An Application of Legendre Wavelet in Fractional Electrical Circuits

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1 Global Journal of Pure and Applied Mahemaics. ISSN Volume, Number (7), pp. 8- Research India Publicaions hp:// An Applicaion of Legendre Wavele in Fracional Elecrical Circuis Riu Arora* and N. S. Chauhan** *Deparmen of Mahemaics, Kanya Guruul (Haridwar Campus) Guruula Kangri Universiy, Haridwar-4944 (UK), India. **Deparmen of Mahemaics and Saisics, Faculy of Science Guruula Kangri Universiy, Haridwar-4944 (UK), India. **Corresponding Auhor Absrac In his paper, Legendre wavele mehod is presened for solving fracional elecrical circuis. We firs derive he operaional marix of fracional order inegraion. he fracional inegraion is described in he Riemann-Liouville sense. he operaional marix of fracional order inegraion is used o ransform fracional elecrical circui s equaion ino he sysem of algebraic equaions. he uniform convergence heorem and accuracy esimaion are also derive for proposed mehod. In addiion, using ploing ools, we compare approximae soluions of each equaion wih is classical soluion. Keywords: Fracional Elecrical circui models, Legendre waveles, Operaional marix of fracional inegraion, Bloc pulse funcion. Subjec Classificaion AMS: 6599, 47N7, 4A8, 65L, 6A. INRODUCION In recen years, waveles ools are one of he growing and predominanly a new echnique. oday many classical physics models are being analysed using wavele mehods. Here, we discuss he basic equaions of elecric circuis involving resisor wih a resisance R measured on ohms, an inducor wih inducance L measured in

2 84 Riu Arora and N. S. Chauhan henries, and a capacior wih capaciance C measured in farads. We invesigae he following equaions for four differen ypes of circuis J ( ) J( ) LC () '' CV ( ) V ( ) R ' () LQ ( ) RQ ( ) Q( ), () C '' ' LJ ' ( ) RJ ( ) V (4) where J( ), V( ) and Q () are he curren, volage and elecric charge in he shell of capacior wih respec o ime respecively. Equaion () represens he LC (inducor-capacior) circui, equaion () represens he RC (resisor-capacior) circui, equaion () represens he LR (inducor-resisor) circui and equaion (4) represens he RLC (resisor- inducor -capacior) circui. Some numerical resuls of consider circuis are presened in [-5]. he mahemaical foundaion of fracional calculus (FC) was esablished over years ago, alhough he applicaion of FC has araced ineres in recen decades. FC, involving derivaives and inegrals of non-ineger order, is he naural generalizaion of he classical calculus [6-9]. FC is a highly efficien and useful ool in he modelling of many sors of scienific phenomena including image processing earhquae, engineering, physics ec []. Waveles are used in signal analysis, medical science, approximaion heory, echnology and many oher areas. Waveles analysis possesses several useful properies such as compac suppor, orhogonaliy, exac represenaion of polynomials o a cerain degree, and abiliy o represen funcions a differen levels of resoluion [, ]. he ouline of his paper is as follows: In secion second, we discuss some noaions, definiions and preliminary facs of he fracional calculus heory. In secion hird, we presen wavele and Legendre wavele and heir properies. In secion four, we show ha he funcion approximaion and collocaion poins. Secion fifh is devoed o he operaional marix of inegraion. In secion six, we give convergence and mean square error heorems for our mehod. Finally, we solve fracional elecric circui s equaions given in secion seven, o illusrae he performance of our mehod. OVERVIEW ON FRACIONAL CALCULUS In his secion, we presen some noaions, definiions and preliminary facs of he fracional calculus heory. Using fracional calculus, one has many choices for definiions of fracional derivaive as well as fracional inegral. Since, we wan o ransform he elecric circui differenial equaions in o he algebraic sysem of

3 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis 85 equaions, and consider he fracional inegral and fracional derivaive in he Riemann-Liouville sense. Definiion. he Riemann-Liouville fracional inegral operaor I of order on he usual Lebesgue space L [, ] a b is given [] by I f f( ),. I has he following properies: (i) I I I, (ii) I I I I, s f ( s) ds, (5) (iii) v (iv) I a a I I f ( ) I I f ( ), ( v ) ( v ) v, (6) where f L [ a, b],, and v. Definiion. For a funcion f given on inerval [ ab, ], he Capuo definiion of fracional order derivaive of order n n of f is defined [] by n d n ( ) ( ) ( ), Da f s f s ds ( n ) d (7) where, n is a ineger. I has he following wo basic properies for n f L [ a, b]. D I f ( ) f ( ) and a nand n ( ) ( a) I D f ( ) f ( ) f ( ),. (8)! For more deails on he mahemaical properies of fracional derivaives and inegrals see [5-7]. WAVELES AND LEGENDRE WAVELES (LW) Waveles consiue a family of funcions consruced by performing ranslaion and dilaion on a single funcion, where is a moher wavele. We define family of coninuous waveles [7] by / b a, b() a a, a, br; a, where a is called scaling parameer and b is ranslaion parameer. If we resriced he parameers a and b o discree values as a a, b nb a, where

4 86 Riu Arora and N. S. Chauhan a, b, and, n are posiive inegers. We have he following family of discree waveles: / ( ) a a nb n,, n Z ; where he funcion n() form a wavele basis for L ( R). In paricular, when a and b, he funcion n() form an orhonormal basis. he Legendre waveles ( ) (, ˆ n, m, ) have four argumens; nˆ n, n,,,...,, can assume any posiive ineger, m is he order for Legendre polynomials and is he normalized ime. he Legendre waveles are defined on he inerval [,]as [4-6] / ˆ ˆ n n m. P ˆ m n, for, () (9), oherwise, where m,,,..., M, M is a fixed posiive ineger and n,,,...,. he coefficien m / is for orhnormaliy, he dilaion parameer is a and ranslaion parameer is b nˆ. Pm () is he well-nown Legendre polynomial of order m defined on he inerval [, ] and can be deermined wih he help of he following recurrence formulae: m m P ( ), P ( ), Pm ( ) Pm ( ) Pm ( ), m m where m,,,.... () FUNCION S APPROXIMAION A funcion f() defined over [,] and approximaed as f ( x) g ( x) () n m n m n m where g f ( ), ( ). If he infinie series in equaion () is runcaed n m hen equaion () can be wrien as M n m n m f ( ) gn m n m( ) G ( ), () where indicaes ransposiion and, G and () vecors given by are mˆ M column

5 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis 87 and G [ g, g,..., g( M ), g,..., g( M ),..., g,..., g ] ; ( M ). () ( ) [,,...,,,...,,...,,..., ] ( M) ( M) ( M ) aing he collocaion poins as following: i,,,...,. (4) i i M M Now, aing define he Legendre wavele marix ˆ ˆ[7] as: 5 m m ˆ mˆ... m m m m (5) For example, when M and he Legendre wavele is expressed as m ˆ mˆ he Legendre marix m ˆ mˆ is an inverible marix [8], he coefficien vecor G is obained by G f mˆ mˆ m m ˆ. (6) LEGENDRE WAVELE OPERAIONAL MARIX OF FRACIONAL INEGRAL he fracional inegral of order in he Riemann-Liouville sense of he vecor (), defined in equaion (5), can be approximaed by Legendre wavele series wih Legendre wavele coefficien marix P. ˆ ˆ (7) I ( ) P ( ), mm where he P is called m ˆ m ˆ Legendre wavele operaional marix of inegral of mˆ mˆ order. I show ha he operaional marix P can be approximae as [9]: ˆ ˆ P ˆ ˆ ˆ ˆ, m m m mf mˆ mˆ m m (8) where ˆ ˆ m m F is he operaional marix of fracional inegraion of order of he bloc-pulse funcion (BPFs), which given in []

6 88 Riu Arora and N. S. Chauhan F mˆ mˆ where j ( j ) j ( j ), [6]. m m m m4, CONVERGENCE AND MEAN SQUARE ERROR HEOREMS FOR MEHOD In his secion we discus convergence and mean square error of Legendre wavele mehod. heorem (Convergence heorem): Le he funcion D f ( x) C [,] bounded and D f ( x) exiss and can be expressed as in equaion () and he runcaed series given in equaion () converges owards he exac soluion. Proof: Le D f ( x) be a funcion in he inerval [,]such ha D f ( x) K, where K is a posiive consan. (9) From equaion (), we approximae D f ( x) as where g D f ( x), ( x) M D f ( x) g ( x), () n m / ˆ ˆ n and ( ) ( ˆ), n n x m Pm x n for x and Pm are well nown Legendre polynomials. Now, we consider g D f ( x), ( x) D f ( x) ( x) dx () ( nˆ)/ ( nˆ)/ ( nˆ)/ ( nˆ)/ g D f ( x) ( x) dx D f ( x) ( x) dx D f ( x) ( x) dx ( nˆ)/ ( nˆ)/ / m ( nˆ)/ ( nˆ)/ g D f ( x) ( x) dx D f ( x) m P x nˆ dx,

7 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis 89 Le x nˆ ˆ ˆ g m D f P m D f P d, / n d / n m m () We consider righ hand side of equaion () and inegrae by par wo imes wih respec o, yield Now, / nˆ Pm ( ) Pm ( ) Pm ( ) Pm ( ) g m D f m m m ˆ P ( ) P ( ) P ( ) P ( ) m D f d / n m m m m m m m ˆ P ( ) P ( ) P ( ) P ( ) g m D f d / n m m m m m m m () nˆ P ( ) P ( ) P ( ) P ( ) g m D f d / m m m m m m m 5 / Pm ( ) Pm ( ) Pm ( ) Pm ( ) nˆ g m d D f m m m (4) g 5 / m m m m nˆ m Pm ( ) m Pm ( ) m Pm ( ) m Pm ( ) d D f g 5 / m K m m m m P ( ) m P ( ) m P ( ) m P ( ) d m m m m g 5 / m m m K m P ( ) m P ( ) m P ( ) m P ( ) d m m m m Le R m P ( ) 4m P ( ) m P ( ) m m m (5) g 5 / K R d. (6) m m m

8 9 Riu Arora and N. S. Chauhan R d m P ( ) 4m P ( ) m P ( ) d. Using Cauchy-Schwarz s inequaliy, we ge m m m R d d m Pm ( ) 4m Pm ( ) m Pm ( ) d, 4 m Pm ( ) 4m Pm ( ) m Pm ( ) d, m 4., m m R d. (7) m Using equaion (7) in equaion (6), we ge g m 5 / K, m m m m 5 / K 6 m m m, Hence desire. g 6K m m m 5 /. (8) heorem (Mean square error): Le he funcion D f ( x) C [,], and D f ( x) exiss bounded second derivaive hen we have he following accuracy esimaion. Where n 6K 5/ n mm n m m m. M n g ( x) g ( x) dx. n m n m Proof: Le us consider he quaniy M n m n m g ( x) g ( x) dx, (9)

9 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis 9 n mm n mm () g ( x) dx g ( x) dx, Using he propery of orhonormal waveles ( x ) ( x ) dx I. () g n mm From equaion (8), we have, Hence desire. n 6K 5/ n mm n m m m. ELECRICAL CIRCUIS In his secion, we apply Legendre wavele collocaion mehod and find he approximae soluion of fracional circuis and comparing heir soluions wih he corresponding classical soluions. Soluion of LC circui Consider he LC circui, only charged capacior and inducor are presen in he circui and is differenial equaion is given as follows. J ( ) J( ). LC () '' wih J() J and J ' (). he classical soluion of equaion () is J( ) J Cos( ), () LC where / LC. Now, we analyse equaion () using fracional calculus, we replace J '' () by D J() where (, )., In he sense of Riemann-Liouville derivaive, we ge he fracional order LC circui and is differenial equaion as D J( ) J( ). (4)

10 9 Riu Arora and N. S. Chauhan wih J() J and D J(). (5) We use equaion () o approximae D J() as M D J( ) u ( ) U ( ), (6) n m Inegraing equaion (6) wih respec o over he inerval[, ], we ge Using condiion (5), yield DJ ( ) DJ () U P ( ), mˆ mˆ mˆ mˆ J( ) J U P ( ). (7) Subsiuing equaions (5-6) in equaion (), we obain mˆ mˆ U ( ) J U P ( ). (8) From equaion (6), we can approximae J as:,,..., ˆ ˆ( ). (9) J J J J mˆ mm Subsiuing equaion (9) in equaion (8), we have U ( ) U P ( ) [ J, J,..., J ] ( ), (4) mˆ mˆ mˆ mˆ U I P [ J, J,..., J ], mˆ mˆ mˆ mˆ mm U [ J, J,..., J ] I P. (4) Hence required mˆ mˆ ˆ ˆ J( ) [ J, J,..., J ] ( ) [ J, J,..., J ] I P P ( ). (4) mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ By solving he above sysem (4) of linear equaions, we can find he value of vecor U. subsiuing value of U in equaion (7), hence we required he numerical resuls of he LC circui for differen value of Mand,. Here, soluion obained by he proposed LWM approach for.5,.75,.999 and and M is graphically show in figure. As i can be clearly seen, for.999 and and M, he fracional RC circui graph behave similar o he classical soluion graph for. I is show ha he proposed LWM approach is more close o he exac soluion. able describes he efficiency of he proposed mehod by comparing wih he classical soluion a. able show ha very high accuracies are obained for and M by he presen mehod.

11 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis Curren Curren Curren Curren Figure.Curren versus ime graph L, C, J. and.5,.75 and.999. able. Numerical resuls of LC circui for L, C, J. and.5,.75 and LW LW LW CS Soluion of RC circui Consider he RC circui differenial equaion given in equaion (), only charged capacior and resisor are presen o he circui and is differenial equaion is given as follows CV ( ) V ( ). R ' (4) ' wih condiion V() V and V (). (44)

12 94 Riu Arora and N. S. Chauhan he classical soluion of equaion (4) is RC V ( ) V e. RC (45) Now, we consider equaion (4) using fracional calculus, we replace V ' () by D V () where (,). In he sense of Riemann-Liouville derivaive, we ge he fracional order RC circui and is differenial equaion as D V ( ) V ( ) RC, (46) wih condiion V() V and D V(). (47) We use equaion () o approximae D V () as M D V ( ) w ( ) W ( ), (48) n m Inegraing equaion (48) wih respec o, over [, ], we ge V( ) V() W P ( ), (49) mˆ mˆ Similarly equaion (9), we can approximae V as So, equaion (49) become ˆ ˆ (5) V() V V, V,..., V ( ). mˆ mm ˆ ˆ ˆ ˆ (5) V( ) V, V,..., V ( ) W P ( ), mˆ Subsiuing equaions (48) and equaion (5) in equaion (46), we obain mm mm W ( ) V, V,..., V ˆ mˆ mˆ ( ) W Pm ˆ mˆ ( ) m RC W ( ) W Pm ˆ mˆ ( ) [ V, V,..., V ] mˆ mˆ ( ), RC RC W I Pm ˆ mˆ [ V, V,..., V ] mˆ mˆ, RC RC Hence required W [ V, V,..., V ] m ˆ mˆ. I Pm ˆ mˆ. RC RC (5) m ˆ mˆ m ˆ mˆ mˆ mˆ mˆ mˆ V ( ) [ V, V,..., V ] ( ) [ V, V,..., V ]. I P P ( ). RC RC We solving he above sysem (5)of linear equaions and obain he value of vecor w. Subsiuing he value of vecor W in equaion (5), hence we require he approximae resuls of he RC circui model for differen value of Mand,. Here, we use he proposed LWM approach for.5,.75,.999 and and M. his has been seen (5)

13 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis 95 from figure ha he obain soluion for.999 and and M, he fracional RC circui graph behave similar o he classical soluion graph for. I is show ha he proposed LWCM approach is more close o he exac soluion. able describes he good organizaion of he proposed mehod by comparing wih he classical soluion a. able also shows ha, very high accuracies are obained for and M by he presen mehod. Volage Volage Volage Volage Figure.Volage versus ime graph R, C, V and.5,.75 and.999. able.numerical resuls of RC circui for R, C, J and.5,.75 and LW LW LW CS

14 96 Riu Arora and N. S. Chauhan Soluion of RLC circui Here, we analyze RLC circui differenial equaion given in equaion (). LCR circui consising of hree inds of circui elemens: a resisor, an inducor and a capacior is differenial equaion is given as follows LQ ( ) RQ ( ) Q( ). C (54) '' ' ' Wih he condiion Q() Q, Q (). (55) he classical soluion of equaion (54) is R R L Q( ) RLC Q e Cos. LC 4L Now, we analyse equaion (54) using fracional calculus, we replace '' Q () by D Q() where (,). In he sense of Riemann-Liouville derivaive, we ge he fracional order RLC circui and is differenial equaion as L D Q ( ) R D Q ( ) Q( ). C (56), (57) Wih he condiion Q() Q and D Q(). (58) We use equaion () o approximae D Q() as D Q M y Y n m ( ) ( ) ( ), (59) Inegraing equaion (59) wih respec o, over [, ] and using condiion (58), we ge and mˆ mˆ mˆ mˆ D Q ( ) D Q () Y P ( ) Y P ( ), (6) Q( ) Q() Y P ( ). (6) mˆ mˆ Similarly equaion (9), we can approximae Q as Q() Q [ Q, Q,..., Q ] ( ). (6) mˆ mˆ From equaion (6) and equaion (6), we have Q( ) [ Q, Q,..., Q ] ( ) Y P ( ). (6) mˆ mˆ mˆ mˆ Subsiuing equaions (59,6) and equaion (6) in equaion (57), we obain R Y ( ) Y Pm ˆ mˆ ( ) [ Q, Q,..., Q ] mˆ mˆ ( ) Y Pm ˆ mˆ ( ), (64) L LC

15 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis 97 Le LC and R, L hen equaion (64) become Y ( ) Y P ( ) [ Q, Q,..., Q ] ( ) Y P ( ) mˆ mˆ mˆ mˆ mˆ mˆ Y Y P Y P [ Q, Q,..., Q ], mˆ mˆ mˆ mˆ mˆ mˆ Y I P P [ Q, Q,..., Q ], mˆ mˆ mˆ mˆ mˆ mˆ Y [ Q, Q,..., Q ]. I P P. (65) Hence required mˆ mˆ mˆ mˆ mˆ mˆ Q( ) [ Q, Q,..., Q ] ( ) [ Q, Q,..., Q ] I P P P ( ). (66) mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ We manipulae he above sysem (65) of linear equaions and obain he unnown vecor Y. Having hese value of vecor Y in equaion (6), we ge numerical resuls of RLC circui for differen value of Mand,. Here, soluion obained by he proposed LWM approach for.5,.75,.999 and and M is graphically shown in figure. As i can be clearly seen, for.999 and and M, he fracional RLC circui graph behave similar o he classical soluion graph for. I is show ha he proposed LWCM approach is more close o he exac soluion. able describes he effeciveness of he proposed mehod by comparing wih he classical soluion a. able shows ha very high accuracies are obained for and M by he presen mehod. Charge Q Charge Q Charge Q Charge Q Figure.Charge Q () versus ime graph R, L, C, V. and.5,.75 and.999.

16 98 Riu Arora and N. S. Chauhan able. Numerical resuls of RC circui for R, L, C, V. and.5,.75 and LW LW LW CS Soluion of RL circui Finally, we consider RL circui differenial equaion given in equaion (4). RL circui consiss only resisor, inducor and a non-varian volage source are presen in he circui and is differenial equaion is given as follows wih J() J and V is he consan volage source. he classical soluion of equaion (67) is LJ ' ( ) RJ ( ) V. (67) R VL VL L J( ) I e. R R (68) Now, we analyse equaion (67) using fracional calculus, we replace J ' () by D J(), where (,). In he sense of Riemann-Liouville derivaive, we ge he fracional order RL circui and is differenial equaion as R L V L ( ) R V D J J( ). L L Le and, hen equaion (69) become D J J (69) ( ) ( ). (7)

17 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis 99 We use equaion () o approximae D J ( ) as M D J ( ) z ( ) Z ( ), (7) n m Inegraing equaion (7) wih respec o, over[, ], we ge J( ) J() Z P ( ), (7) mˆ mˆ Similarly equaion (9), we can approximae J and as J() J [ J, J,..., J ] ( ) and [,,..., ] ( ). (7) mˆ mˆ mˆ mˆ mˆ mˆ Subsiuing equaions (7-7) in equaion (7), we obain Z ( ) [ J, J,..., J ] ( ) Z P ( ) [,,..., ] ( ), mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ Z ( ) [ J, J,..., J ] ( ) Z P ( ) [,,..., ] ( ) mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ Z [ J, J,..., J ] Z P [,,..., ] mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ Hence required mˆ mˆ m ˆ mˆ mˆ mˆ Z [,,..., ] [ J, J,..., J ]. I P. (74) J( ) [ J, J,..., J ] [,,..., ] [ J, J,..., J ]. I P. P ( ). (75) mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ mˆ By solving he above sysem (74) of linear equaions, we can find he value of coefficien vecor Z. Insering he value of vecor Z in equaion (7), hence we obain he numerical resuls for differen value of Mand,. Here, soluion obained by he proposed LWM approach for.5,.75,.999 and and M is graphically show in figure 4. As i can be clearly seen, for.999 and ; M, he fracional RL circui graph behave similar o he classical soluion graph for. I is show ha he proposed LWM approach is more close o he exac soluion. able 4 shows he capabiliy of he presened mehod and also shows ha very high accuracies are obained for and M.

18 Riu Arora and N. S. Chauhan Curren Curren Curren Curren Figure 4.Curren versus ime graph R, L, V, V. and.5,.75 and.999. able 4. Numerical resuls of RL circui for R, L, V, V. and.5,.75 and.999. x LW LW LW CS

19 An Applicaion of Legendre Wavele in Fracional Elecrical Circuis CONCLUSION In his paper, he Legendre wavele mehod (LWM) is applied o obain approximae analyical soluions of he fracional elecrical circui models. I can be concluded ha, LWM is very powerful and efficien echnique for finding approximae soluions for many real life problems [4-6]. he main advanage of he mehod is is fas convergence o he soluion. I has been shown in he heorem 6., by increasing and order m of Legendre polynomial, he Legendre wavele series converges very fas. See, in he figure, figure and figure approximae soluions graph behave as similar o he classical soluion bu.999 he Capuo Fabrizio approach [] shows damping and behave very differenly for LC circui. As similar proposed mehod presen good approximaed resuls for RC, LCR and RL a.999 bu he Capuo fracional derivaive [] graph for RL circui coincide wih he classical soluion bu diverges o very large posiive values as ime progresses. he numerical resuls obained here, conform o is high degree of accuracy. Such analysis can be furher applied o oher physical models o develop a beer undersanding of use of waveles in real life problems. he implemenaion of his mehod is a very easy accepable and valid. he soluions of he elecrical circui equaions are presened graphically and in abular form. ACKNOWLEDGMEN he auhors are very hanful o respeced Dr. Sag Ram Verma, Deparmen of Mahemaics and Saisics, Guruula Kangri Universiy, Haridwar, for encouragemen and suppor. REFERENCES [] Alsaedi A, Nieo J, Venesh V, Fracional elecrical circuis, advances in mechanical engineering 5; 7():-7. [] Gomez F, Rosales J, Guia M, RLC elecrical circui of non-ineger order, Cenral European J. of Phy ;(): [] Aangana A. and Nieo JJ, Numerical soluion for he model of RLC circui via he fracional derivaive wihou singular ernel, Adv. Mech. Eng, Epup ahead of prin 9 ocober 5. doi:.77/ [4] Kaczore, posiive elecrical circuis and heir reachabiliy, Arch Elec. Eng. ;6():8-. [5] Kaczore and Rogowsi K, fracional linear sysems and elecrical circuis, Springer, London; 7. [6] Oldham KB, Spanier J, he fracional calculus, Academic Press, New Yor; 974. [7] Miller KS, Ross B, An inroducion o he Fracional calculus and fracional differenial equaions, Wiley, New Yor; 99. [8] Podlubny I, Fracional differenial equaions, Academic Press, New Yor; 999.

20 Riu Arora and N. S. Chauhan [9] Abbas S, Benchohra M and N Guereaa GM, opics in fracional differenial equaions, New Yor, Springer;. [] Diehelm K, he analysis of farcional differenial equaions: an applicaionoriened exposiion using differenial operaors of capuo ype, 4 (Lecure noes in Mahemaics), Berlin: Springer-Verlag;. [] Daubechies I, en Lecures on Wavele, Philadelphian, SIAM; 99. [] Chui CK, Waveles: A mahemaical ool for signal analysis, Philadelphia PA, SIAM; 997. [] Wang Y, Fan Q, he second ind chebyshev wavele mehod for solving fracional differenial equaions, Applied Mahemaics and Compu. ; 8(): [4] Jafari H, Yousefi S, Firoozjaee M, Momani S, Khalique CM, Applicaion of Legendre waveles for solving fracional differenial equaions, Comp. and Mah. wih Applic. ;6():8-45. [5] M. Razzaghi, S. Yousefi, Legendre wavele direc mehod for variaional problems, Mah. Compu. Simula. ;5():85-9. [6] Razzaghi M, Yousefi S, Legendre wavele mehod for consrained opimal conrol problems, Mah. Mehod Appl. Sci. ;5():59-9. [7] Li Y, Solving a nonlinear fracional differenial equaion using chebyshev waveles, Commun. Nonlinear Sci. Numer. Simula. ;5():84-9. [8] Balaji S, Legendre wavele operaional marix mehod for soluion of fracional order Riccai differenial equaion, J. of he Egyp. Mah. Soc. 5;():6-7. [9] Heydari MH, Hooshmandasl MR, Ghaini FMM, Mohummadi F, Wavele collocaion mehod for solving muli order fracional differenial equaions, J. Appl. Mah.. doi:.55//544. [] Rehman M, Khan RA, he Legendre waveles mehod for solving fracional differenial equaions, Commun. Nonlinear Sci. Numer. Simula. ;6():46-7.