AN EFFICIENT METHOD FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS USING BERNSTEIN POLYNOMIALS

Transcription

1 Journal of Fracional Calculus and Applicaions, Vol. 5(1) Jan. 2014, pp ISSN: hp://fcag-egyp.com/journals/jfca/ AN EFFICIENT METHOD FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS USING BERNSTEIN POLYNOMIALS RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM Absrac. In his paper we propose an efficien numerical echnique for solving fracional iniial value problems. I is based on he Bernsein polynomials. We derive an explici form for he Bernsein operaional marix of fracional order inegraion. Numerical resuls are presened. In order o show he efficiency of he presened mehod, we compare our resuls wih some operaional marix echniques. 1. Inroducion Fracional calculus is a branch of mahemaics ha deals wih a generalizaion of he well-known operaions of differeniaions and inegraions o arbirary fracional orders. Fracional derivaive provides an excellen insrumen for he descripion of memory and herediary properies of various maerials and processes. This is he main advanage of fracional derivaives in comparison wih he classical ineger-order models in which such effecs are in fac negleced. Fracional calculus found many applicaions in various fields of physical sciences such as elecrochemical process [1-2], dielecric polarizaion [3], earhquakes [4], fluid-dynamic raffic model [5], solid mechanics [6], bioengineering[7-9] and economics[10]. Fracional derivaives and inegrals also appear in heory of conrol of dynamical sysems, when he conrolled sysem and he conroller are described by a fracional differenial equaion. In recen years, a number of mehods have been proposed and applied successfully o approximae various ypes of fracional differenial equaions such as Adomian decomposiion mehod [11-13] and [45], Variaional ieraion mehod [14-15] and [40-42], Homoopy perurbaion mehod [16-17] and [43], Homoopy analysis mehod [18], fracional differenial ransform mehod [19-23], power series mehod [24], and oher mehods [25-29], [38-39], and [46-48] Mahemaics Subjec Classificaion. 35R11, 26A33, 35B50, 34L15. Key words and phrases. Bernsein polynomial, Operaional marix of inegraion, Nonlinear fracional differenial equaion. Submied July 17, 2013 Revised December 13,

2 130RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) Recenly, waveles operaional marix are used o find he soluion of he fracional differenial equaions. Li e al. [30] derived he Haar waveles operaional marix of fracional order inegraion wih he Block pluse funcions. Li [31] used chebyshev wavele operaional marix o approximae he soluion of he same problem. Saadamandi and Dehghan [32] used he Legendre operaional marix of differeniaion o solve such problems. Bernsein polynomials have been used for solving parial differenial equaions [33]. More recenly, we used Bernsein s approximaion o find he sable soluion of he problem of Abel inversion [34-35]. Then we sudied Abel a inegral equaion arising in classical heory of elasiciy [36]. In his paper we presen an efficien numerical mehod for solving linear and nonlinear fracional differenial equaions. Bernsein operaional marix of fracional order inegraion is developed and is applied for solving fracional differenial equaions. Some illusraive examples are given o demonsrae he validiy and he effeciveness of he proposed mehod. Finally, we compare our resuls wih some operaional marix mehods such as chebyshev wavele and Haar waveles mehods. 2. Bernsein polynomials and funcion approximaion 2.1. Bernsein polynomials. A Bernsein polynomial is a linear combinaion of Bernsein basis polynomials. The Bernsein basis polynomials of degree are defined by µ () = (1 ) = (1) Le be he linear space ha is consising of all polynomials of degree less han or equal o in <[]-he ring of polynomials over he field <. Then, { () : = } is a basis for. For simpliciy, we assume ha = 0 if 0 or. Thus, any polynomial () in can be wrien as X () = () (2) =0 In his case, () is called a polynomial in Bernsein form and he coefficiens are called Bernsein coefficiens. I is easy o verify he following properies: (1) (0) = 0 and (1) =,where is he Kronecker dela funcion. (2) () has one roo, each of mulipliciy and, a = 0 and = 1 respecively. (3) () 0 for [0 1] and (1 ) = (). (4) For 6= 0, has a unique local maximum µ in [0 1] a = and he maximum value is ( ). (5) P =0 () = 1. (6) 1 () = () () (7) Le () [0 1], hen ()() = P =0 () converges o () uniformly on [0 1] as

3 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING 131 (8) Le () () [0 1] hen () () () () () () () 0 as,wherekk ishesupremumnormand µ () = 1 0 µ 1 1 µ 1 1 is an eigenvalue of. For more deails, see [44] Funcion approximaion. Using Gram-Schmid orhonormalizaion process, we can normalize he Bernsein basis polynomials. The new se of orhonormal polynomialsisdenoedby{ () : = }. Any funcion in 2 [0 1] can be in erms of { () : = } as X () = lim () (3) where, = h i = Z 1 0 =0 () (). If he series (2.3) is runcaed a = 1, henwehave () X = = () (4) =0 where and () are 1 marices which are given by = [ 0 1 ] (5) and () = [ 0 () 1 () ()] (6) If he domain is [0] where 1, weusedefine () = [ 0 () 1 () ()] where ( )= ( ) 3. Bernsein operaional marix of fracional order inegraion 3.1. Fracional inegral and derivaive. In his secion, we review he definiion and some preliminary resuls of he fracional derivaives. Definiion 1. The Rimann-Liouville fracional inegral operaor of order 0 on he usual Lebesgue space 1 [0 1] is given by () = Z 1 Γ() 0 () (7) ( ) 1 0 () = () (8)

4 132RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) R where Γ() = 1 is he Euler gamma funcion. 0 In he nex definiion we define he Capuo fracional derivaive of order Definiion 2. The Capuo fracional derivaive of order is defined by () = () = Z 1 Γ( ) 0 provided ha he inegral exiss, where =[] +1 [] posiive real number 0 The following properies hold: () () (9) ( ) +1 is he ineger par of he ( Γ( +1) ) = Γ( + +1) + (10) and 1 X () =() () (0 + )! (11) =0 for 1 [0 1]. From now on all fracional derivaives are in Capuo sense Block Pulse Funcions and operaional marix of fracional inegraion. Aseof Block Pulse Funcions (BPF) on [0 1) are defined as follows: ½ 1 +1 () = (12) 0 where =01 1 These funcions are disjoin and orhogonal, i.e., ½ 0 6= () () = (13) () = and Z 1 ½ 0 6= () () = (14) = 0 1 Kilicman and Al Zhour [37] have obained he Block Pulse operaional marix of he fracional order inegraion as follows: ( )() = () (15) where () = [ 0 () 1 () 1 ()], = ( +1) ( 1) +1 and = Γ( +2) : If he domain of he soluion in he fracional differenial equaion is [0] where 1, we can use he same definiion for ()

5 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING Bernsein operaional marix of he fracional inegraion. In his secion, we derive he Bernsein polynomials operaional marix of he fracional order inegraion. Firs, we rewrie he Riemann Liouville fracional order inegraion as follows: () = 1 Γ() Z 0 ( ) 1 () = 1 Γ() 1 () (16) where 0 and 1 () denoes he convoluion produc of he funcions 1 and (). The operaional marix of inegraion of () which is defined in equaion (2.6), can be obained as Z 0 () = () (17) where is marix. Orhonormal Bernsein polynomials can be wrien in erms of he Block Pulse funcions as () = Φ () (18) where () = [ 0 () 1 () 1 ()] Le be he Bernsein polynomials operaional marix of he fracional order inegraion. Then () = () (19) Equaions (3.9) and (3.12) imply ha () = Φ () =Φ () = Φ () (20) From equaions (3.12), (3.13) and (3.14) we ge () = Φ () = Φ () (21) Then, he Bernsein polynomials operaional marix of he fracional order inegraion is given by = Φ Φ 1 (22) 4. Resuls and discussions In his secion we consider six examples o demonsrae he performance and efficiency of he presen mehod. Comparison wih Haar wavele operaional marix mehod (HWOM mehod) and Chebyshev wavele operaional marix mehod (CWOM mehod). Examples 4.1 Consider he linear fracional differenial equaion, [25], () = () 0 2 (23) subjec o (0) = 1 0 (0) = 0 The exac soluion of he above problem is () = ( ) where X () = Γ( +1) (24) =0

6 134RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) I is easy o see ha for =1 he exac soluion is () = If 0 1 we use only he condiion (0) = 1 For 1 2 we use boh iniial condiions. Le () = () hen () = ()+1= ()+1 (25) Using equaion (3.12), we have () = Φ ()+1 (26) From Equaion (4.1) and Equaion (4.3), we have Φ ()+ Φ ()+1=0 or Φ ()+ Φ ()+1=0 (27) For solving Equaion (4.5), we use he Malab funcion fsolve. Figure 1 represens he graphs of he exac soluion and he approximae soluions using he proposed mehod for differen values of which are =05 (red), 075 (green), 095 (blue), and 1 (yellow) for =24. Figure 2 represens he graphs of he exac soluion and he approximae soluions using he proposed mehod, HWOM mehod, and HWOM mehod for =1and =24 I is worh menion ha he graphs of he proposed mehod, HWOM mehod, and HWOM mehod for =1are coincide. Figure 3 represens he graphs of he exac soluion and he approximae soluions using he proposed mehod for differen values of which are =125 (blue), 15 (green), 175 (red), and 195 (yellow) for =24. Figure 4 represens he graphs of he absolue error of he proposed mehod for =125 and = y Figure 1: Proposed soluion ( ) and Exac soluion (-o-o-)

7 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING y Figure 2: Proposed soluion ( ) wih exac one (-o-o-) y Figure 3: Proposed soluion ( ) and Exac soluion (-o-o-) Abs error Figure 4: Absolue error of he proposed mehod for =24 Example 4.2 Consider he nonlinear fracional differenial equaion, [31], 2 ()+ 2 ()+ 1 ()+ 3 () = () (28)

8 136RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) subjec o (0) = 0 (0) = 0 where () = 2 Γ(2) + 2 Γ(4 2 ) Γ(4 1 ) The exac soluion of Problem (4.6) is () = Le hen 2 () = () 2 () = 2 2 () 1 () = 2 1 () () = 2 () = 2 () = 2 Φ () (29) Similarly, () can be expanded in erms of he orhonormal Bernsein polynomials as follows or Assume ha Equaion (3.7) implies ha Equaions (3.12) and (4.6)-(4.11) give us ()= () (30) ()= Φ () (31) 2 Φ =[ 1 2 ] (32) 3 () = () (33) Φ ()+ 2 2 Φ ()+ (34) 2 1 Φ () () Φ () =0 In his example, we chose =1 =1 =1 =1 1 =0333 and 2 =1234. For solving Equaion (4.12), we use he Malab funcion fsolve. Figure 5 represens he graphs of he exac soluion (Red) and he approximae soluions using he proposed mehod (green), HWOM mehod(blue), and HWOM mehod (yellow ) for =24 I is worh menion ha he graphs of he proposed mehod, HWOM mehod, and HWOM mehod for =1are coincide.

9 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING y Figure 5: Exac soluion, proposed mehod, HWOM mehod, and HWOM mehod In Table 1 we compare he absolue error of our resuls wih he absolue error of he resuls of Li [31]. I is worh menion ha he proposed mehod gives beer resuls ha Li s resuls wih fewer number of Bernsein polynomials. Table (1) Proposed mehod (m=16) Li[31] (m=24) E E E E E E E E E E E E E E E E E E-4 Example 4.3 Consider he nonlinear fracional differenial equaion, [30], 22 ()+ 2 ()+ 1 ()+ 3 () = () (35) subjec o (0) = 0 (0) = 00 (0) = 0 where () = 2 2 Γ(18) 08 + Γ(4 2 ) The Exac soluion of problem (4.13) is () = Le hen 22 () = () 2 () = 22 2 () 1 () = 22 1 () () = 22 Φ 2 Γ(4 1 ) ()

10 138RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) Using he same procedure as in Example 4.2, we ge Φ () Φ ()+ (36) 22 1 Φ () () Φ () =0 In his example, we chose =1 =1 =1 =1 1 =075 2 =125. For solving Equaion (4.14), we use Malab funcion fsolve. Figure 6 represens he graphs of he exac soluion (red) and he approximae soluions using he proposed mehod (green), HWOM mehod(blue), and CWOM mehod (yellow) for = y Figure 6: Exac soluion, proposed mehod, HWOM mehod, and CWOM mehod In Table 2 we compare he absolue error of our resuls wih absolue error of he resuls of Li [30]. Table 2 Proposed mehod (m=12) Li [30] (m=16) Proposed mehod (m=16) Li [30] (m=32) E E E E E-4 5.0E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-4 Example 4.4 Consider he nonlinear fracional differenial equaion, [30], 20 ()+ () 2 subjec o ()+ () ()+ () 1 ()+ () () = () (0) = 2 0 (0) = 0

11 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING 139 where () = () Γ (3 2 ) 2 2 () + () = 05 () = 13 () = 14 () = 15 () Γ (3 1 ) () µ and The Exac soluion of problem (4.15) is () = Le hen 2 () = () 2 () = 2 2 () () = 1 () 1 () = 2 1 () Using he same procedure as in Example 4.2, we ge () = 2 ()+2 (37) ()+ () 2 2 ()+ () 1 () + (38) () 2 1 ()+() 2 ()+2 = () In his example, we chose = 1 1 = 0333 and 2 = For solving Equaion (4.17), we use Malab funcion fsolve. Figure 7 represens he graphs of he exac soluion (red) and he approximae soluions using he proposed mehod green), HWOM mehod (blue), and CWOM mehod (yellow) for =24 Figure 8 represens he graphs of he absolue error of he proposed mehod for = y Figure 7: Exac soluion, proposed mehod, HWOM mehod, and CWOM mehod

12 140RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) Abs. Error Figure 8: Absolue error of he proposed mehod for =24 Example 4.5 Consider he nonlinear fracional differenial equaion, [25], () = () 3 2 () (39) subjec o (0) = 0 0 (0) = 0 where () = () = Γ (9 ) 8 Γ 5+ 2 Γ Γ ( +1)+( ) 2 The Exac soluion of problem (4.18) is () = Le hen () = () 2 () = 2 2 () () = 1 () 1 () = 2 1 () () = () ()= () (40) Using he same procedure as in Example 4.2, we generae a sysem of nonlinear equaions which can be solve by Malab. Figure 9 represens he graphs of he exac soluion and he approximae soluions using he proposed mehod for differen values of which are =05 (red), 075 (green), 125 (blue), 15 (yellow), and 175 (black) for =24. I is worh menion ha he absolue error of he proposed mehod for = and =24is less ha

13 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING 141 y Figure 9: Proposed soluion ( ) and Exac soluion (-o-o-) Example 4.6 Consider he nonlinear fracional differenial equaion, [27], 05 () = () Γ (25) 15 (41) subjec o (0) = 0 The Exac soluion of problem (4.18) is () = 2 Using he same procedure as in Example 4.2, we generae a sysem of linear equaions which can be solve by Malab. Figure 10 represens he graphs of he exac soluion and he approximae soluions using he proposed mehod for =24. Figure 11 represens he graphs of he absolue error of he proposed mehod for = y Figure 10: Proposed soluion ( ) and Exac soluion (-o-o-)

14 142RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) Figure 11: Absolue error of he proposed mehod for =24 In Table 3 we compare he absolue error of our resuls wih absolue error of he resuls of Ford [27]. Table 3 Time Error obained by Diehelm & Ford [27] ob- using Error ained proposed mehod h=0.1 h= Conclusion In his paper we presen an efficien numerical mehod for solving linear and nonlinear fracional differenial equaions. We develop and apply he Bernsein operaional marix of fracional order inegraion for solving fracional differenial equaions. We presen six numerical examples o demonsrae he validiy and he effeciveness of he proposed mehod. In addiion, we compare our resuls wih Ford [27], HWOM mehod [30], and CWOM mehod [31], see Figures (5)-(7). Also, we compare our resuls wih he exac soluion of he fracional iniial value problems whicharepresenedinexamples(1)-(6),seefigures (1)-(3), (5)-(7),(9), (10). From Figures (4)-(8) and (11), we see ha he absolue error of he proposed mehod is wihin 10 3 The main advanage of he proposed mehod is small size of he Bernsein operaional marix of fracional order inegraion produces high accuracy, see ables (1)-(3). Also, he complexiy of he proposed mehod is small comparing wih he complexiy of he CWOM mehod and HWOM mehod. This mehod is compuer oriened and i is easy o program i. Finally, one can generalize hese echniques o sysem of fracional differenial equaions. We will leave his for he fuure work. Acknowledgemen Auhors would like o hank he reviewers for heir suggesions o improve he qualiy of he paper. Auhors are hankful o Mr. Koushlendra K. Singh for his conribuion in he preparaion of his manuscrip.

15 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING 143 References [1] M. Ichise, Y. Nagayanagi, and T. Kojima, An analog simulaion of non-ineger order ransfer funcions for analysis of elecrode processes, J. Elecronical. Chem. Inerfacial Elecrochem. 33 (1971) pp [2] H.H. Sun, B. Onaral, and Y. Tsao, Applicaion of posiive realiy principle o meal elecrode linear polarizaion phenomena, IEEE Trans. Biomed. Eng. BME-31 (10) (Ocober 1984) pp [3] H.H. Sun, A.A. Abdelwahab, and B. Onaral, Linear approximaion of ransfer funcion wih a pole of fracional order, IEEE Trans. Auoma. Conrol AC-29 (5) (1984) pp [4] J.H. He, Nonlinear oscillaion wih fracional derivaive and is applicaions, in: Inernaional Conference on Vibraing Engineering 98, Dalian, China, 1998, pp [5] J.H. He, Some applicaions of nonlinear fracional differenial equaions and heir approximaions, Bull. Sci. Technol. 15 (2) (1999) pp [6] Y.A. Rossikhin, and M.V. Shiikova, Applicaions of fracional calculus o dynamic problems of linear and nonlinear herediary mechanics of solids, Appl. Mech. Rev. 50 (1997) pp [7] R.L. Magin, Fracional calculus in bioengineering, Cri. Rev. Biomed. Eng. 32 (1) (2004) pp [8] R.L. Magin, Fracional calculus in bioengineering-par 2, Cri. Rev. Biomed. Eng. 32 (2) (2004) pp [9] R.L. Magin, Fracional calculus in bioengineering-par 3, Cri. Rev. Biomed. Eng. 32 (3/4) (2004) pp [10] R.T. Baillie, Long memory processes and fracional inegraion in economerics, J. Economerics 73 (1996) pp [11] Z. Odiba, and S. Momani, Numerical mehods for nonlinear parial differenial equaions of fracional order, Appl. Mah. Model. 32 (2008) pp [12] S. Momani, and Z. Odiba, Numerical approach o differenial equaions of fracional order, J. Compu. Appl. Mah. 207 (2007) pp [13] S.A. El-Wakil, A. Elhanbaly, and M.A. Abdou, Adomian decomposiion mehod for solving fracional nonlinear differenial equaions, Appl. Mah. Compu.182 (2006) pp [14] N.H. Sweilam, M.M. Khader, and R.F. Al-Bar, Numerical sudies for a muli-order fracional differenial equaion, Phys. Le. A 371 (2007) pp [15] S. Das, Analyical soluion of a fracional diffusion equaion by variaional ieraion mehod, Compu. Mah. Appl. 57 (2009) pp [16] S. Momani, and Z. Odiba, Homoopy perurbaion mehod for nonlinear parial differenial equaions of fracional order, Phys. Le. A 365 (2007) pp [17] N.H. Sweilam, M.M. Khader and R.F. Al-Bar, Numerical sudies for a muli-order fracional differenial equaion, Phys. Le. A 371 (2007) pp [18] I. Hashim, O. Abdulaziz and S. Momani, Homoopy analysis mehod for fracional IVPs, Commun.NonlinearSci.Numer.Simul.14(2009)pp [19] A. Arikoglu and I. Ozkol, Soluion of fracional differenial equaions by using differenial ransform mehod, Chaos Solions Frac. 34 (2007) pp [20] A. Arikoglu, and I. Ozkol, Soluion of fracional inegro-differenial equaions by using fracional differenial ransform mehod, Chaos Solions Frac. 40(2009) pp [21] P. Darania, and A. Ebadian, A mehod for he numerical soluion of he inegro-differenial equaions, Appl. Mah. Compu. 188 (2007) pp [22] V.S. Erurk, and S. Momani, Solving sysems of fracional differenial equaions using differenial ransform mehod, J. Compu. Appl. Mah. 215 (2008) pp [23] V.S. Erurk, S. Momani, and Z. Odiba, Applicaion of generalized differenial ransform mehod o muli-order fracional differenial equaions, Comm. Nonlinear Sci. Numer. Simula. 13 (2008) pp [24] Z. Odiba, and N. Shawagfeh, Generalized Taylor s formula, Appl. Mah. Compu. 186 (2007) pp [25] P. Kumar, and O.P. Agrawal, An approximae mehod for numerical soluion of fracional differenial equaions, Signal Processing 86 (2006) pp [26] F. Liua, V. Anh, and I. Turner, Numerical soluion of he space fracional Fokker-Planck equaion, J. Compu. Appl. Mah. 166 (2004) pp

16 144RAJESH K. PANDEY, ABHINAV BHARDWAJ, AND MUHAMMED I. SYAM JFCA-2014/5(1) [27] K. Diehelm, N. J. Ford, Analysis of Fracional Differenial Equaions, Journal of Mah. Analysis and Applicaions, 265 ( 2002) pp [28] K. Diehelm, N. J. Ford, Muli-order fracional differenial equaions and heir numerical soluion, Appl. Mah. Compu., 154(3) (2004) pp [29] K. Diehelm, J. M. Ford, N. J. Ford, and Marc Weilbeer, Pifalls in fas numerical solvers for fracional differenial equaions, Journal of Compu. and Applied Mah., 186 (2), (2006) pp [30] Y. Li, and W. Zaho, Haar wavele operaional marix of fracional order inegraion and is applicaions in solving he fracional order differenial equaions, Appl. Mah. Compu., 216 (2010) pp [31] Y. Li, Solving a nonlinear fracional differenial equaion using Chebyshev waveles, Communicaions in Nonlinear Science and Numerical Simulaion, 15 (9) (2010) pp [32] A. Saadamandi, and M. Dehghan, A new operaional marix for fracional-order differenial equaions, Compu. Mah. Appl., 59 (3) (2010) pp [33] D. Bhaa. Use of modified Bernsein polynomials o solve Kdv-Burgers equaion numerically, Appl. Mah. Compu., 206 (2008) pp [34] O. P. Singh, V. K. Singh, and R. K. Pandey, A sable numerical inversion of Abel s inegral equaions using almos Bernsein operaional marix. J. Quaniaive Specroscopy & Radiaive Transfer, 111 (2010) pp [35] V. K. Singh, R. K. Pandey, and O. P. Singh, New sable numerical soluion singular inegral equaion of Abel ype using normalized Bernsein polynomials, Applied Mah. Sciences, 3 (2009) pp [36] R. K. Pandey, and B. Mandal, Numerical soluion of sysem of generalized Abel inegral equaions using Bernsein s polynomials, Journal of Advanced Research in Scienific Compuing 2 (2010) pp [37] Kilicman A, and Al Zhour ZAA. Kronecker operaional marices for fracional calculus and some applicaions. Appl. Mah. Compu. 187(1) (2007) pp [38] Q. Al-Mdallal, M. Syam, M. Anwar, A collocaion-shooing mehod for solving fracional boundary value problems,communicaions in Nonlinear Science and Numerical Simulaion, 15)12) (2010) pp [39] Q. Al-Mdallal, M. Syam, An efficien mehod for solving non-linear singularly perurbed wo poins boundary-value problems of fracional order, Communicaions in Nonlinear Science and Numerical Simulaion 17(6) (2012) pp [40] A. Wazwaz, The variaional ieraion mehod: A reliable analyic ool for solving linear and nonlinear wave equaions, Compuers & Mahemaics wih Applicaions, 54(7 8) (2007), pp [41] A. Wazwaz, The variaional ieraion mehod for solving linear and nonlinear sysems of PDEs, Compuers & Mahemaics wih Applicaions, 54(7 8) (2007), pp [42] H. Siyyam, M. Hamdan, Firs-order accurae finie difference schemes for boundary voriciy approximaions in curvilinear geomeries Applied Mahemaics and Compuaion, 215(6) (2009), pp [43] F. Awawdeh, H. Jarada, O. Alsayyed, Solving sysem of DAEs by homoopy analysis mehod, Chaos, Solions & Fracals, 42(3) (2009), pp [44] A. K. Singh, V. K. Singh, O. P. Singh, The Bernsein Operaional Marix of Inegraion, Applied Mahemaical Sciences, 3(49) (2009), pp [45] D.B.Dhaigude,G.A.Birajdar,Numericalsoluionofsysemoffracionalparialdifferenial equaions by discree Adomian decomposiion mehod, Journal of fracional calculus and applicaions, 3(12) (2012), pp [46] A. Cernea, On a fracional differenial inclusion wih srip boundary condiions, Journal of fracional calculus and applicaions, 4(2) (2013), pp [47] A. M. A. El-Sayed, S. M. Salman, On a discreizaion process of fracional order Riccai differenial equaion, Journal of fracional calculus and applicaions, 4(2) (2013), pp [48] S. Zhang, Exisence and uniqueness resul of soluions o iniial value problems of fracional differenial equaions of variable order, Journal of fracional calculus and applicaions, 4(1) (2013), pp Rajesh K. Pandey, PDPM Indian Insiue of Informaion Technology,, Design and Manufacuring Jabalpur, M. P , India

17 JFCA-2014/5(1) AN EFFICIENT METHOD FOR SOLVING 145 Abhinav Bhardwaj, PDPM Indian Insiue of Informaion Technology,, Design and Manufacuring Jabalpur, M. P , India :Muhammed I. Syam, Deparmen of Mahemaical Sciences, Unied Arab Emiraes Universiy,, P.O.Box 15551, Al Ain, UAE address: