Research Article The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method

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1 Absrac ad Applied Aalysis Volme 3, Aricle ID 3875, 6 pages hp://dx.doi.org/.55/3/3875 Research Aricle The Aalyical Solio of Some Fracioal Ordiary Differeial Eqaios by he Smd Trasform Mehod Hasa Bl, Haci Mehme Baskos, ad Fehi Bi Mhammad Belgacem 3 Deparme of Mahemaics, Fira Uiversiy, Elazig, Trkey Deparme of Comper Egieerig, Tceli Uiversiy, Tceli, Trkey 3 Deparme of Mahemaics, Facly of Basic Edcaio, PAAET, Shamiya, Kwai Correspodece shold be addressed o Hasa Bl; hbl@fira.ed.r Received 6 March 3; Acceped 3 Jly 3 Academic Edior: Saa Saha Ray Copyrigh 3 Hasa Bl e al. This is a ope access aricle disribed der he Creaive Commos Aribio Licese, which permis resriced se, disribio, ad reprodcio i ay medim, provided he origial work is properly cied. We irodce he rdimes of fracioal calcls ad he coseqe applicaios of he Smd rasform o fracioal derivaives. Oce his coecio is firmly esablished i he geeral seig, we r o he applicaio of he Smd rasform mehod (STM) o some ieresig ohomogeeos fracioal ordiary differeial eqaios (FODEs). Fially, we se he solios o form wo-dimesioal (D) graphs, by sig he symbolic algebra package Mahemaica Program 7.. Irodcio The Smd rasform was firs defied i is crre shape by Wagala as early as 993, which he sed o solve egieerig corol problems. Alhogh he migh have had ideas for i sooer ha ha (989) as some coferece proceedigs showed, he sed i o corol egieerig problems [, ]. Laer, Wagala exeded i he Smd rasform o wo variables [3]. The firs applicaios o differeial eqaios ad iversio formlae were doe by Weerakoo i wo papers i 994 ad 998 [4, 5]. The Smd rasform was also firs defeded by Weerakoo agais Deaki s defiiio who claimed ha here is o differece bewee he Smd ad he Laplace ad who remidedweerakoohahesmrasformisreallyhe Carso or he S-mliplied rasform disgised [6, 7]. The applicaios followed i hree cosecive papers by Asir dealig wih he covolio-ype iegral eqaios ad he discree dyamic sysems [8, 9]. A his poi, Belgacem e al. sig previos refereces ad coecios o he Laplace rasform exeded he heory ad he applicaios of he Smd rasform i [ 7] o varios applicaios. I he meaime, sbseqe o exchages bewee Belgacem ad oher scholars, he followig papers sprag p i he las decade [8 ]. Moreover, he Smd rasform was also sed o solve may ordiary differeial eqaios wih ieger order [3 9]. The applicaio of STM rs o o be pragmaic i geig aalyical solio of he fracioal ordiary differeial eqaios fas. Noably, implemeaios of differece mehods sch as i he differeial rasform mehod (DTM), he Adomia decomposiio mehod (ADM) [3 33], he variaioal ieraio mehod (VIM) [34 4] empowered s o achieve approximae solios of varios ordiary differeial eqaios. STM [4 44] which isewlysbmieohelierareisasiableechiqefor solvig varios kids of ordiary differeial eqaios wih fracioal order (FODEs). I his sese, i is esimaed ha his ovel approach ha is sed o solve homogeeos ad ohomogeeos problems will be pariclarly valable as a ool for scieiss ad applied mahemaicias.. Fdameal Properies of Fracioal Calcls ad STM.. Fdameal Facs of he Fracioal Calcls. Firsly, we meio some of he fdameal properies of he fracioal calcls. Fracioal derivaives (ad iegrals as well) defiiios may differ, b he mos widely sed defiiios are hose of Abel-Riema (A-R). Followig he omeclare

2 Absrac ad Applied Aalysis i [45], a derivaive of fracioal order i he A-R sese is defied by D α [f ()] d { Γ [m α] = m f (τ) dτ, α m+ m <α m, ( τ) { d m { m f (), α=m, () where m Z + ad α R +.D α is a derivaive operaor here, ad D α [f ()] = Γ [α] ( τ) α f (τ) dτ, <α. () O he oher had, accordig o A-R, a iegral of fracioal order is defied by implemeig he iegraio operaor J α i he followig maer: J α [f ()] = Γ [α] ( τ) α f (τ) dτ, >, α>. (3) Whe i comes o some of he fdameal properies of fracioal iegraio ad fracioal differeiaio, hese have bee irodced o he lierare by Podlby [46]. Amog hese, we meio J α [ ]= D α [ ]= Γ [+] Γ [++α] +α, Γ [+] Γ [+ α] α. Aoher mai defiiio of he fracioal derivaive is ha of Capo [46, 47] who defied i by C D α [f ()] { Γ = [m α] f (m) (τ) dτ, m < α < m, α m+ ( τ) { d m { m f (), α = m. (5) A fdameal feare of he Capo fracioal derivaive is ha [7] J α [ C D α f ()] =f() k= (4) f (k) ( + ) k k!. (6).. Fdameal Facs of he Smd Trasform Mehod. The Smd rasform is defied i [, ] as follows. Over he se of fcios A={f() M, τ,τ >, (7) f () <Me /τ i, if ( ) j [, )}, he Smd rasform of f() is defied as F () =S[f()] = f () e, ( τ,τ ). (8) Theorem. If F()is he Smd rasform of f(),oekows ha he Smd rasform of he derivaives wih ieger order is give as follows [46 49]: df () S[ ]= [F () f()]. (9) Proof. Le s ake he Smd rasform [46 49] off () = df()/ as follows: S[ df () ]= df () e p = lim p = lim [ p e (/) f () = lim [ p e (/) f () p p df () e + p e (/) f () ] + ( p e (/) f () )] = lim [ p f () + ( p e (/) f () )] = f () + F (). () Eqaio () gives s he proof of Theorem. Whewe coie i he same maer, we ge he Smd rasform of he secod-order derivaive as follows [46 49]: S [ d f () ] = () [F () f() df ]. () = Ifwegoohesameway,wegeheSmrasform of he -order derivaive as follows: S[ d f () ]= [F () k= k d f () ]. () = Theorem. If F() is he Smd rasform of f(), oe ca ake io cosideraio he Smd rasform of he Riema- Lioville fracioal derivaive as follows [7]: S[D α f ()] = α [F () k= α k [D α k (f ())] = ], < α<. (3) Proof. LesakeheLaplacerasformoff () = df()/ as follows: L[D α f ()] =s α F (s) =s α F (s) k= k= s k [D α k (f ())] = s k [D α k (f ())] =. (4)

3 Absrac ad Applied Aalysis 3 Therefore, whe we sbsie / for s, we ge he Smd rasform of fracioal order of f() as follows: S[D α f ()] = α [F () k= α k [D α k (f ())] = ]. (5) Now, we will irodce he improveme form of STM for solvig FODEs. We ake io cosideraio a geeral liear ordiary differeial eqaio wih fracioal order as follows: α U () α = U () + U () +U() +c, (6) beig sbjec o he iiial codiio U () =f(). (7) The, we will obai he aalyical solios of some of he fracioal ordiary differeial eqaios by sig STM. Whe we akehe Smrasformof (6) der he erms of () ad(5), we obai he Smd rasform of (6) as follows: S[ α U () α ]=S[ U () ]+S[ α [F () F () k= U () ]+S[U()] +S[c], α k [D α k (U ())] = ] = () [F () f() f + [F () f()]+f() +c, k= α k [D α k (U ())] = = α [F () f() ] = U () ] = + α [F () f()] + α F () +c α, F () = α F () α U () + α k [D α k (U ())] = α U () k= + α F () α f () + α F () +c α, F () α F () α F () α F () = α f () + α k [D α k (U ())] = α U () k= α U () +c α, F () = α α α [g() α U () α U () +c α ], = = (8) where g() is defied by k= α k [D α k (U())] = α ( U()/) =. Whe we ake he iverse Smd rasform of (8) by sig he iverse rasform able i [, 7], we ge he solio of (6) by sig STM as follows: U () =S [ α α α [g() α U () α U () +c α ]]. (9) 3. Applicaios of STM o Nohomogeeos Fracioal Ordiary Differeial Eqaios I his secio, we have applied STM o he ohomogeeos fracioal ordiary differeial eqaios as follows. Example 3. Firsly, we cosider he ohomogeeos fracioal ordiary differeial eqaio as follows [5]: D α [U ()] = U() + Γ [3 α] α Γ [ α] α () +, >, <α, Wih he iiial codiio beig U () =. () I order o solve () by sig STM, whe we ake he Smd rasform of boh sides of (), we ge he Smd rasformof () as follows: S[D α U ()]+S[U()] =S[ Γ [3 α] α Γ [ α] α + ], S[D α U ()]+F() =S[ Γ [3 α] α ] S[ Γ [ α] α ]+S[ ] S[], F () α Dα [U ()] +F() = = Γ [3 α] S[ α ] Γ [ α] S[ α ] +S[ ] S[], F () α +F() = Γ [3 α] α Γ [3 α] Γ [ α] α Γ [ α] +, ( + α )F() = α α +, ( + α )F() = + +α +α, ( + α )F() =( ) + α ( ), F () = ( ), F () =. ()

4 4 Absrac ad Applied Aalysis Whe we ake he iverse Smd rasform of () by sig he iverse rasform able i [], we ge he aalyical solio of () by STM as follows: U () =. (3) Remark 4. If we ake he correspodig vales for some parameers io cosideraio, he he solio of () isi fll agreeme wih he solio of (3) meioed i [5]. To or kowledge, he aalyical solio of FODEs ha we fidihispaperhasbeeewlysbmieohelierare. Example 5. Secodly, we cosider he ohomogeeos fracioal ordiary differeial eqaio as follows [5]: D.5 U () +U() = + Γ [3] Γ [.5].5, >, (4) Wih he iiial codiio beig U () =. (5) I order o solve (4) by sig STM, whe we ake he Smd rasform of boh sides of (4), we ge he Smd rasformof (4) as follows: S[D.5 U ()]+S[U()] =S[ ]+ Γ [3] Γ [.5] S[.5 ], S[D.5 U ()]+S[U ()] =S[ ] +.545S [.5 ], F ().5 Dα [U ()] +F() = = +.5 F ().5 +F() = +.5, ( )F() = +.5 ( +.5 )F() = +.5, (6) F () = ( +.5 ) = =. Whe we ake he iverse Smd rasform of (6) by sig he iverse rasform able i [48], we ge he aalyical solio of (4) by sig STM as follows: U () =. (7) Remark 6. The solio (7) obaiedbysighesmd rasform mehod for (4) has bee checked by he Mahemaica Program 7. To or kowledge, he aalyical solio ha we fid i his paper has bee ewly sbmied o he lierare. 4. Coclsio ad Fre Work Prior o his sdy, varios approaches have bee performed o obai approximae solios of some fracioal differeial eqaios [5, 5]. I his paper, ohomogeeos Aalyical solio by STM Figre : The D srfaces of he obaied solio by meas of STM for (3)whe<< Aalyical solio by STM Figre : The D srfaces of he obaied solio by meas of STM for (7)whe<<3. fracioal ordiary differeial eqaios have bee solved by sig he Smd rasform afer givig he relaed formlae for he fracioal iegrals, he derivaives, ad he Smd rasform of FODEs. The Smd echiqe ca be sed o solve may ypes sch as iiial-vale problems ad bodary-vale problems i applied scieces, egieerig fields, aerospace scieces, ad mahemaical physics. The Smd rasform mehod has bee sed for The discree fracioal calcls i [43].ThisechiqehasbeeivesigaediermsofhedobleSmrasformi[44]. Coseqely, his ew approach has bee implemeed wih sccess o ieresig fracioal ordiary differeial eqaios. As sch ad pragmaically so, i eriches he library of iegral rasform approaches. Wiho a dob, ad based o or fidigs sch as Figres ad, hestm echiqe remais direc, robs ad valable ool for solvig same fracioal differeial eqaios. Refereces [] G. K. Wagala, Smd rasform: a ew iegral rasform o solve differeial eqaios ad corol egieerig

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7 Advaces i Operaios Research Advaces i Decisio Scieces Applied Mahemaics Algebra Probabiliy ad Saisics The Scieific World Joral Ieraioal Differeial Eqaios Sbmi yor mascrips a Ieraioal Advaces i Combiaorics Mahemaical Physics Complex Aalysis Ieraioal Mahemaics ad Mahemaical Scieces Mahemaical Problems i Egieerig Mahemaics Discree Mahemaics Discree Dyamics i Nare ad Sociey Fcio Spaces Absrac ad Applied Aalysis Ieraioal Sochasic Aalysis Opimizaio