fracoal (arbrary orders dffereal equaos, Appl. Mah. Copu. 6, 5, o. 3, [3] H.A. El-Saa, E. Ahed, M.I. Shehaa, ad A.M.A. El-Sayed, O sably, perssece, ad Hopf bfurcao fracoal orderdyacal syses., Nolear Dy. 56, 9, o. -, -6. [4] Daele Fuaro, Polyoal approxao of dffereal equaos., Lecure Noes Physcs. New Seres : Moographs. 8. Berl: Sprger- Verlag., 99. [5] Robero Garrappa, O soe explc Adas ulsep ehods for fracoal dffereal equaos., J. Copu. Appl. Mah. 9, 9, o., [6] W. Gausch, Orhogoal polyoals: copuao ad approxao, Oxford Uversy Press, New Yor, 4. [7] R. Goreflo ad F. Maard, Fracoal calculus: egral ad dffereal equaos of fracoal order, Fracals ad fracoal calculus couu echacs (Ude, 996, CISM Courses ad Lecures, vol. 378, Sprger, Vea, 997, pp [8] P Kuar ad O P. Agrawal, A cubc schee for uercal soluo of fracoaldffereal equaos, : Proceedgs of he Ffh EUROMECH Nolear Dyacs Coferece, Edhove Uversy of Techology, Edhove, The Neherlad, Augus, 5, 7-. [9] R. L ad F. Lu, Fracoal hgh order ehods for he olear fracoal ordary dffereal equao, Nolear Aal. 66, 7, o. 4, [] F. Maard, Fracoal calculus: soe basc probles couu ad sascal echacs, Fracals ad fracoal calculus couu echacs (Ude, 996, CISM Courses ad Lecures, vol. 378, Sprger, Vea, 997, pp [] Yldr A. Moa, S., Aalycal approxae soluos of he fracoal coveco-dffuso equao wh olear source er by he's hooopy perurbao ehod, Ieraoal Joural of Copuer Maheacs. 87,, o. 5, [] Kuraz A. Kes Y. Ourac, G., A ew aalycal approxae ehod for he soluo of fracoal dffereal equaos, Ieraoal Joural of Copuer Maheacs. 85, 8, o., 3-4. [3] Igor Podluby, Fracoal dffereal equaos, Maheacs Scece ad Egeerg, vol. 98, Acadec Press Ic., Sa Dego, CA, 999. [4] Igor Podluby, Marx approach o dscree fracoal calculus, Frac. Calc. Appl. Aal. 3,, o. 4,
9 J. Basc. Appl. Sc. Res., ( , [5] Trgeassou J Poo T, Modelg ad sulao of fracoal syses usg a o eger egraor, : Proceedgs of DETC3, 3 ASME Desg Egeerg Techcal Cofereces, Sepeber ï ½-6, Chcago, Illos 3, ï ½-6. [6] Abbas Saadaad ad Mehd Dehgha, A ew operaoal arx for solvg fracoal-order dffereal equaos, Copuers & Maheacs wh Applcaos. 59,, o. 5, [7] J. Sabaer, O P. Agrawal, ad J. A. Terero Machado, Advaces fracoal calculus, Sprger, Dordrech, 7. [8] B. Saovc T.M. Aaacovc, O a uercal schee for solvg dffereal equaos of fracoal order, Mechacs Research Coucaos. 35, 8, o. 7, [9] She S.-L. Wu, X.-J., Chaos he fracoal-order lorez syse, Ieraoal Joural of Copuer Maheacs. 86, 9, o. 7, [3] Moa S. Yldr, A., Seres soluos of a fracoal oscllaor by eas of he hooopy perurbao ehod, Ieraoal Joural of Copuer Maheacs. 87,, o. 5, 7-8. [3] Lxa Yua ad O P. Agrawal, A uercal schee for dyac syses coag fracoal dervaves, Joural of vbrao ad acouscs, 4,, o.,
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