Studes n Nonlner Scences (): 5-, ISSN -9 IOSI Publctons, Numercl Soluton of Nonlner Mult-order Frctonl fferentl Equtons by Implementton of the Opertonl Mtr of Frctonl ervtve M.M. Khder eprtment of Mthemtcs, Fculty of Scence, Benh Unversty, Benh, Egypt Abstrct: he mn m of ths rtcle s to generlze the Legendre opertonl mtr to the frctonl dervtves nd mplemented t to solve the nonlner mult-order frctonl dfferentl equtons. In ths pproch, truncted Legendre seres together wth the Legendre opertonl mtr of frctonl dervtves re used. he mn chrcterstc behnd the pproch usng ths technque s tht t reduces such problems to those of solvng system of lgebrc equtons thus gretly smplfyng the problem. he proposed pproch ws bsed on the shfted Legendre tu nd shfted Legendre collocton methods. In the lmt, s pproches n nteger vlue, the scheme provdes soluton for the nteger-order dfferentl equtons. he frctonl dervtves re descrbed n the Cputo sense. A comprson between the proposed method nd the Adomn ecomposton Method (AM) s gven. he study s conducted through llustrtve emple to demonstrte the vldty nd pplcblty of the presented method. he results revel tht the proposed method s very effectve nd smple. Moreover, only smll number of shfted Legendre polynomls s needed to obtn stsfctory result. Key words: Cputo's frctonl dervtve opertonl mtr. Legendre polynomls collocton method Adomn decomposton method INROUCION he number of scentfc nd engneerng problems nvolvng frctonl clculus s lredy very lrge nd stll growng nd perhps the frctonl clculus wll be the clculus of twenty-frst century. For emple, frctonl clculus s ppled to model frequency dependent dmpng behvor of mny vscoelstc mterls [], contnuum nd sttstcl mechncs, economcs [] nd others [-5]. Frctonl dfferentls nd frctonl ntegrls provde more ccurte models of systems under consderton. One of the most recent wors on the subect of frctonl clculus,.e., the theory of dervtves nd ntegrls of frctonl (non-nteger) order, s the boo of Podlubny [6-8], whch dels prncplly wth frctonl dfferentl equtons. And tody, there re mny wors on frctonl clculus [6, 9]. Most frctonl dfferentl equtons do not hve ect solutons, so ppromton nd numercl technques must be used, such s Adomn's decomposton method [-], He's vrtonl terton method [9], homotopy perturbton method [, 4], collocton method [5], Glern method nd other methods [6]. Orthogonl functons hve receved consderble ttenton n delng wth vrous problems. he mn chrcterstc behnd the pproch usng ths technque s tht t reduces these problems to those of solvng system of lgebrc equtons thus gretly smplfyng the problem. In ths method, truncted orthogonl seres s used for numercl ntegrton of dfferentl equtons, wth the gol of obtnng effcent computtonl solutons. Severl ppers hve ppered n the lterture concerned wth the pplcton of shfted Legendre polynomls [, 8]. In ths pper we ntend to etend the pplcton of Legendre polynomls to solve frctonl dfferentl equtons. Our mn m s to generlze Legendre opertonl mtr to frctonl clculus. It s worthy to menton here tht, the method bsed on usng the opertonl mtr of n orthogonl functon for solvng dfferentl equtons s computer orented. he orgnzton of ths pper s s follows. In the net Secton, Legendre opertonl mtr of the frst dervtve s obtned. Secton summrzes the Legendre opertonl mtr of the frctonl dervtve. In Secton 4, the opertonl mtr of frctonl dervtve for nonlner mult-order frctonl dfferentl equton s mplemented. In Secton 5, some numercl results re gven to clrfy the method. An mplementton of the Adomn decomposton method s ntroduced n Secton 6. Also concluson s gven n Secton. Note tht we hve computed the numercl results usng Mthemtc progrmmng. Correspondng Author: r. M.M. Khder, eprtment of Mthemtcs, Fculty of Scence, Benh Unversty, Benh, Egypt 5
Studes n Nonlner Sc., (): 5-, In ths secton some necessry defntons nd mthemtcl prelmnres of the frctonl clculus theory requred for our subsequent development re gven [8]. efnton : he Cputo frctonl dervtve opertor α of order α s defned n the followng form: (m) f (t) f() dt, G(m ) ( t) m + > where m-<α<m, m N, >. Smlr to nteger-order dfferentton, Cputo frctonl dervtve opertor s lner operton: ( λ f() + µg() ) λ f() + µ g() where λ nd µ re constnts. For the Cputo's dervtve we hve [8]: n C, C s constnt (), for n N nd n<α G(n+ ) n-, for n N nd n α G(n ) + We use the celng functon α to denote the smllest nteger greter thn or equl to α. Also N {,,, }. Recll tht for α N, the Cputo dfferentl opertor concdes wth the usul dfferentl opertor of nteger order. For more detls on frctonl dervtves defntons nd ts propertes [6-8]. LEGENRE OPERAIONAL MARIX OF HE FIRS ERIVAIVE he well nown Legendre polynomls re defned on the ntervl [-,] nd cn be determned wth the d of the followng recurrence formul [9]: + L (z) zl (z)- L (z),,,... + + - + - where L (z) nd L (z) z. In order to use these polynomls on the ntervl [,] we defne the so clled shfted Legendre polynomls by ntroducng the chnge of vrble z -. Let the shfted Legendre polynomls L (-) be denoted by P (). hen P () cn be obtned s follows: () 6 ( + )(-) P + () P()- P -(),,,... + + where P () nd P () -. he nlytc form of the shfted Legendre polynomls P () of degree gven by: ( + )! P () ( ) + ( )(!) Note tht P () (-) orthogonlty condton s:, for P()P()d +, for () (4) nd P (). he he functon u() squre ntegrble n [,], my be epressed n terms of shfted Legendre polynomls s: u() cp() where the coeffcents c re gven by: c (+ ) u()p()d,,,... In prctce, only the frst (m+)-terms shfted Legendre polynomls re consdered. hen we hve: m u () cp() C Ψ() m where the shfted Legendre coeffcent vector C nd the shfted Legendre vector Ψ() re gven by: C [c,c,...,c m], () [P (),P(),...,P m()] (5) Ψ () In the followng theorem we wll defne the dervtve of the vector Ψ(). heorem : he dervtve of the vector Ψ() cn be epressed by: d Ψ() () Ψ () (8) d
Studes n Nonlner Sc., (): 5-, where () s the (m+) (m+) opertonl mtr of dervtve gven by:,,...,m, f m odd (+ ), for -, (d ),,...,m-, f m even, otherwse () For emple for even m we hve:................ 5... m-... m- () 5... In the followng seton we generlze the opertonl mtr of dervtve of shfted Legendre polynomls gven n (8) for frctonl dervtve. LEGENRE OPERAIONAL MARIX OF HE FRACIONAL ERIVAIVE By usng Eq.(8), for n N, t s cler tht: d n Ψ() () n (n) n ( ) Ψ () Ψ (), n,,,... (9) d heorem : Let Ψ() be shfted Legendre vector defned n () nd lso suppose α> then: ( ) α α Ψ () Ψ () () where (α) s the (m+) (m+) opertonl mtr of frctonl dervtve of order α n the Cputo sense nd s defned s follows:.... w w. w,,,,,m, ( α).... w w. w,,,,,m,.... m m m w w. w m,, m,, m,m, () where w,, s gven by: + + + l ( ) (+ )! ( l+ )! w,, (+ ) l ( )!! G( + ) ( l)! ( l!) (+ l + ) () Note tht n (α), the frst α rows, re ll zero []. Proof: Let P () be shfted Legendre polynoml then, by usng (), () nd (4) we cn fnd tht: α P(),,,,..., α -, α> () Also, by usng () nd (4) we cn obtn (for α,, m): + + ( ) ( )! ( ) ( )! α + + P() ( ) α α (4) ( )! (!) ( )!! G( + )
Studes n Nonlner Sc., (): 5-, Now, ppromte α- by m+-terms of shfted Legendre seres, we hve: where m d P(), (5) + l + l ( ) ( l+ )! + l α ( ) ( l+ )! d (+ ) P()d (+ ) d (+ ), l ( l)! ( l!) l ( l)! ( l!) (+ l + ) Employng Eqs. (4)-(6), we get: m + ( ) ( )! m + P() d P() w P(),,...,m,,, ( )!! G( + ) () where w,, s gven n Eq. (). Rewrte Eq. () s vector form we hve: P() [ w, w,..., w ] (),,...,m Ψ,,,,,m, (8) Also ccordng to Eq. (), we cn wrte: α P() [,,...,] ψ (),,,,..., α - (9) A combnton of Eqs. () nd () leds to the desred result. Remr: If α n N, then heorem gves the sme result s Eq. (9). In the followng secton, n order to show the hgh mportnce of opertonl mtr of frctonl dervtve, we pply t to solve the nonlner multorder frctonl dfferentl equton. he estence nd unqueness nd contnuous dependence of the soluton to ths problem re dscussed n []. OPERAIONAL MARIX OF FRACIONAL ERIVAIVE FOR NONLINEAR MULI-ORER FRACIONAL IFFERENIAL EQUAION Consder the nonlner mult-order frctonl dfferentl equton: ß ß u() F(,u(), u(),..., u() ) () wth ntl condtons: () u () d,,,...,n () 8 where n<α n+, <β <β < <β <α nd α denotes the Cputo frctonl dervtve of order α. It should be noted tht F cn be nonlner n generl. o solve problem ()-() we ppromte u(), α β u() nd u(), for,,, by the shfted Legendre polynomls s: m u() cp() C Ψ() () α α ( α) u() C Ψ () C Ψ () () β β β u() C Ψ () C Ψ (),,,..., (4) where vector C [c, c,,c m ] s n unnown vector. By substtutng these equtons n Eq.() we get: ( α) C Ψ() F(,C Ψ(), (ß ) (ß ) C Ψ(),...,C Ψ() ) Also, by substtutng Eqs. () n Eq. () we obtn: u() C Ψ () d u () () C () Ψ () d,,,...,n (5) (6) In order to obtn the soluton u(), we frst collocte Eq. (5) t m-n ponts. For sutble
Studes n Nonlner Sc., (): 5-, collocton ponts we use the frst m-n shfted Legendre roots of P m+ (). hese equtons together wth Eq. (6) generte m+ nonlner equtons whch cn be solved usng Newton's tertve method. Consequently u() gven n Eq. () cn be clculted. NUMERICAL IMPLEMENAION Consder the followng nonlner dfferentl equton: 4 u() + u() + u () 9 () subect to the ntl condtons: () () () u() u () u (), u () 6 (8) o solve the bove problem, by pplyng the technque descrbed n Secton 4 wth m 4, we ppromte soluton s: u() cp() + cp() + cp() + + Ψ cp() cp() 4 4 C () ( ) 4 44-448 -96 π π π π π Usng Eq. (5) we hve: C c c c c c 4 ( ) (4) C Ψ () + C Ψ () + (C Ψ ()) 9 (9) Now we collocte Eq. (9) t the frst root of P 5 (),.e., Here we hve: () () 6 6 4 6 4 4 45 4.65 4 Also by usng Eq. (6) we get: C Ψ () c c+ c c + c 4 () C Ψ () c 6c + c c 4 () C Ψ () c 6c + 8c 4 () C Ψ () c 84c 6 4 By solvng Eqs. (9) nd () we obtn: () () (4) 84 68 9 c, c, c, c, c 4 4 4 herefore 9 u(),,,, 6 6 + 4 4 + 4 4 + 9 + whch s the ect soluton of the problem. 9
Studes n Nonlner Sc., (): 5-, It s cler tht n ths emple the present method cn be consdered s n effcent method. where the frst A n Adomn's polynomls tht represent the nonlner term N(u) u re gven by: AOMIAN ECOMPOSIION MEHO FOR NONLINEAR MULI-ORER FRACIONAL IFFERENIAL EQUAION In order to obtn the soluton of the problem ()-(8) by mens of AM we pply the nverse opertor on () nd usng the ntl condtons (8), we cn derve: u() f() + L ( 9 ) L [u() u + ()] () where ϕ() s the soluton of the homogenous dfferentl equton 4 u() nd the opertor L s n ntegrl opertor nd gven by L (.) (.)dddd. he AM [, ] ssumes tht the unnown soluton cn be epressed by n nfnte seres of the form: u() u n () () n nd the nonlner opertor term N(u) u cn be decomposed by n nfnte seres of polynomls gven by: N(u) A n n () where the components u n (), n wll be determned recurrently nd A n re the so-clled Adomn's polynomls of u, u, u, defned by: d n n- A n [ n N ( λ u) ] λ, n n! dλ (4) these polynomls cn be constructed for ll nonlnerty ccordng to lgorthm set by Adomn. Substtutng from (), () nto () nd equtng the smlr terms n both sdes of the equton, we get the followng recurrence relton: u () L ( 9 + ) u () L ( / n+ un + A n), n (5) A u, A u u, A (u u u + u ), A u 6u uu u + + u,... other polynomls cn be generted n le mnner. Now from the recurrence reltonshp (5) we obtn the followng resultng components: u ().585 + u().585.54 / 8.69 - u ().54 / 4.65 4 + + 8.69 +.85 4/ u () 4.65 4.85 9/.85-4/.4-4,... Hvng u (),,,,,n, the soluton s s follows: n u() f () u() (6) n he convergence of AM s ntroduced n mny ppers, for emple []. he behvour of the ect soluton u() nd the ppromte soluton by mens of AM u AM () ϕ () s presented n the Fg..??? u??.5..5..5. Fg. : he behvour of u() (dshed lne) nd u AM () ϕ () (contnuous lne)
CONCLUSION Frctonl clculus hs been used to model physcl nd engneerng processes tht re found to be best descrbed by frctonl dfferentl equtons. For tht reson we need relble nd effcent technque for the soluton of frctonl dfferentl equtons. hs pper dels wth the ppromte soluton of clss of mult-order frctonl dfferentl equtons. he frctonl dervtves re descrbed n the Cputo sense. Our mn m s to generlze the Legendre opertonl mtr to the frctonl clculus. In ths pproch, truncted Legendre seres together wth the Legendre opertonl mtr of frctonl dervtves re used for numercl ntegrton of frctonl dfferentl equtons. he mn chrcterstc behnd the pproch usng ths technque s tht t reduces such problems to those of solvng system of lgebrc equtons thus gretly smplfyng the problem. he method s ppled to solve nonlner frctonl dfferentl equtons. Illustrtve emple s ncluded to demonstrte the vldty nd pplcblty of the presented technque. he comprson certfes tht our method gves good results. All results obtned by usng Mthemtc verson 6. REFERENCES. Bgley, R.L. nd P.J. orv, 984. On the ppernce of the frctonl dervtve n the behvor of rel mterls. J. Appl. Mech., 5: 94-98.. Blle, R.., 996. Long memory processes nd frctonl ntegrton n econometrcs. J. Econometrcs, :5-59.. Swelm, N.H., M.M. Khder nd R.F. Al-Br,. Numercl studes for mult-order frctonl dfferentl equton. Physcs Letters A, : 6-. 4. Swelm, N.H. nd M.M. Khder,. A Chebyshev pseudo-spectrl method for solvng frctonl order ntegro-dfferentl equtons. Accepted n ANZIM, o Apper. 5. Swelm, N.H., M.M. Khder nd A.M. Ngy,. Numercl soluton of two-sded spcefrctonl wve equton usng fnte dfference method. Journl of Computtonl nd Appled Mthemtcs, 5: 8-84. 6. s, S., 8. Functonl Frctonl Clculus for System Identfcton nd Controls, Sprnger, New Yor.. Mller, K.S. nd B. Ross, 99. An Introducton to the Frctonl Clculus nd Frctonl fferentl Equtons, Wley, New Yor. Studes n Nonlner Sc., (): 5-, 8. Podlubny, I., 999. Frctonl fferentl Equtons, Acdemc Press, New Yor. 9. He, J.H., 998. Appromte nlytcl soluton for seepge ow wth frctonl dervtves n porous med. Computer Methods n Appled Mechncs nd Engneerng, 6 (-): 58.. ftrdr, G.V. nd H. Jfr,. Solvng mult-order frctonl dfferentl equton usng Adomn decomposton. Appled Mthemtcs nd Computton, 89: 54-548.. Jfr, H. nd V. ftrdr-ge, 6. Solvng lner nd nonlner frctonl dffuson nd wve equtons by Adomn decomposton. Appl. Mth. nd Comput., 8: 488-49.. Sh, R.S. nd R.K. Ber, 5. An ppromte soluton of nonlner frctonl dfferentl equton by domn decomposton method. Appl. Mth. Comput., 6: 56-5.. Swelm, N.H., M.M. Khder nd R.F. Al-Br, 8. Homotopy perturbton method for multdmensonl nonlner coupled system of prbolc nd hyperbolc equtons. opologcl Methods n Nonlner Anlyss, : 95-4. 4. Swelm, N.H. nd M.M. Khder, 9. Ect solutons of some coupled nonlner prtl dfferentl equtons usng the homotopy perturbton method. Computers nd Mthemtcs wth Applctons, 58: 4-4. 5. Khder, M.M.,. On the numercl solutons for the frctonl dffuson equton. Communctons n Nonlner Scence nd Numercl Smulton, 6: 55-54. 6. Rwshdeh, E.A., 6. Numercl soluton of frctonl ntegro-dfferentl equtons by collocton method. Appl. Mth. Comput., 6:.. Enelund, M. nd B.L. Josefson, 99. medomn fnte element nlyss of vscoelstc structures wth frctonl dervtves consttutve reltons, AIAA J., 5 (): 6. 8. He, J.H., 999. Some pplctons of nonlner frctonl dfferentl equtons nd ther ppromtons. Bull. Sc. echnol., 5 (): 86-9. 9. Kumr, P. nd O.P. Agrwl, 6. An ppromte method for numercl soluton of frctonl dfferentl equtons. Sgnl Processng, 86: 6.. Meerschert, M.M. nd C. dern, 6. Fnte dference ppromtons for two-sded spce frctonl prtl dfferentl equtons. Appl. Numer. Mth., 56: 8-9.. Momn, S. nd K. A-Khled, 5. Numercl solutons for systems of frctonl dfferentl equtons by the decomposton method. Appl. Mth. Comp., 6 (): 55.
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