Analytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function

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1 Anlyic soluion of liner frcionl differenil equion wih Jumrie derivive in erm of Mig-Leffler funcion Um Ghosh (), Srijn Sengup (2), Susmi Srkr (2b), Shnnu Ds (3) (): Deprmen of Mhemics, Nbdwip Vidysgr College, Nbdwip, Ndi, Wes Bengl, Indi; Emil: (2):Deprmen of Applied Mhemics, Universiy of Clcu, Kolk, Indi Emil (2b): (3)Scienis H+, RCSDS, Recor Conrol Div. (complex) BARC Mumbi Indi Senior Reserch Professor, Dep. of Physics, Jdvpur Universiy Kolk Adjunc Professor. DIAT-Pune Ex- UGC Visiing Fellow. Dep of Appl. Mhemics; Univ. of Clcu Emil (3): Absrc There is no unified mehod o solve he frcionl differenil equion. The ype of derivive here used in his pper is of Jumrie formulion, for he severl differenil equions sudied. Here we develop n lgorihm o solve he liner frcionl differenil equion composed vi Jumrie frcionl derivive in erms of Mig-Leffler funcion; nd show is conjugion wih ordinry clculus. In hese frcionl differenil equions he one prmeer Mig-Leffler funcion plys he role similr s exponenil funcion used in ordinry differenil equions. Keywords Jumrrie frcionl derivive, Riemnn-Liouvelli frcionl derivive, Mig-Leffler funcion, Frcionl differenil equions. Inroducion The nlyicl soluions of he frcionl differenil equion re emerging brnch of pplied science lso in bsic science. Differen mehods re developing o solve he frcionl differenil equions. Since he definiion of frcionl derivive is modifying o rele i wih he clssicl derivive. Mhemicins re rying o develop he formuls of frcionl clculus bu geomery of frcionl derivive hs no concree shpe []. Depending on differen ype of derivives differen mehods of soluion re developing [2-7]. Riemnn-Liouville definiion he frcionl derivive of consn is non-zero which crees difficuly o rele beween he clssicl clculus. To overcome his difficuly Jumrie [2]-[5] modified he definiion of frcionl derivive of Riemnn-Liouvell ype nd wih his new formulion, we obin he derivive of consn s zero. Thus using his definiion linking beween he frcionl nd clssicl clculus becomes esier. There is no unique mehod o solve he liner frcionl differenil equions. Using he Jumrie modified definiion of frcionl derivive we obin he derivive of Mig- Leffler funcion s Mil-Leffler funcion, s in cse of clssicl whole-order derivive he derivive of exp( x ) is iself exponenil funcion. Thus vi use of Jumrie modified Riemnn- Liouvelli derivive, here exiss conjugion wih clssicl clculus, which eses in mny cses o solve frcionl differenil equion composed wih Jumrie frcionl derivive. Here we wn o develop n lgorihm o solving he liner frcionl differenil equion using he

2 Mig-Leffler funcion. We hve obined pplied his mehod o homogeneous frcionl differenil equions nd go corresponding fundmenl soluion. Orgnizion of he pper is s follows. In secion 2. some definiion of frcionl derivive is reproduced wih essenil exmples. In secion 3. nd 4. some properies of Mig-Leffler funcion is described. Finlly in secion 5. he mehods for solving he liner frcionl differenil equion composed by Jumrie frcionl derivive is developed using he Mig- Leffler funcion. 2. Some definiions of frcionl There re mny definiion of frcionl derivive. Grunwld-Lenikov frcionl derivive [6], Liouville frcionl derivive [8], Riemnn-Liouville frcionl derivive [], Cpuo frcionl derivive [8-], Kolwnker-Gngl locl frcionl derivive [2-6], Jumrie modified frcionl derivive [2]. Here we use he Riemnn-Liouville frcionl derivive nd is modified form by Jumrie [2]. 2. Riemnn-Liouville definiion of frcionl derivive Le he funcion f ( ) is one ime inegrble hen he inegro-differenil expression m+ d m D f ( ) = ( τ ) f ( τ ) dτ Γ( + m + ) d is known s he Riemnn-Liouville (R-L)definiion of frcionl derivive [6] wih m s ineger wih condiion m < m +. In Riemnn-Liouville definiion he funcion, f ( ) is inegred ( m ) fold nd hen differenie m + imes. We cn re-wrie he bove s follows The lef R-L frcionl derivive is defined by And he righ R-L derivive is k d k ( ) = ( ) ( ) D f τ f τ dτ Γ( k ) d k b d k ( ) = ( ) ( ) Db f τ f τ dτ Γ( k ) d Where in bove k is ineger such h ( k ) < k h is jus greer hn frcionl number. Using he lef R-L derivive we ge he frcionl derivive of he funcion f ( ) = K s nonzero, s demonsred below. 2

3 Similrly he righ R-L derivive of f ( ) d D f ( ) = ( ξ ) Kdξ Γ( ) d K d ( ξ ) = Γ( ) d ( ) = K Γ( ) = K is ( b ) Db f ( ) = K. Γ( ) This shows h he frcionl derivive of consn (K) is non-zero bu in clssicl clculus derivive of consn is zero which is conrdicion beween he clssicl derivive nd he frcionl derivive of consn. To overcome his difference Jumrie [2] modified he lef R-L frcionl derivive. We ge he R-L lef derivive of power funcion s 2. 2 Jumrie modified definiion of he frcionl derivive is ( ξ ) f ( ξ ) dξ, <. Γ ( ) def d D f ( ) = ( ξ ) [ f ( ξ ) f ()] dξ, < <. Γ ( ) d ( n ) ( n ) [ f ( )], n < n +, n. Using his definiion we ge D { K} =, <. The bove formul in line-, becomes frcionl order inegrion if we replce by which is ( ) = ( ) ( ) D f τ f τ dτ Γ( ) (2) Using he bove formul we ge for f ( ) = ( ) γ, he frcionl inegrl for order γ γ ( ) = ( ) ( ) Γ( ) D τ τ dτ 3

4 Using he subsiuion τ = + ξ( ) we hve for; τ =, ξ = nd for τ =, ξ = ; dτ = ( ) dξ, ( τ ) = ξ ( ) = ( )( ξ ) ; ( τ ) = ξ ( ), we ge he following γ γ ( ) = ( ) ( ) Γ( ) D τ τ dτ γ γ ( ) ( ) ( ) ( ) = ξ ξ dξ Γ( ) γ + ( ) = ( ) Γ( ) γ ξ ξ ξ γ + ( ) = B(, γ + ) Γ( ) Γ ( γ + ) γ + = ( ), ( <, γ > ) Γ ( γ + + ) d We used Be-funcion Γ( ) Γ ( + ) B(, γ + ) = ξ ( ξ ) dξ = defined s Γ ( + γ + ) γ γ def p q Γ( p) Γ( q) B( p, q) = u ( u) du = Γ ( p + q) Applying he bove obined resul he frcionl inegrl of order ( υ), wih υ < is Γ ( γ + ) D ( ) = ( ) Γ ( γ + 2 υ) Tking one whole derivive of he bove we ge ( υ ) γ γ + υ D D ( ) = D ( ) ( υ ) γ υ γ d Γ ( γ + ) γ + υ = ( ) d ( γ 2 υ) Γ + Γ ( γ + ) = ( γ + υ)( ) Γ( γ υ + 2) Γ ( γ + ) = ( γ + υ)( ) ( γ υ + ) Γ ( γ + υ) Γ ( γ + ) = ( ) Γ ( γ + υ) We hve herefore clculed frcionl derivive by R-L lef formul forυ such h υ < hus our neres ineger is one h is k = nd we wrie h below 4 γ υ γ υ γ υ

5 υ d υ D f ( ) = ( τ ) f ( τ ) dτ Γ( υ) d D D f ( ) ( υ ) = Thus for =, he frcionl RL derivive of f ( ) = γ is D υ γ Γ ( γ + ) = Γ ( γ + υ) For consn funcion f ( ) =, puing in bove expressionγ =, we ge D υ [ ] = Γ ( υ) Le us now see wh Jumrrie derivive is from bove R-L derivive obined for f ( ) composiion of he Jumrie derivive, wih sr poin of inegrion s γ υ υ ( ) γ d γ γ f [ ] = ( ) d Γ( ) d ξ ξ ξ D D D = D D ( ) γ ( ) γ γ γ = γ. The = nd f ( ) = γ is The bove expression show h his Jumrrie derivive is composed of wo RL derivives hose re D γ minus RL derivive of consn D γ. Going by similr seps s done for D γ, we ge firs he frcionl inegrl in erms of incomplee Gmm funcion s γ + ( ) / γ + ( ) γ γ = ( ) d = B η (, γ + ) η = Γ( ) Γ( ) D z z z The frcionl derivive of γ following is by king one whole derivive of bove expression we ge he D γ The frcionl derivive of γ is Therefore = B η (, γ + ) η = d Γ( ) γ + d γ D = ( ) Γ( ) γ 5

6 ( ) γ d γ γ f [ ] = ( ) d Γ( ) d ξ ξ ξ = D D D = D D ( ) γ ( ) γ γ γ γ + γ d = B η (, γ + ) ( ) d Γ( ) Γ( ) For =, we hve ( ) γ d γ f [ ] = ( ) d Γ( ) d Γ ( γ + ) = D D = D = ( ) Γ ( γ + ) ( ) γ γ γ 6 ξ ξ ξ We will be using Jumrie derivive for power funcion γ wih sr poin of differeniion s =, in subsequen secions. When sr poin of differeniion is non-zero we will be shifing he origin o h non-zero poin nd use he bove formul. 3. Some properies Mig-Leffler funcion nd is pplicion In 93 Mig-Leffler [7]-[9] inroduce funcion defined by n infinie series def E ( ) = Γ ( + ) Γ ( + 2 ) Γ ( + 3 ) is he one prmeer Mig-Leffler funcion. Using Jumrie derivive of order, wih < wih sr poin s = for f ( ) n =, h ( n ) is ( ) Γ ( n + ) / Γ [ ( n ) + ], for n =,2,3,... ; nd lso using Jumrie derivive of consn s zero, we ge he following very useful ideniy. In ll he subsequen secions we will sy D is he Jumrie derivive wih zero s sr poin Thus D ( E ( )) = D Γ ( + ) Γ ( + 2 ) Γ ( + 3 ) Γ ( + ) Γ ( + 2 ) Γ ( + 3 ) = Γ() Γ ( + ) Γ ( + 2 ) Γ ( + ) Γ ( + 3 ) Γ ( + 2 ) = Γ ( + ) Γ ( + 2 ) Γ ( + 3 ) = E ( )

7 D ( E ( )) = E ( ) () This shows h AE ( ) is soluion is soluion of he frcionl differenil equion Where A is rbirry consn. Therefore wih y () = hs soluion D y = y (2) D y = y y = E ( ). Using his propery of he Mig-Leffler one cn esily prove he following heorem. Theorem : The Mig-Leffler funcion E ( ) sisfies he relion Proof: Le y = E ( ) E ( b ) hen E ( ) E ( b ) = E (( + b) ) y = E ( ) E ( b ) D y = E ( b ) D ( ) + E ( ) D ( b ) D y = E ( ) E ( b ) + be ( ) E ( b ) D y = ( + b) y = ( + b) E ( ) E ( b ) (3) Using he soluion of he equion (2) we ge he soluion of he equion (3) in he following form ([ ] ) y = AE + b From he definiion of y we ge y () =. Therefore we hve y = E (( + b) ). Thus we ge E (( + b) ) = E ( ) E ( b ) (4) We ge useful propery of one prmeer Mig-Leffler funcion. Using he bove propery of Mig-Leffler funcion we ge 7

8 E ( ) E ( ) = or E ( ) =. (5) E ( ) E ( ) E ( ) = E (2 ) (6) 4. Complex Mig-Leffler funcion nd is properies Jumrie [28] defined he complex Mig-Leffler in he following form def = + E ( i ) cos ( ) isin ( ) 2k E ( i ) + E ( i ) k cos ( ) = = ( ) 2 (2 k)! k = (2k + ) E ( i ) E ( i ) k sin ( ) = = ( ) 2 (2 k + )! On he oher hnd Jumrie [28] defined period ( M ) of he funcion E ( i ) in he following form, king ( ) E ( i M ) = nd herefore k = cos ( + M ) = cos ( ) sin ( + M ) = sin ( ) cos (( ) ) = cos ( ) sin (( ) ) = ( ) sin ( ). The series presenion of cos ( ) is cos ( ) = Γ ( + 2 ) Γ ( + 4 ) Γ ( + 6 ) Tking erm by erm Jumrie derivive we ge Γ ( + 2 ) Γ ( + 4 ) Γ ( + 6 ) D [cos ( )] = Γ ( + 2 ) Γ ( + ) Γ ( + 4 ) Γ ( + 3 ) Γ ( + 6 ) Γ ( + 5 ) 3 = +... = sin ( ) Γ ( + ) Γ ( + 3 ) 5. Soluion of liner second order frcionl differenil equion Le us consider he funcion y = AE ( ) + BE ( b ) wih A nd B is consns. Differeniing imes wih respec o, for < <, wih Jumrie derivive we ge 8

9 D y = AE ( ) + BbE ( b ) D y y = AE ( ) + BbE ( b ) y ( ) = AE ( ) + BbE ( b ) AE ( ) + BE ( b ) = B( b ) E ( b ) D y y = B( b ) E ( b ) Differeniing bove by Jumrrie derivive nd re-rrnging, we ge 2 D y D y = Bb b E b 2 D y b D y by ( ) ( ) ( + ) + = This shows h he frcionl differenil equion hs soluion in he form ( + ) + = 2 D y b D y by y = AE ( ) + BE ( b ). On he oher hnd consider he differenil equion i cn be express in he following form 2 D y b D y by ( + ) + = Le, ( D b) y( ) = x( ) hen equion (7) reduce o he form ( D )( D b) y( ) =. (7) ( D ) x( ) = or D x( ) = x( ) Soluion of he bove equion is sme s he soluion of he equion (2) which is x( ) = A E ( ) ( D b) y( ) = A E ( ) D y by = A E ( ) E ( b )( D y by) = A E ( ) E ( b ) A D ye b D E E b b On inegring boh side we ge h is pplying D ( ( )) = ( ( ) ( )) on boh sides of bove, we ge 9

10 A ye ( b ) = ( E ( ) E ( b )) + B b A y = AE + BE b A = b ( ) ( ) where. Therefore y AE ( = ) + BE ( b ) is soluion of he differenil equion. Thus we cn se he following heorem Theorem 2: The frcionl differenil equion ( D )( D b) y( ) = hs soluion of he form y = AE ( ) + BE ( b ) where A nd B re consns. Proof of he heorem is follows from he previous rgumens. Similrly one cn generlized he soluion of he differenil equion in he following form If ( D )( D )( D )...( D ) y( ) = 2 3 wih ll i ' s re disinc be frcionl differenil equion wih < hen soluion of he differenil equion will be n = i= y A E ( ) where A i re rbirry consns nd E ( i ) is one prmeer Mig-Leffler funcion. Le us consider he funcion Where A nd B re consns. Then i y = ( A + B) E ( ) i n

11 D y = Γ ( + ) AE ( ) + ( A + B) E ( ) D y y = Γ ( + ) AE ( ) + ( A + B) E ( ) {( A + B) E ( )} 2 D y D y = Γ + AE 2 2 D y D y y = Γ ( + ) AE ( ) ( ) ( ) = ( D y y) 2 + = Thus soluion of he differenil equion is A nd B re consns. 2 2 D y D y y 2 + = y = ( A + B) E ( ) On he oher hnd consider he differenil equion ( D ) y = or ( D 2 D + ) y = 2 2 D y D y y 2 + = (8) Le ( D ) y = v hen equion (8) reduce o he form Soluion of his differenil equion is v( ) = A E ( ) A nd B re consns. ( D ) y = A E ( ) Thus he following heorem cn be sed Theorem 3: The frcionl differenil equion ( D ) v = (9) E ( )( D y y) = A E ( ) E ( ) A D [ ye ( )] = A = D Γ ( + ) A ye ( ) = + B Γ ( + ) A y = ( A + B) E ( ) where A = Γ ( + )

12 2 2 D y D y y 2 + = hs soluion of he form y = ( A + B) E ( ) where A nd B re consns. The proof of he heorem is lredy explined in he previous rgumens. Theorem 4: Soluion of he frcionl differenil equion is of he form 2 2 D y D y b y 2 + ( + ) = y = E ( )[ Acos ( b ) + B sin ( b )]. Proof: The given differenil equion cn be wrien in he following form 2 2 (( D ) b ) y or ( D ib)( D ib) y + = + = () Using heorem 3 we ge he soluion of he frcionl differenil () cn be wrien in he following form y = A E (( + ib) ) + B E (( ib) ) y = A E ( ) E ( ib ) + B E ( ) E ( ib ) = E ( )[ A {cos ( b ) + isin ( b )} + B {cos ( b ) i sin ( b )}] = E ( )[ Acos ( b ) + ibsin ( b )] Where A = A + B nd B = A B. Thus we ge useful resuls. 6. Conclusions There re severl mehods o solve frcionl differenil equions, nd he soluion depends on he ype of frcionl derivive used. Here we develop n nlyicl mehod o find he soluions of liner frcionl differenil equion, composed by Jumrie frcionl derivive in erms of one prmeer Mig-Leffler funcion. Some well known properies of Mig-Leffler hve been used o find soluion of he frcionl differenil equions. The soluions obined re similr s he soluions obined usul clculus obined in erms he exponenil funcion. This conjugion wih ordinry clculus when Jumrie ype frcionl derivive is used o compose he frcionl differenil equions is useful in severl physicl problems. 2

13 7. References [] de Oliveir, E. C., J. A. T. Mchdo. A review of definiions of frcionl derivives nd Inegrl. Mhemicl Problems in Engineering. Hindwi Publishing Corporion [2] Jumrie, G. Modified Riemnn-Liouville derivive nd frcionl Tylor series of nondifferenible funcions Furher resuls, Compuers nd Mhemics wih Applicions, 26. (5), [3] G. Jumrie, On he soluion of he sochsic differenil equion of exponenil growh driven by frcionl Brownin moion, AppliedMhemics Leers, vol. 8, no. 7, pp , 25. [4] G. Jumrie, An pproch o differenil geomery of frcionl order vi modified Riemnn- Liouville derivive, Ac Mhemic Sinic, vol. 28, no. 9, pp , 22. [5] G. Jumrie, On he derivive chin-rules in frcionl clculus vi frcionl difference nd heir pplicion o sysems modelling, Cenrl Europen Journl of Physics, vol., no. 6, pp , 23. [6] X. J. Yng, Locl Frcionl Funcionl Anlysis nd Is Applicions, Asin Acdemic Publisher Limied, Hong Kong, 2. [7] X. J. Yng, Advnced Locl Frcionl Clculus nd Is Applicions, World Science, New York, NY, USA, 22. [8] Miller KS, Ross B. An Inroducion o he Frcionl Clculus nd Frcionl Differenil Equions.John Wiley & Sons, New York, NY, USA; 993. [9 ] Ds. S. Funcionl Frcionl Clculus 2 nd Ediion, Springer-Verlg 2. [] Podlubny I. Frcionl Differenil Equions, Mhemics in Science nd Engineering, Acdemic Press, Sn Diego, Clif, USA. 999;98. [ ] M. Cpuo, Liner models of dissipion whose q is lmos frequency independen-ii, Geophysicl Journl of he Royl Asronomicl Sociey,, 967. vol. 3, no. 5, pp [2] K M Kolwnkr nd A D. Gngl. Locl frcionl Fokker plnk equion, Phys Rev Le [3 ] Abhy Prve, A. D. Gngl. Clculus of frcls subse of rel line: formulion-; World Scienific, Frcls Vol. 7, 29. [4] Abhy Prve, Seem sin nd A.D.Gngl. Clculus on frcl curve in R n rxiv:96 oo76v ; lso in Prmn-J-Phys. [5] Abhy prve, A. D. Gngl. Frcl differenil equion nd frcl ime dynmic sysems, Prmn-J-Phys, Vol 64, No. 3, 25 pp

14 [6]. E. Sin, Abhy Prve, A. D. Gngl. Fokker-Plnk Equion on Frcl Curves, Seem, Chos Solions & Frcls-52 23, pp [7]. Erdelyi A. Asympoic expnsions, Dover (954). [8]. Erdelyi.A. (Ed). Tbles of Inegrl Trnsforms. vol., McGrw-Hill, 954. [9]. Erdelyi.A. On some funcionl rnsformion Univ Poiec Torino 95. Acknowledgemen Acknowledgmens re o Bord of Reserch in Nucler Science (BRNS), Deprmen of Aomic Energy Governmen of Indi for finncil ssisnce received hrough BRNS reserch projec no. 37(3)/4/46/24-BRNS wih BSC BRNS, ile Chrcerizion of unrechble (Holderin) funcions vi Locl Frcionl Derivive nd Deviion Funcion. 4