Efficient Solution of Fractional Initial Value Problems Using Expanding Perturbation Approach

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1 Journal of mahemaics and compuer Science 8 (214) Efficien Soluion of Fracional Iniial Value Problems Using Expanding Perurbaion Approach Khosro Sayevand Deparmen of Mahemaics, Faculy of Science, Malayer Universiy, Malayer, Iran sayehvand@malayeru.ac.ir Aricle hisory: Received February 213 Acceped May 213 Available online July 213 Absrac In his aricle, expanding perurbaion approach is applied for solving he iniial value problems wih fracional coordinae derivaives. The fracional derivaive is described in he Capuo sense. The response expressions are wrien in erms of he Miag-Leffler funcions. Convergence of he approach is proved. Comparisons are made o confirm he reliabiliy and effeciveness of he presen ideas. Keywords: Expanding perurbaion approach, fracional iniial value problems, Capuo derivaive. 1. Inroducion During he las years, numerous papers deal wih analyical mehods for solving differenial equaions of fracional order [1-1 ] appeared in several inernaional journals, e.g. he (G /G)- mehod [11], he parameer-expanding mehod [12], Fan Sub-equaion Mehod [13],he Adomian s decomposiion mehod [14,15], he homoopy perurbaion echnique [16-19]. The homoopy analysis mehod [2] and he variional ieraion mehod [21]. In his paper, we recommend a robus analyical mehod based on expanding perurbaion approach [22, 23] o solve he fracional iniial value problems. We will show ha he suggesed sraegy inroduce a powerful improvemen for solving his ype of problem. 359

2 Khosro Sayevand / J. Mah. Compuer Sci. 8 (214) Fracional calculus Recenly, here has been a grea deal of ineres in fracional calculus (ha is, calculus of inegrals and derivaives of any arbirary real or complex order). Researchers have found many of he physical phenomena can be modeled accuraely using he fracional derivaives. For more informaion abou applicaions of his ineresing heory see [1, 2, 3]. Here under some preliminaries and noaions regarding fracional calculus presened. For more deails see [2]. There are several definiions of a fracional derivaive of order [1, 3]. The wo mos commonly used definiions are he Riemann-Liouville and Capuo. Each definiion uses Riemann-Liouville inegraion. Riemann-Liouville fracional inegraion of order is defined as 1 ( ) I ( ) d,,, I ( ) = ( ), 1 ( ) ( ) 1 (,,...,, x,,..., ) I (,,..., ) dx, n x 1 2 n 1 ( ) ( x ) (2) where ( ) is he well-nown Gamma funcion. The nex wo equaions define Riemann- Liouville and Capuo fracional derivaives of order respecively, m d m D ( ) I ( ), m 1 m, m N, m d ( ), 1,, m ( m) I m m m N D* () m d d m ( ), m. (3) (4) In his paper we have chosen o use he Capuo definiion, which is a modificaion of he Riemann - Liouville definiion, because when we inerpre he fracional derivaive in Eq. (1) as a Capuo fracional derivaive wih suiable condiions on he forcing funcion ( x1, x 2,..., x n, ), he Capuo fracional derivaive allows radiional iniial and boundary condiions o be included in he formulaion of he problem. The reader is advised o consuls he geomeric and physical inerpreaion for fracional derivaives and fracional inegrals of boh he Riemann- Liouville and Capuo ypes in [2]. 36

3 Khosro Sayevand / J. Mah. Compuer Sci. 8 (214) Fracional iniial value problems One very imporan class of differenial equaions of fracional order is he fracional iniial value problems wrien in he following form D ( ) f (, ( )), (),,1,2,..., n 1, (5) ( ) ( ) Where f is an arbirary funcion, D specified iniial condiions. denoes he fracional derivaive and ( ) are he Theorem The Eq. (5) is equivalen o he nonlinear Volerra inegral equaion of he second ind m 1 ( ) 1 f ( x, ( x )) ( ) dx. 1! ( ) ( x ) (6) Proof. See [22; Sayevand and Golbabai, pp. 5] 1.3. Expanding perurbaion approach: Some basic noaions Definiion The funcion f ( x, ) is an approximaion o ux (, ) uniformly [23, 24] valid o order ( ) if u( x, ) f ( x, ) lim uniformly for x R. (7) ( ) Definiion Le f ( x ) and g( x ) be wo funcions defined on he real numbers. We se f ( ) O( g ( )), if f ( ). (8) lim g ( ) Definiion We se f ( ) o( g ( )), if Now, consider he following funcion f ( ) lim. (9) g ( ) f ( ) an n( ),, (1) n Such han( ) o( n 1( )). We say ha, f has an asympoic perurbaion expansion. 361

4 Khosro Sayevand / J. Mah. Compuer Sci. 8 (214) Theorem Under he assumpion of he relaions (9), (1), we have a n f lim n 1 ( ) am m( ) m ( ). (11) n Proof. From Eq. (1) we obain f ( ) a ( ) ( ) a a..., (12) ( ) ( ) Whence, (12) implies ha f ( ) a ( ) (13) lim a1, 1( ) Coninuing, he process, he proof can be compleed. 2. The idea of he suggesed scheme In his secion, we shall illusrae he applicabiliy of our proposed scheme (EPM) o fracional iniial value problems. Consider he following iniial value problem D ( ) f (, ( )), (), B (, ),,1,2,..., n 1, ( ) ( ) (14) Where B is boundary operaor and he derivaives is described in he Capuo sense. Eq. (14) can be wrien in he following form f (, ( )) g (, ( )) L[ f (, ( ))] N [ f (, ( ))], (15) where, L is a linear par, N is nonlinear par and g is a nown analyical funcion. By he same manipulaion as given in Secion (1) and Theorem We obain: ( ( )), ( ( )) L ( ) S ( ( )) g ( ), 1 ( ( )) L ( ) S ( ( ),..., ( )), j 2,3,... j 1 j 2 j 2 j 2 (16) Where N [ f (, ( ))] S ( a ( )) a S, n n n n n n (17) 362

5 Khosro Sayevand / J. Mah. Compuer Sci. 8 (214) and S S ( ( ), ( ),..., ( )). (18) l l 1 l Consequenly, we obain he following relaions: ( ), ( ) ( ) ( ( )) ( ) (), (19)! n 1 n ( ) 1 LI S g I j 1 ( ) LI ( ) S ( ( ),..., ( )) I (), j 2,3,...! n 1 n ( ) j 1 j 2 j 2 j 2 j 1 3. Applicaion and numerical resuls To give a clear overview of he suggesed scheme, he following examples are presened. All he resuls are calculaed by using he symbolic calculus sofware Mahemaica. Example 3.1. Consider he following fracional Riccai equaion subjec o he iniial condiion The exac soluion of Eq. (18) for 1, is 2 ( ) 1 ( ), 1, (2) (). (21) 2 e 1 ( ). 2 e 1 (22) By he same manipulaion as given in pervious secion we obain 1 (2 1) 16 (2 ) (4 ) ( 1) (3 1) (3 1) (5 1) 3 5 ( )... 2 (23) Now, assume ha he leading erm has he form 1 ( ) ( )( ),,, R. (24) Where is a suiable funcion. Subsiuing (24) in o (23) cause ha we have some differen approximaion. For insance, he following expression can be obained easily when 1, ( ) O( ). (25) 363

6 Khosro Sayevand / J. Mah. Compuer Sci. 8 (214) Figure 3.1. Shows ha he soluion obained by our scheme (EPM) which are in excellen agreemens wih exac soluion. Figure 3.1. Miag-Leffler funcion E ( z) is defined by he following series represenaion, valid in he whole complex plane [25-28] E z ( z), (1 ) (26) Eq. (25) is a convergence series. I is easily verified ha our obained soluion in Eq. (22) are convergence and Miag-Leffler sable. Example 3.2. Consider he following iniial value problem Subjec o he iniial condiion D ( ) D ( ) ( ), (27) * * 2 ( ), (28) 2 Using our approach, we ge he () as approximae soluion, which is in excellen agreemen wih he analyical soluion obained by Jafari e al. [27]. 4. Concluding remars In his sudy, he approximae soluion of he fracional iniial value problems, are obained using expanding perurbaion approach. The fracional derivaive are described in he Capuo sense. The soluions are given in he Miag-Leffler sable series. The obained resuls show ha our soluion is in high agreemen wih he analyical soluion obained by Adomian decomposiion mehod and homoopy perurbaion echnique. Finally, our ou pus, reveal ha his proposed sraegy provides highly accurae soluions. 364

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