FRACTIONAL-order differential equations (FDEs) are

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1 Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch Hong Kong Comprison of Anlyicl nd Numericl Soluions of Frcionl-Order Bloch Equions using Relible Mehods Sekson Sirisubwee Absrc In his pper we solve Cpuo frcionl-order Bloch equions. The Bloch equions re model for nucler mgneic resonnce NMR which is physicl phenomenon rising in engineering medicine nd he physicl sciences. A revised vriionl ierion mehod nd n Adms-Bshforh- Moulon ype predicor-correcor scheme re used o obin nlyicl soluions nd numericl soluions respecively for he frcionl-order equions. We compre he nlyicl nd numericl soluions for seleced frcionl orders. Index Terms Revised vriionl ierion mehod Adms- Bshforh-Moulon ype predicor-correcor mehod Cpuo frcionl-order Bloch equions. I. INTRODUCTION FRACTIONAL-order differenil equions FDEs re bsed on vrious definiions of frcionl derivives. They re generlizions of clssicl differenil equions nd hey hve been of ineres o mhemicins scieniss nd engineers for more hn hiry yers becuse hey cn be used o describe he memory nd herediry properies of vrious physicl processes [1]. FDEs hve been efficienly used o model mny rel phenomen in fields such s engineering [2] physics [3] pplied mhemics [4] nd disese models [5]. In generl mos nonliner FDEs do no hve exc soluions nd hence pproxime nlyic soluions nd numericl soluions re usully required. Some echniques for obining pproxime nlyic soluions of FDEs include he Lplce-Adomin decomposiion mehods LADM [6] he Dun-Rch modified decomposiion mehod [7] he homoopy nlysis mehod HAM [8] nd he Lplce-vriionl ierion mehod LVIM [9]. The LADM nd HAM re bsed on he ssumpion h he soluions of FDEs re in he form of infinie series wheres he LVIM consrucs correcion funcionl by generlized Lgrnge muliplier mehod. Numericl mehods for obining pproxime soluions of FDEs re bsed on discreizion of he independen vrible nd include he Adms-Bshforh-Moulon ype predicor-correcor or PECE Predic Evlue Correc nd Evlue mehod [1] he simulink model [11] nd he finie elemen mehod [12]. The im of his pper is o invesige nd compre nlyicl nd numericl soluions of he following Cpuo Mnuscrip received Jnury 4 218; revised Jnury Sekson Sirisubwee is lecurer in he Deprmen Mhemics King Mongku s Universiy of Technology Norh Bngkok Bngkok 18 nd resercher wih he Cenre of Excellence in Mhemics CHE Si Ayuhy Rod Bngkok 14 Thilnd emil:sekson.s@sci.kmunb.c.h. frcionl-order Bloch equion CD α = ω CD α = ω CD α = M wih iniil condiions 1 = M x = M y = M z. 2 Sysem 1 is generlizion of he clssicl sysem of firsorder Bloch equions [13] wih iniil condiions given =. The frcionl-order sysem 1 is developed from he firs-order sysem by replcing he firs-order ime derivives wih he Cpuo frcionl derivives of order α 1] denoed by C D α. I is imporn o minin consisen se of unis for boh sides of ech equion in he sysem vi frcionl ime consns. In Eq. 1 he ses nd represen he sysem mgneizion in x y nd z componens respecively. The menings of he prmeers in he sysem re s follows: ω is he resonn frequency is he spin-lice relxion ime is he spin-spin relxion ime nd M is he equilibrium mgneizion. The pplicions of sysem 1 in some specific fields cn be found in [13] [15]. In his pper we obin n pproxime nlyicl soluion of he iniil vlue problem in Eqns. 1 2 using he revised vriionl ierion mehod RVIM [16] nd numericl soluion using he Adms-Bshforh-Moulon ype predicorcorrecor scheme PECE [17]. These wo mehods re currenly regrded s he mos relible nd efficien nlyicl nd numericl mehods for solving FDEs. The pper is orgnized s follows. In secion 2 preliminry definiions nd necessry properies re given. In secion 3 descripion of he mehods used in he pper re briefly given. In secion 4 he soluions of he IVP 1 2 re obined using he nlyicl nd numericl mehods. Finlly secion 5 includes discussion nd conclusions. II. PRELIMINARY DEFINITIONS AND PROPERTIES In his secion we provide necessry definiions of frcionl-order operors including he Riemnn-Liouville frcionl inegrl nd he Cpuo frcionl derivive. The imporn properies of he operors re briefly given. A funcion f > is sid o be in he spce C α α R if i cn be wrien s f = p g for some p > α where g is coninuous in [. The funcion is lso sid ISBN: ISSN: Prin; ISSN: Online IMECS 218

2 Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch Hong Kong o be in he spce Cα m if f m C α m N for more deils see [1]. Definiion 2.1: [1]. The Riemnn-Liouville frcionl inegrl operor of order α > of funcion f C α wih is defined s RLJ α f = 1 Γα τ α 1 fτdτ > 3 where Γ is he gmm funcion. If α = hen RL J α f = f. Definiion 2.2: [1]. Given posiive rel number α he Cpuo frcionl derivive of order α wih is defined in erms of he Riemnn-Liouville frcionl inegrl i.e. C Dα f = RL J m α f m where m 1 < α < m m N or i cn be expressed s CD α f = 1 Γ m α f m τ α m1 dτ 4 τ where nd f C m 1. If α = m hen C D α f = f m. Remrk 2.1: [1] For f C n 1 α β m 1 < α m α β n where m n N nd γ 1 we hve he following imporn properies 1. RL J α RLJ β f = RL J β RLJ α f = RL J αβ f 2. RL J α γ Γγ 1 = Γγ α 1 γα 3. RL J α CD α f = f k= f k k. k! III. DESCRIPTIONS OF THE METHODS In his secion we give descripions of he RVIM nd he PECE mehods h we use o solve he IVP 1-2. A. The revised vriionl ierion mehod We firs provide he generl principle of he vriionl ierion mehod VIM [16] [18] for solving frcionlorder differenil equion. Then we describe he RVIM for solving sysem of FDEs. Consider he following frcionl-order differenil equion: CD α u Nu = f < α 1 5 where N is nonliner operor wih respec o u nd f is source funcion. According o he VIM he correcion funcionl for Eq. 5 is consruced s follows: u n1 = u n RL J α [λ C D α u n Nu n f] = u n 1 τ α 1 λτ Γα C D α u n τ Nu n τ fτ dτ 6 where λ is he Lgrnge muliplier which cn be opimlly deermined vi vriionl heory [19]. Some pproximions re required o idenify he Lgrnge muliplier. The correcion funcionl equion 6 cn be pproximed by he following equion: u n1 = u n λτ u nτ Nũ n τ fτ dτ. ISBN: ISSN: Prin; ISSN: Online 7 If we now pply resriced vriions ũ n o he erm Nu hen we cn esily deermine he muliplier. Assuming he foremenioned funcionl o be sionry i.e. δũ n = we obin δu n1 = δu n δ λτ u nτ fτ dτ. 8 This yields he Lgrnge muliplier λ = 1. Subsiuing λ = 1 ino he funcionl equion 6 we obin he following ierion formul: u n1 = u n RL J α C D α u n Nu n f. 9 The iniil pproximion u cn be seleced o sisfy he iniil condiions of he problem. Finlly we cn pproxime he soluion u = lim n u n by he Nh erm pproximion u N. Now we describe he RVIM for solving sysem of frcionl-order differenil equions. Consider he following incommensure sysem of frcionl-order differenil equions: CD α i u i N i u 1... u m = f i < α i 1 i = m 1 where N i re operors of u j j = m nd f i re known funcions. The correcion funcionls for his cse re s follows: u in1 = u in RL J α i C Dα i u in N i u 1n... u mn f i i = m. 11 Similrly he iniil pproximions u i i = m cn be independenly seleced s long s hey sisfy he iniil condiions of he sysem. The N h order erms u in i = m cn hen be used o represen pproximions of he soluions u i = lim n u in i = m of he sysem. The ierive formul for he RVIM cn be consruced by modifying he ierion formul 11 obined by he sndrd VIM s bove. The modificion cn be done by replcing u 1n... u i 1n in he formul of u in1 wih he upded vlues u 1n1... u i 1n1 respecively. In consequence he recursive formul for he RVIM employed o solve sysem 1 is expressed s u in1 = u in RL J α i C Dα i u in N i u 1n1... u i 1n1 u in... u mn f i i = m. 12 The modified echnique in he RVIM formul 12 cn ccelere he convergence of ierive pproxime soluions compring o he pproxime soluions obined using he sndrd VIM. Hence he correced soluion u in1 of he RVIM is more ccure hn he soluion u in1 from he sndrd VIM becuse upded vlues re used o compue he RVIM soluion. B. The predicor-correcor scheme The Adms-Bshforh-Moulon ype predicor-correcor scheme or he PECE mehod [1] is now widely used o solve FDEs. Hence we will use his echnique o numericlly solve he IVP 1 2. The formuls of he mehod re briefly given s follows. IMECS 218

3 Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch Hong Kong Consider he frcionl-order iniil vlue problem CD α u = f u T u k = u k k = 1...m 1 α m 1 m 13 where f is nonliner funcion nd m is posiive ineger. The IVP 13 cn be convered o he following Volerr inegrl equion u = k= u k k k! 1 Γα τ α 1 fτ uτdτ. 14 To esime he inegrl in 14 we uniformly discreize he whole ime T by uniform grid { n = nh : n = 1...N} for some ineger N wih he sep size h := T/N. Le u h n denoe he pproximion o u n. Suppose h we hve lredy clculed pproximions u h j j = n hen he pproximion u h n1 of he IVP 13 cn be compued using he PECE mehod s follows: u h n1 = k= Γα 2 k n1 u k k! Γα 2 f n1 u P h n1 jn1 f j u h j 15 j= where n α1 n αn 1 α if j = n j 2 jn1 = α1 n j α1 2n j 1 α1 if 1 j n 1 if j = n The preliminry pproximion u P h n1 in Eq. 15 is clled predicor nd is given by u P h n1 = where k= k n1 u k 1 k! Γα b jn1 f j u h j j= 17 b jn1 = hα α n 1 jα n j α. 18 IV. MAIN RESULTS In his secion we exhibi he use of he RVIM nd he PECE s described bove o solve he FIVP in Eqs. 1-2 wih he sring poin = =. However we firs demonsre he exc soluion which cn be discovered in [13] of he problem for α = 1 s follows. = e / M x cosω M y sinω = e /T2 M y cosω M x sinω = M z e /T1 M 1 e /T1. 19 The iniil condiions nd prmeer vlues which re employed in our simulions re s follows. M x = M y = 1 M z = ω = 6 π = 1 = M = 1. 2 A. The pplicion of he RVIM The curren secion is devoed o he use of he RVIM o obin n nlyicl soluion of he IVP 1-2. Applying he ierion formul 12 of he RVIM wih λτ = 1 o he problem we obin he following ierion formuls: [ Mx n1 = Mx n RL J α λτ D α Mx n τ C ω My n τ M ] x n τ = Mx n RL J α ω My n τ n τ [ λτ My n1 = My n RL J α ] n τ Mz n1 = Mz n RL J α C 21 D α My n τ ω Mx n1 τ ω Mx n1 τ n τ = My n RL J α [ λτ D α Mz n τ ] M Mz n τ = M n z RL J α C M M n z τ 22 n = in which he iniil pproximions re chosen o be M x = M x M y = M y nd M z = M z. B. The pplicion of he PECE mehod Applying he PECE mehod in Eqs o he IVP 1-2 we discreize he ime inervl wih poins { n } nd obin he formuls for pproximions Mx hn = Mx h n My hn = My h n Mz hn = Mz h n s follows: Mx hn1 = Mx ω My Phn1 Phn1 Γα 2 Γα 2 j= My hn1 = My Γα 2 Γα 2 M hn1 z = M z in which Γα 2 1jn1 ω M hj y ω M Phn1 x 2jn1 ω Mx hj j= Γα 2 j= M Phn1 x = M x 1 Γα M Phn1 y = M y 1 Γα M Phn1 z = M z 1 Γα hj M Mz Phn1 M Mz hj 3jn1 j= b 1jn1 ω M hj y b 2jn1 ω Mx hj j= 24 Phn1 hj M x hj hj M Mz hj b 3jn1 29 j= ISBN: ISSN: Prin; ISSN: Online IMECS 218

4 Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch Hong Kong where for l = n α1 n αn 1 α if j = n j 2 ljn1 = α1 n j α1 2n j 1 α1 3 if 1 j n 1 if j = n 1 nd b ljn1 = hα α n 1 jα n j α j n. 31 We will use he discreized formuls o obin he numericl soluions for he IVP 1-2 in he following secion. C. Simulion resuls Given he prmeer vlues described in Eq. 2 nd he sring poin = we will exhibi he simulion resuls of he IVP 1-2 obined using he formuls of he exc soluion in Eq. 19 he RVIM in Eqns nd he PECE mehod in Eqns for α = In priculr he bsolue errors of he numericl resuls genered by he RVIM nd he PECE mehod compred o hose obined using he exc soluion re clculed for α = 1. Moreover he bsolue differences of he numericl resuls clculed by he RVIM nd he PECE mehod re mesured for α =.9.8. The following resuls re for α = 1. Using he ierion formuls we obin he 8h erms of he pproximions Mx 8 My 8 nd Mz 8 s follows. M 8 x = M 8 y = M 8 z = The simulion resuls of he problem for α = 1 using ll of he mehods i.e. he exc formuls 19 he RVIM in Eq. 32 nd he PECE mehod in Eqns wih he sep size h = 1 4 re shown in Fig. 1. I cn be esily observed h he numericl simulions obined by he RVIM nd he PECE mehod re in very good greemen wih he exc soluions. The bsolue errors beween numericl soluions which re clculed using he RVIM nd he PECE mehod nd he exc soluions re shown in Tble I nd II respecively. The following numericl conclusions obined from Tble I II re s below. The pproxime soluion componen obined by he wo mehods is he mos ccure when compred wih is corresponding exc soluion. When is lrger he pproxime soluion componens chieved by he RVIM re significnly less ccure compred wih heir corresponding exc soluions. However he flucuion of he bsolue errors in he PECE mehod is pprecibly lower hn h of he RVIM when is incresing. From he simulion resuls of he RVIM nd he PECE mehod compred wih he exc soluions he wo mehods re relible nd efficien ools for obining pproxime soluions of he problem for α =.9.8 s well. Nex we will simule numericl resuls of he problem for α =.9.8 s follows. Applying he RVIM o he problem vi he ierion formuls he 8h erm of Exc of Exc of Exc Fig. 1. Simulion comprisons of he soluions for he IVP 1-2 using he exc soluions formuls he RVIM nd he PECE mehod for α = 1. pproximions of he soluions for α =.9 re expressed s M 8 x = M 8 y = M 8 z = The obined 8h erm pproximions using he RVIM for ISBN: ISSN: Prin; ISSN: Online IMECS 218

5 Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch Hong Kong TABLE I THE ABSOLUTE ERRORS OF NUMERICAL RESULTS OBTAINED BY THE RVIM WITH N = 8 COMPARED WITH THE EXACT SOLUTIONS FOR THE IVP 1-2 WHEN α = 1 Exc-RVIM E E E E E E E E E-6 3.3E E E E E TABLE II THE ABSOLUTE ERRORS OF NUMERICAL RESULTS OBTAINED USING THE PECE METHOD WITH h = 1 4 COMPARED WITH THE EXACT SOLUTIONS FOR THE IVP 1-2 WHEN α = Exc-PECE E E E E E E E E E E E E E E E α =.8 re M 8 x = M 8 y = M 8 z = In similr fshion he numericl simulions resuls of he problem for α =.9.8 uilizing he RVIM in Eq nd he PECE mehod in Eqns wih he sep size h = 1 4 re grphiclly porryed in Figs. 2 3 respecively. I is no difficul o observe from such figures h he numericl resuls from he RVIM nd he PECE mehod re sill in good greemen for α =.9.8. The bsolue differences of he resuls obined by he wo mehods re numericlly shown in Tble III nd IV for α =.9 nd α =.8 respecively. We cn conclude from he ls wo bles h he wo mehods provide he numericl d which re quie close o ech oher for he specified vlues of. Especilly he soluion componen obined vi he wo mehods hs he lowes discrepncy. TABLE III THE ABSOLUTE DIFFERENCES AT THE SPECIFIED VALUES OF COMPARED USING THE RESULTS OF THE RVIM N = 8 AND THE PECE METHOD h = 1 4 FOR THE IVP 1-2 WHEN α =.9 RVIM-PECE E-3 1.8E E E-4 1.2E E E E E E E E Fig. 2. Simulion comprisons of he soluions for he IVP 1-2 using he RVIM nd he PECE mehod for α =.9. V. CONCLUSION In his pper we hve obined pproxime soluions of he frcionl-order iniil vlue problem 1 2 for he NMR Bloch equions. Approxime nlyicl soluions hve been compued vi he RVIM nd pproxime numericl soluions hve been compued vi he PECE mehod. The simulions of he soluions clculed by he wo mehods hve been compred for ineger order α = 1 nd frcionl orders α =.9.8. The exc soluions of he problem for α = 1 hve been used o mesure he ccurcy of he soluions obined by he wo pproxime mehods. A comprison of he simulion resuls for he wo pproxime mehods wih he exc soluion for α = 1 show h he pproxime mehods give quie ccure soluions. Comprisons of he pproxime nlyicl nd numericl resuls hve shown h he pproxime soluions re in ISBN: ISSN: Prin; ISSN: Online IMECS 218

6 Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch Hong Kong TABLE IV THE ABSOLUTE DIFFERENCES AT THE SPECIFIED VALUES OF COMPARED USING THE RESULTS OF THE RVIM N = 8 AND THE PECE METHOD h = 1 4 FOR THE IVP 1-2 WHEN α =.8 RVIM-PECE.5 5.1E-2 6.8E E E E-2 1.6E E E-4 8.8E E E E E E-5 2.5E E E E E-3 2.1E E Fig. 3. Simulion comprisons of he soluions for he IVP 1-2 using he RVIM nd he PECE mehod for α =.8. good greemen for α =.9.8. Finlly we believe h he RVIM nd PECE mehods cn be relibly nd efficienly pplied o solve frcionl-order differenil equion sysems from oher engineering nd pplied science problems. ACKNOWLEDGMENT The uhors would like o hnk he ediors nd he nonymous referees for heir vluble suggesions on he improvemen of his pper. The uhor ws finncilly suppored by he Fculy of Applied Science King Mongku s Universiy of Technology Norh Bngkok conrc no REFERENCES [1] I. Podlubny Frcionl differenil equions. Acdemic press 1998 vol [2] A. D. Obembe H. Y. Al-Yousef M. E. Hossin nd S. A. Abu- Khmsin Frcionl derivives nd heir pplicions in reservoir engineering problems : A review Journl of Peroleum Science nd Engineering vol. 157 no. Supplemen C pp [3] O. Guner nd A. Bekir The Exp-funcion mehod for solving nonliner spce?ime frcionl differenil equions in mhemicl physics Journl of he Associion of Arb Universiies for Bsic nd Applied Sciences vol. 24 no. Supplemen C pp [4] S. Sirisubwee S. Koonprser C. Khopn nd W. Pork Two relible mehods for solving he 3 1-dimensionl spce-ime frcionl Jimbo-Miw equion Mhemicl Problems in Engineering vol [5] A. Crvlho nd C. M. Pino A dely frcionl order model for he co-infecion of mlri nd HIV/AIDS Inernionl Journl of Dynmics nd Conrol vol. 5 no. 1 pp [6] N. Dogn Numericl soluion of choic genesio sysem wih mulisep Lplce Adomin decomposiion mehod Kuwi Journl of Science vol. 4 no [7] S. K. Sekson Sirisubwee nd S. Kew Dun-Rch modified decomposiion mehod for solving some ypes of nonliner frcionl muli-poin boundry vlue problems Journl of elecricl engineering vol. 12 no. 1 pp [8] B. Ghznfri nd F. Veisi Homoopy nlysis mehod for he frcionl nonliner equions Journl of King Sud Universiy - Science vol. 23 no. 4 pp [9] O. Mrin A modified vriionl ierion mehod for he nlysis of viscoelsic bems Applied Mhemicl Modelling vol. 4 no. 17 pp [1] K. Diehelm N. J. Ford nd A. D. Freed A predicor-correcor pproch for he numericl soluion of frcionl differenil equions Nonliner Dynmics vol. 29 no. 1 pp [11] I. Peráš Frcionl-order feedbck conrol of dc moor Journl of elecricl engineering vol. 6 no. 3 pp [12] E. Zho T. Cho S. Wng nd M. Yng Finie-ime formion conrol for muliple fligh vehicles wih ccure linerizion model Aerospce Science nd Technology vol. 71 no. Supplemen C pp [13] R. Mgin X. Feng nd D. Blenu Solving he frcionl order bloch equion Conceps in Mgneic Resonnce Pr A vol. 34 no. 1 pp [14] Y. Qin C. Lu nd L. Li Muli-scle cyclone civiy in he Chngjing River Huihe River vlleys during spring nd is relionship wih rinfll nomlies Advnces in Amospheric Sciences vol. 34 no. 2 pp [15] D. Blenu R. L. Mgin S. Bhlekr nd V. Dfrdr-Gejji Chos in he frcionl order nonliner Bloch equion wih dely Communicions in Nonliner Science nd Numericl Simulion vol. 25 no. 1 pp [16] C. Ünlü H. Jfri nd D. Blenu Revised vriionl ierion mehod for solving sysems of nonliner frcionl-order differenil equions in Absrc nd Applied Anlysis vol [17] M. Zyernouri nd A. Mzvinos Frcionl Adms- Bshforh/Moulon mehods: An pplicion o he frcionl Keller-Segel chemoxis sysem Journl of Compuionl Physics vol. 317 no. Supplemen C pp [18] J.-H. He Approxime nlyicl soluion for seepge flow wih frcionl derivives in porous medi Compuer Mehods in Applied Mechnics nd Engineering vol. 167 no. 1-2 pp [19] M. Inokui H. Sekine nd T. Mur Generl use of he lgrnge muliplier in nonliner mhemicl physics Vriionl mehod in he mechnics of solids vol. 33 no. 5 pp ISBN: ISSN: Prin; ISSN: Online IMECS 218