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1 Compers and Mahemaics wih Applicaions 59 (00) Conens liss available a ScienceDirec Compers and Mahemaics wih Applicaions jornal homepage: Solving fracional bondary vale problems wih Dirichle bondary condiions sing a new ieraive mehod Varsha Dafardar-Gejji, Sachin Bhalekar Deparmen of Mahemaics, Universiy of Pne, Ganeshkhind, Pne , India a r i c l e i n f o a b s r a c Keywords: Bondary vale problems Capo derivaive Anomalos diffsion Diffsion-wave eqaion Dirichle bondary condiions Linear/nonlinear fracional diffsion-wave eqaions on finie domains wih Dirichle bondary condiions have been solved sing a new ieraive mehod proposed by Dafardar-Gejji and Jafari V. Dafardar-Gejji, H. Jafari, An ieraive mehod for solving nonlinear fncional eqaions, J. Mah. Anal. Appl. 36 (006) ]. 009 Elsevier Ld. All righs reserved.. Inrodcion Time fracional diffsion-wave eqaion has been of considerable ineres in he lierare 4]. This eqaion has nmeros applicaions in varios branches of Science and Engineering. Nigmallin 5] sed fracional diffsion-wave eqaion o model elecromagneic acosic and mechanical responses. Giona e al. 6] sdied he relaaion phenomena in comple viscoelasic maerial sing fracional diffsion eqaions. Mainardi 7] has shown ha he fracional wave eqaion governs he propagaion of mechanical diffsive waves in viscoelasic media. Mezler and Klafer 8] have demonsraed ha fracional diffsion eqaion describes a non-markovian diffsion process wih a memory. Varios mehods have been sed o solve linear eqaions. On he conrary for solving nonlinear fracional differenial eqaions, one mainly has o depend pon nmerical or ieraive mehods. Ieraive mehod sch as Adomian decomposiion 9] has proven sccessfl in dealing wih boh linear as well as nonlinear problems as his mehod is free from ronding off errors. I is compaionally less demanding in erms of memory and power. Fracional bondary vale problems (FBVPs) have been solved by varios mehods. Agrawal 0] has sed finie sine ransform echniqe for solving fracional diffsion eqaion defined on a bonded domain, Dafardar-Gejji and Jafari ] have employed separaion of variables mehods. Separaion of variables mehod has been eended o solve mli-erm diffsion-wave eqaions by Dafardar-Gejji and Bhalekar ]. Fracional diffsion-wave eqaions wih iniial condiions have been solved by Dafardar-Gejji and Bhalekar 3]. Recenly El-Sayed and Gaber 4] have solved FBVPs involving Dirichle bondary condiions by Adomian decomposiion mehod (ADM). Odiba and Momani 5] sed ADM o solve ime fracional wave eqaion. In he presen paper we ilize a new ieraive mehod (NIM) 6] for solving FBVPs over finie domain. NIM proposed by Dafardar-Gejji and Jafari 6] is simple in principles and comper friendly. In he case of nonlinear problems, in ADM one has o compe Adomian polynomials involving edios calclaions, whereas a cople of comper commands are sfficien in he case of NIM. Resls obained by NIM are in high agreemen 7] wih resls obained by ADM. In his aricle we discss he bondary vale problem (BVP) D α = + A(), > 0, 0 < < l, (.) Corresponding ahor. addresses: vsgejji@mah.nipne.erne.in (V. Dafardar-Gejji), sachin.mah@yahoo.co.in (S. Bhalekar) /$ see fron maer 009 Elsevier Ld. All righs reserved. doi:6/j.camwa

2 80 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) where m < α < m, m =, and A() a given nonlinear fncion of ogeher wih he following condiions k (, 0) = p k(), k = 0,..., (m ), (.) k (0, ) = f 0 (), (l, ) = f l (). (.3) In he presen paper we solve he FBVP (.) (.3) sing a new ieraive mehod (NIM) proposed by Dafardar-Gejji and Jafari 6]. Frher illsraive eamples are solved wih NIM and resls are compared wih hose by ADM.. Preliminaries and noaions In his secion we se p noaions and recall some basic definiions. Definiion.. Riemann Lioville fracional inegraion of order α is defined as I α f () = Γ (α) 0 ( y) α f (y)dy, > 0. (.) Definiion.. Capo fracional derivaive of order α is defined as ( d m f () D α f () = Im α d m Noe ha for 0 m < α m, a 0 and γ >, I α () γ = Γ (γ + ) Γ (γ + α + ) ()γ +α, ( I α ) m Dα f () = f () f (k) (0) k k!. ), 0 m < α m. (.) (.3) (.4) 3. New ieraive mehod Dafardar-Gejji and Jafari 6] have considered he following nonlinear fncional eqaion = f + L() + N(), (3.) where f is a given fncion, L and N are given linear and nonlinear fncions of respecively, = (,,..., n ). (3.) is assmed o have a solion of he form = i. Becase L is linear ( ) L i = L( i ). Frher define G 0 = N( 0 ), ( ) m G m = N i N ( m ) i. (3.) (3.3) The recrrence relaion is defined as 0 = f = L( 0 ) + G 0 (3.5) m+ = L( m ) + G m, m =,,.... (3.6) (3.4)

3 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) Hence, ( ) ( ) m+ m m i = L i + N i, i= (3.7) and ( ) ( ) i = f + L i + N i. (3.8) The k-erm approimae solion of (3.) is given by = k. Convergence of NIM We presen below he condiion for convergence of he series i. For deails we refer he reader o Bhalekar, Dafardar- Gejji, Convergence of he New Ieraive Mehod, (sbmied)]. Theorem 3.. If N is C ( ) in a neighborhood of 0 and N (n) ( 0 ) L, for any n and for some real L > 0 and i M < e, i =,,..., hen he series n=0 G n is absolely convergen and moreover, G n LM n e n (e ), n =,,.... Theorem 3.. If N is C ( ) and N (n) ( 0 ) M e, n, hen he series n=0 G n is absolely convergen. Comparison wih ADM In ADM is assmed o be of he form i. In his mehod 0 = f, = L( 0 ) + A 0, m+ = L( m ) + A m, m =,,... (3.9) (3.0) (3.) where, d m A m = m! dλ m ( )] m N k λ k λ=0 are Adomian polynomials. Taylor series epansion of N() arond 0 is N() = N( 0 ) + N ( 0 ) ] + N ( 0 )! + N(3) ( 0 ) 3!, m = 0,,,... (3.) ] ] 3 +. (3.3) In ADM he erms on he righ-hand side of (3.3) are groped as ( ) ( ) N( 0 ) + }{{} N ( 0 ) + }{{}! N ( 0 ) + N 3 ( 0 ) + 3! N(3) ( 0 ) + N ( 0 ) + 3 N ( 0 ) +, (3.4) A 0 A } {{ } A whereas in NIM, N() is decomposed as N( 0 ) + N( }{{} 0 + ) N( 0 ) + N( }{{} ) N( 0 + ) }{{} G 0 G G } {{ } A 3 + N( ) N( ) }{{} +. (3.5) G 3 Ths in ADM and NIM, he erms of Taylor series of N() arond 0 are groped differenly and he recrrence relaions are se accordingly.

4 804 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) Fracional BVP The righ inverse operaor T of / is defined as 8,9]: T = 0 0 d d l l 0 0 d d. (4.) Applying I α on boh sides of (.), sing (.) and he iniial condiions (.) we ge = m k p k k! + I α + I α A(). Applying T on boh sides of (.) and sing he bondary condiions (.3) we ge (4.) = f 0 + l (f l f 0 ) + T ( D α ) T (A()). (4.3) (4.) and (4.3) ogeher yield { ] = m k p k + f 0 + k! l (f l f 0 ) Seing 0 = m k p k k! + f 0 + l (f l f 0 ) ], + I α + T ( D α )] + I α A() T (A())]}. (4.4) (4.5) we ge L() = N() = I α + T ( D α )], I α A() T (A())] = 0 + L() + N(). Now Eq. (4.8) can be solved by NIM. (4.6) (4.7) (4.8) 5. Nmerical eamples In his secion we solve some linear and nonlinear fracional BVPs eplicily by NIM. Eample. Consider he linear fracional diffsion eqaion along wih iniial and bondary condiions D α =, 0 < < l, > 0, 0 < α, (5.) (, 0) = + +, (0, ) = α Γ (α + ) +, (l, ) = l + l + + In view of (4.5) 0 = α Γ (α + ). (5.) (5.3) (5.4) α ] Γ (α + ) + l, (5.5) and sing (3.5), (3.6) and (4.6) n = α ] + n+ Γ (α + ) l, n =,,.... (5.6) Hence, he solion of (5.) (5.3) rns o o be (, ) = i = α (5.7) Γ (α + )

5 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) Fig.. (Eample ): For-erm approimae solion of (5.8). ADM NIM Fig.. (Eample ): Comparison of ADM and NIM for = 0.. Eample. Consider he nonlinear fracional diffsion eqaion, D = +, 0 < <, > 0, wih (, 0) =, (0, ) =, (, ) = 0. In view of (4.5) (5.8) 0 = + ( )], = ( ) ( )( ) ( ) 96 ( ) + 48 ( ). For-erm approimae solion of (5.8) is ploed in Fig.. In Fig. ADM solion is compared wih NIM solion for = 0.. Eample 3. Consider he linear fracional wave eqaion, D α =, 0 < <, > 0, < α, (5.9) (, 0) =, (, 0) =, (5.0) α α (0, ) = Γ (α + ), (, ) = + + Γ (α + ). (5.) Following NIM, 0 = For n =,, n = n ( ) α ] Γ (α + ) +, α + Γ (α + ) ]. (5.) (5.3)

6 806 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) Fig. 3. (Eample 4): Three-erm approimae solion of (5.5) for he case α = ADM 0.9 NIM Fig. 4. (Eample 4): Comparison of ADM and NIM solions for = 0., α =.3. Hence, he solion of (5.9) (5.) rns o o be (, ) = i = α + + Γ (α + ). (5.4) Eample 4. Consider he nonlinear wave eqaion of fracional order D α = +, 0 < <, > 0, < α, (5.5) along wih he iniial and bondary condiions (, 0) =, (, 0) =, (0, ) =, (, ) =. Using algorihm (3.4) (3.6) of NIM, we ge (5.6) (5.7) 0 = ( + ) + ], = ) ( α+ ( + ). 480 Γ (α + 3) Fig. 3 shows hree-erm approimae solion of (5.5) for α =.8. ADM solion is compared wih NIM solion for =, α =.3 in Fig. 4. Eample 5. Consider he following nonlinear wave eqaion D.5 = +, 0 < <, > 0, (5.8)

7 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) Fig. 5. (Eample 5): Three-erm approimae solion of (5.8) ADM NIM Fig. 6. (Eample 5): Comparison of ADM and NIM solions for =. (, 0) =, (, 0) = 0, (0, ) =, (, ) =. (5.9) (5.0) Following algorihm (3.4) (3.6) of NIM 0 = + ], ( = ( ) ) (667 + ( )). In Fig. 5, hree-erm approimae solion is ploed. In Fig. 6 ADM solion is compared wih NIM solion for =. Eample 6. Consider he following nonlinear hea eqaion of fracional order 0 < α D α = +, 0 < <, > 0, (, 0) =, (0, ) =, (, ) = + +. (5.) (5.) (5.3) Eac solion for α = is given by (, ) =. In Fig. 7 we compare he wo-erm solion by NIM (red), ADM (green) + a α = and eac solion (ble) for he case = 0.3. Fig. 8 shows NIM solion for α = whereas Figs. 9 and 0 compare NIM solion and ADM solion respecively for he case α =.

8 808 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) ADM Eac, NIM.0 Fig. 7. (Eample 6): Comparison Fig. 8. (Eample 6): NIM α = Fig. 9. (Eample 6): NIM α =. Acknowledgemens V. Dafardar-Gejji acknowledges he Universiy Grans Commission, N. Delhi, India for he projec (F. No. 3-8/005(SR)). She frher acknowledges he Deparmen of Science and Technology, India for financial sppor o paricipae in he Third IFAC Workshop in Ankara, Trkey. S. Bhalekar acknowledges he Concil of Scienific and Indsrial Research, Delhi, India for a Jnior Research Fellowship. Mahemaica 6.0 has been sed for he compaions in his paper.

9 V. Dafardar-Gejji, S. Bhalekar / Compers and Mahemaics wih Applicaions 59 (00) Fig. 0. (Eample 6): ADM α =. References ] W.R. Schneider, W. Wyss, Fracional diffsion and wave eqaions, J. Mah. Phys. 30 () (989) ] F. Mainardi, Fracional relaaion oscillaion and fracional diffsion-wave phenomena, Chaos Solions Fracals 7 (996) ] Y. Fjia, Cachy problems of fracional order and sable processes, Japan J. Inds. Appl. Mah. 7 (990) ] F. Mainardi, P. Paradisi, Fracional diffsive waves, J. Comp. Acos. 9 (4) (00) ] R. Nigmallin, Realizaion of he generalized ransfer eqaion in a medim wih fracional geomery, Physica Sas (B) Basic Res. 33 () (986) ] M. Giona, S. Cerbelli, H.E. Roman, Fracional diffsion eqaion and relaaion in comple viscoelasic maerials, Physica A 9 (99) ] F. Mainardi, Fracional diffsive waves in viscoelasic solids, in: J.I. Wegner, F.R. Norwood. (Eds.) IUTAM Symposim Nonlinear Waves in Solids, Fairfield, 995, pp ] R. Mezler, J. Klafer, Bondary vale problems for fracional diffsion eqaions, Physica A 78 (000) ] G. Adomian, Solving Fronier Problems of Physics: The Decomposiion Mehod, Klwer, ] O.P. Agrawal, Solion for fracional diffsion-wave eqaion defined in a bonded domain, Nonlinear Dynam. 9 (00) ] V. Dafardar-Gejji, H. Jafari, Bondary vale problems for fracional diffsion-wave eqaion, As. J. Mah. Anal. Appl. 3 (006) 8. ] V. Dafardar-Gejji, S. Bhalekar, Bondary vale problems for mli-erm fracional differenial eqaions, J. Mah. Anal. Appl. 345 (008) ] V. Dafardar-Gejji, S. Bhalekar, Solving fracional diffsion-wave eqaions sing he new ieraive mehod, Frac. Calc. Appl. Anal. () (008) ] A.M.A. El-Sayed, M. Gaber, The Adomian decomposiion mehod for solving parial differenial eqaions of fracal order in finie domains, Phys. Le. A 359 (006) ] Z. Odiba, S. Momani, Approimae solions for bondary vale problems of ime-fracional wave eqaion, Appl. Mah. Comp. 8 (006) ] V. Dafardar-Gejji, H. Jafari, An ieraive mehod for solving non linear fncional eqaions, J. Mah. Anal. Appl. 36 (006) ] S. Bhalekar, V. Dafardar-Gejji, New ieraive mehod: Applicaion o parial differenial eqaions, Appl. Mah. Comp. 03 (008) ] D. Lesnic, A compaional algebraic invesigaion of he decomposiion mehod for ime-dependen problems, Appl. Mah. Comp. 9 (00) ] D. Lesnic, The Cachy problem for he wave eqaion sing he decomposiion mehod, Appl. Mah. Le. 5 (00)