America Joural of Computatioal ad Applied Mathematics 01, (3): 83-91 DOI: 10.593/j.ajcam.01003.03 A Decompositio Algorithm for the Solutio of Fractioal Quadratic Riccati Differetial Equatios with Caputo Derivatives O. S. Odetude 1,*, O. A. Taiwo 1 Departmet of Mathematical Scieces, Olabisi Oabajo Uiversity of Ago-Iwoye, P.M.B 00, Ago-Iwoye, Ogu State, Nigeria Departmet of Mathematics, Uiversity of Ilori, P.M.B 1515, Ilori, Nigeria Abstract A Iterative Decompositio Method is applied to solve Fractioal Quadratic Riccati Differetial Equatios i which the fractioal derivatives are give i the Caputo sese. The method presets solutios as rapidly coverget ifiite series of easily computable terms. Solutios obtaied compared favorably with exact solutios ad solutios obtaied by other kow methods. Keywords Fractioal Riccati Differetial Equatios, Caputo Derivatives, Iterative Decompositio Method 1. Itroductio Fractioal Differetial Equatios, which are a geeralizatio of differetial equatios have bee applied to model may physical models very accurately. They have bee used extesively i such areas as Thermal Egieerig, acoustics, Electromagetism, Cotrol Theory of Dyamical Systems, Robotics, Viscoelasticity, Diffusio, Sigal Processig, Populatio Dyamics ad so o [, 5, 13]. Riccati Differetial Equatios are amed after the Italia Noblema Jacapo Fracesco Riccati (1676-175) []. I [16], the fudametal theories of the Riccati equatios, ad diffusio processes are discussed. The oe-dimesioal Schrodiger equatio is closely related to a Riccati Differetial Equatio [15]. Solitary wave solutio of a oliear partial differetial equatio ca be represeted as a polyomial i two elemetary fuctios satisfyig a projective Riccati differetial equatio [7]. Riccati differetial equatios are kow to be cocered with applicatios i patter formatio i dyamic games, liear systems with Markovia jumps, river flows, ecoometric models, stochastic systems, cotrol theory, ad diffusio problems [10, 15, 16]. May authors ad researchers have studied the aalytical ad approximate solutios of Riccati differetial equatios. Some approximate solutios have bee obtaied from Homotopy Perturbatio method (HPM) [1,, 3, 1], Homotopy Aalysis Method (HAM) [] ad the Variatioal * Correspodig author: tayo@uilori.edu.g (O. S. Odetude) Published olie at http://joural.sapub.org/ajcam Copyright 01 Scietific & Academic Publishig. All Rights Reserved Iteratio Method (VIM) [7]. I this paper, we cosider the fractioal quadratic Riccati differetial equatio D* yx ( )= Ax ( ) Bxyx ( ) ( ) Cxy ( ) ( x), x R, 0 < 1, x> 0 subject to the iitial coditio ( k y ) (0) = y, k = 0,1,,, 1 k (1) () where is the order of the fractioal derivative, x is a iteger, Ax ( ), Bx ( ) ad Cx ( ) are kow fuctios, ad k y is a costat. The derivative is the Caputo-type derivatives. It is well kow that most fractioal differetial equatios do ot have aalytical solutios. Several authors have cosidered the approximate solutio (1)-() usig modificatios of various methods, which have bee applied successfully to iteger-order differetial equatios. For istace, the Fractioal Variatioal Iteratio Method (FVIM) was applied i [8]. I [11], the well-kow Adomia Decompositio Method (ADM) was cosidered for the same problems. I [1], the Homotopy Perturbatio Method (HPM) was modified to solve fractioal order quadratic Riccati differetial equatios. I [], the HPM ad ADM are compared for quadratic Riccati differetial equatios; ad i [10], the Homotopy Aalysis Method is applied to fractioal Riccati equatios. I [15], the Berstei operatioal matrices are applied; motivated probably by the results i [5]. I [], the Differetial Trasform Method (DTM) is applied. I this paper, we propose to apply the Iterative
8 O. S. Odetude et al.: A Decompositio Algorithm for the Solutio of Fractioal Quadratic Riccati Differetial Equatios with Caputo Derivatives Decompositio Method (IDM) which has bee applied successfully for iteger-order differetial equatios (see [16, 17]). The success of the method for iteger-order differetial equatios motivated [17, 18]. The method is devoid of ay form of liearizatio or discretizatio. The rest of the paper is orgaized as follows: I sectio, we give briefly defiitios related to the theory of fractioal calculus. I sectio 3, we preset the solutio procedure of the Iterative Decompositio Method (IDM). Numerical examples are preseted i sectio to illustrate the efficiecy ad accuracy of the IDM. The coclusios are the give i the i the last sectio 5.. Basic Defiitios I this sectio, we give some defiitios ad properties of the fractioal calculus. Defiitio.1: A real fuctio f( x), x > 0, is said to be i the space p µ, such that Cµ, µ R if there exists a real umber > p f( x)= x f ( x ), where 1 ( ) 1 to be i space C µ if ad oly if Clearly Cµ Cβ if β µ., ad it is said ( ) f Cµ, N. Defiitio.: Let 0, ad < 1, N. The operator adt, defied by 1 d t 1 adt f ()= t ( t x) f ( x) dx, a (3) a t b Γ( ) dt a t D f()= t f() t is called the Riema-Liouville Fractioal derivative operator of order. Defiitio.3: The Riema-Liouville fractioal itegral operator defied o L 1 [ ab, ] of order 0 of a fuctio f C µ, µ 1 is defied as 1 1 J f ( x)= ( x t) f () t dt, >0, x >0 Γ( ) () Defiitio.: Let < 1, N, ad f ( ) ( x) L1 [ ab, ]. The operator D* defied by 1 t 1 ( ) D* f ()= t ( t x) f ( x) dx Γ( ) (5) a is called the Caputo Fractioal Derivative Operator of order. Properties of the operator J is foud i [7] ad iclude the followig (6) J f( x)= f( x) J x Γ ( γ = x Γ ( γ γ γ J J f( x)= J f( x) (7) β β (8) β β J J f( x)= J J f( x) (9) Also, if m1< < mm, N ad f C µ, µ 1, the D* J f( x)= f( x) m1 =0 m (10) x ( ) J D* f( x)= f( x) f (0) (11)! 3. Iterative Decompositio Method For the fractioal quadratic Riccati differetial equatio (1), 0< 1, with iitial coditio y(0) = a, by applyig the operator J to both sides of (1), we have yx ( )= a J Ax ( ) Bxyx ( ) ( ) Cxy ( ) ( x) (1) The Iterative Decompositio Method [17, 18] suggests that (1) is the of the form y = f N( y) (13) The solutio is decomposed ito the ifiite series of coverget terms yx ( )= y( x) (1) = Form (13), we have Ny ( )= J Ax ( ) Bxyx ( ) ( ) Cxy ( ) ( x) (15) The operator N is the decomposed as { 1 } { N( y y y ) N( y y )} N( y)= N( y ) N( y y ) N( y ) 1 1 (16)
America Joural of Computatioal ad Applied Mathematics 01, (3): 83-91 85 Let G = N( y ) The, 1 G = N N y, = 1,, (17) i i=0 i=0 G i=0 N( y)= i set y = f. From (13) ad (1), we have The, y G1 =, = 1,, 1 y = N 1 y (18) =0 We ca the approximate the solutio (1) by (18) as N.. Numerical Examples We ow apply the method proposed i sectio 3 to solve some umerical examples to establish the accuracy ad efficiecy of the method. Example.1 [19] Cosider the Riccati Differetial equatio D* y = 1 y, 0 < 1 (19) subject to the iitial coditio y (0) = 0. The exact solutio for the case =1 is ( ) = ta yx x. By the Iterative Decompositio Method (IDM), x y ( x)=, Γ ( 3 Γ ( x y1 ( x)= [ ( 1)] Γ Γ (3 5 7 Γ ( Γ ( x [ Γ ( ] Γ (6 x y( x)= [ ( 1)] 3 (3 1) (5 1) [ ( 1)] [ (3 1)] Γ Γ Γ Γ Γ Γ (7 9 [ Γ ( ] Γ ( Γ (8 x y3( x)= [ ( 1)] 5 [ (3 1)] Γ Γ Γ (5 Γ (9 9 5 [ Γ ( ] Γ (6 Γ (8 [ Γ ( ] [ Γ (3 ] Γ (7 Γ (9 3 11 6 3 [ Γ ( ] Γ (6 Γ (10 (0) [ Γ ( ] [ Γ (3 ] Γ (7 Γ (11 11 6 [ Γ ( ] [ Γ ( ] Γ (10 [ Γ ( ] [ Γ (3 ] [ Γ (5 ] Γ (11 3 13 7 3 [ Γ ( ] Γ ( Γ (6 Γ (1 [ Γ ( ] [ Γ (3 ] Γ (5 Γ (7 Γ (13 15 8 [ Γ ( ] [ Γ (6 ] Γ (1 [ Γ ( ] [ Γ (3 ] [ Γ (7 ] Γ (15
86 O. S. Odetude et al.: A Decompositio Algorithm for the Solutio of Fractioal Quadratic Riccati Differetial Equatios with Caputo Derivatives The, yx ( ) ca be approximated as 3 5 ( )= x Γ ( x Γ ( Γ ( x ( 1) [ ( 1)] (3 1) Γ [ ( 1)] 3 Γ Γ Γ Γ (3 Γ (5 yx 7 7 [ Γ ( ] Γ (6 x Γ ( Γ (6 Γ ( x [ Γ ( ] [ Γ (3 ] Γ (7 [ Γ ( ] Γ (3 Γ (5 Γ (7 9 9 [ Γ ( ] Γ ( Γ (8 x [ Γ ( ] Γ (6 Γ (8 x 5 5 [ Γ ( ] [ Γ (3 ] Γ (5 Γ (9 [ Γ ( ] [ Γ (3 ] Γ (7 Γ (9 3 11 11 [ Γ ( ] Γ (6 Γ (10 x [ Γ ( ] [ Γ ( ] Γ (10 x (1) 6 3 6 [ Γ ( ] [ Γ (3 ] Γ (7 Γ (11 [ Γ ( ] [ Γ (3 ] [ Γ (5 ] Γ (11 3 13 7 3 [ Γ ( ] Γ ( Γ (6 Γ (1 [ Γ ( ] [ Γ (3 ] Γ (5 Γ (7 Γ (13 15 8 [ Γ ( ] [ Γ (6 ] Γ (1 [ Γ ( ] [ Γ (3 ] [ Γ (7 ] Γ (15 For the particular case =1, Example.: [9] Cosider the fractioal Riccati Differetial Equatio with iitial coditio y (0) = 0. The exact solutio for the case =1 is 3 5 7 9 11 13 15 x x 9x 157x 13x x x yx ( )= x () 3 15 630 1130 51975 185 59535 Table 1. Compariso of Solutios of Example 1 for =1 x Exact Approx. Sol By IDM Error.0 0.0000000000 0.0000000000 0.0000000.1 0.10033671 0.100336713 8.16E-10. 0.07100355 0.0709997 1.0580E-7.3 0.30933696 0.309333 1.9050E-6. 0.793187 0.7777155 1.5500E-5.5 0.5630898 0.56176 8.110E-5.6 0.681368083 0.683805690 3.3110E-.7 0.8883805 0.81190 1.130E-3.8 1.096385570 1.061001110 3.5380E-3.9 1.60158180 1.996690 1.1090E-3 1.0 1.55707750 1.593009690.8110E-3 D * = y ( x) 1, 0 < 1 (3) e yx ( )= e t 1 t 1.
America Joural of Computatioal ad Applied Mathematics 01, (3): 83-91 87 Applyig the iverse operator to both sides of (3), y x = Γ ( 3 Γ ( x y1 ( x)= [ Γ ( 1)] Γ (3 5 7 Γ ( Γ ( x [ Γ ( ] Γ (6 x y( x)= [ ( 1)] 3 (3 1) (5 1) [ ( 1)] [ (3 1)] Γ Γ Γ Γ Γ Γ (7 7 Γ ( Γ ( Γ (6 x y3( x)= [ Γ ( 1)] Γ (3 Γ (5 Γ (7 9 [ Γ ( ] Γ (8 Γ (5 Γ (6 Γ ( Γ (7 5 [ Γ ( ] [ Γ (3 ] Γ (5 Γ (7 Γ (9 11 [ Γ ( ] [ Γ (10 ] Γ ( [ Γ (5 ] Γ (6 [ Γ ( ] Γ (7 6 3 [ Γ ( ] [ Γ (3 ] [ Γ (5 ] Γ (7 Γ (11 3 13 7 3 [ Γ ( ] Γ ( Γ (6 Γ (1 [ Γ ( ] [ Γ (3 ] Γ (5 Γ (7 Γ (13 15 8 [ Γ ( ] [ Γ (6 ] Γ (1 [ Γ ( ] [ Γ (3 ] [ Γ (7 ] Γ (15 () The yx ( ) ca be approximated as 3 5 ( )= x Γ ( x Γ ( Γ ( x ( 1) [ ( 1)] (3 1) Γ [ ( 1)] 3 Γ Γ Γ Γ (3 Γ (5 yx Γ ( Γ (6 Γ ( Γ (5 Γ (3 Γ ( [ Γ ( ] Γ (3 Γ (5 Γ (7 7 9 [ Γ ( ] Γ (8 Γ (5 Γ (6 Γ ( Γ (7 5 [ Γ ( ] [ Γ (3 ] Γ (5 Γ (7 Γ (9 11 [ Γ ( ] [ Γ (10 ] Γ ( [ Γ (5 ] Γ (6 [ Γ ( ] Γ (7 6 3 [ Γ ( ] [ Γ (3 ] [ Γ (5 ] Γ (7 Γ (11 (5) 3 13 7 3 [ Γ ( ] Γ ( Γ (6 Γ (1 [ Γ ( ] [ Γ (3 ] Γ (5 Γ (7 Γ (13 15 8 [ Γ ( ] [ Γ (6 ] Γ (1 [ Γ ( ] [ Γ (3 ] [ Γ (7 ] Γ (15
88 O. S. Odetude et al.: A Decompositio Algorithm for the Solutio of Fractioal Quadratic Riccati Differetial Equatios with Caputo Derivatives Table. Approximate Solutios for Example. for the case = 0.5 ad 1 x Exact =1 Approx. by IDM =1 Approx.VIM [8] =1 Approx. by IDM =0.5 Approx by VIM [8] =0.5.0 0.000000 0.000000 0.000000 0.000000 0.000000.1 0.099667 0.099668 0.099667 0.083613 0.086513. 0.197375 0.197373 0.197375 0.1575 0.16158.3 0.9131 0.9195 0.9130 0.17856 0.3856. 0.37998 0.379933 0.380005 0.30056 0.3153.5 0.6117 0.617 0.6375 0.00376 0.1368.6 0.53709 0.536867 0.53793 0.50976 0.5155.7 0.60367 0.60678 0.606768 0.61969 0.6603.8 0.66036 0.66765 0.669695 0.737509 0.7578.9 0.71697 0.71766 0.78139 0.859919 0.87007 1.0 0.76159 0.778 0.7816 0.979 0.998176 For the particular case =1, 3 5 7 9 11 13 15 x x 17x 38x 31x x x yx ( )= x (6) 3 15 63 835 113000 185 59535 Example.3 Cosider the Fractioal Quadratic Riccati Differetial Equatio [5]. with iitial coditio y (0) = 0 The exact solutio for the case =1 is D yx ( )= yx ( ) y( x) 1, o< 1 (7) 1 ( 1) yx ( ) = 1 tah x log 1 Applyig the iverse operator to both sides of (7) ad usig the iitial coditio give x Takig y ( x)= Γ ( yx ( )= J yx ( ) y( x) Γ (, by the IDM, we have 3 x Γ ( x y1( x)= Γ ( [ Γ ( ] Γ (3 3 x [ Γ ( ] Γ ( Γ (3 y( x)= x (3 1) Γ [ Γ ( ] Γ ( Γ ( 5 Γ ( Γ ( 1 Γ (3 3 [ Γ ( ] Γ (3 Γ (5 6 7 Γ (5 x [ Γ ( ] Γ (6 x [ Γ ( ] Γ (3 Γ (6 [ Γ ( ] [ Γ (3 ] Γ (7 (8)
America Joural of Computatioal ad Applied Mathematics 01, (3): 83-91 89 The, yx ( ) ca be approximated as x yy(xx) = Γ( xx Γ( Γ( xx 3 [Γ( ] Γ(3 xx3 Γ(3 [Γ( ] Γ( Γ(3 [Γ( ] xx Γ( Γ( Γ( Γ( {1 Γ(3 } [Γ( ] 3 xx Γ(3 Γ(5 5 Γ(5 xx 6 [Γ( ] Γ(3 Γ(6 [Γ( ] Γ(6 xx 7 [Γ( ] [Γ(3 ] Γ(7 For the particular case =1, we have X y(x) Exact =1 yy(xx) = xx xx xx3 3 xx 3 15 xx5 xx6 9 xx7 63 Table 3. Approximate Solutios of Example.3 for values of y(x) Approx.IDM =1 y(x) IDM =0.5 y(x) =0.5 [10] y(x) IDM =0.75 y(x) =0.75 [10] 0.0 0 0 0 0 0 0 0.1 0.11095 0.11067 0.967 0.57731 0.33598 0.60 0. 0.1976 0.1778 0.9059 0.9165 0.7051 0.69709 0.3 0.39510 0.39500 1.15389 1.16653 0.6539 0.698718 0. 0.56781 0.566839 1.353673 1.35359 0.899065 0.9319 0.5 0.75601 0.755839 1.83765 1.8633 1.13787 1.13795 0.6 0.953566 0.95309 1.55858 1.559656 1.33075 1.3316 0.7 1.1596 1.1565 1.579990 1.58998 1.70879 1.97600 0.8 1.36363 1.370 1.60039 1.578559 1.61951 1.6303 0.9 1.56911 1.5688 1.671108 1.53008 1.699098 1.739 1.0 1.68998 1.68389 1.779976 1.8805 1.769830 1.7765 Example. Cosider the Fractioal Quadratic Riccati Differetial Equatio [5] with iitial coditio y (0) = 1 The exact solutio for the case =1 is 1 1 Applyig the iverse operator to both sides of (9), x Takig D yx ( )= x y( x), o< 1 (9) t 1 t 3 tj [ 3( ) Γ ( ) J3] Γ yx ( )= t 1 t 3 tj [ ( ) Γ( ) J ] Γ yx ( )=1 J y( x) Γ ( 3) x y ( x)=1, Γ ( 3) (30)
90 O. S. Odetude et al.: A Decompositio Algorithm for the Solutio of Fractioal Quadratic Riccati Differetial Equatios with Caputo Derivatives we have { } y 1 ( x)= J y ( x) 3 = x x Γ ( 5) x Γ ( Γ ( 3) [ ( 3)] Γ Γ (3 3) 3 3 3 x Γ ( x 8x Γ ( 3) x y( x)= Γ ( [ Γ ( ] Γ (3 Γ (3 3) Γ ( Γ ( 3) Γ (3 3) 8 Γ (3 3) x 16 Γ (3 5) x Γ ( Γ ( 3) Γ ( 3) Γ ( 3) Γ ( 3) Γ ( 5) 5 8 Γ ( 5) x 16 Γ ( 5) x [ Γ ( 3)] Γ ( 5) [ Γ ( 3)] Γ (5 5) 5 5 6 8 Γ ( 5) Γ ( 5) x 16 Γ ( 5) Γ ( 7) x Γ ( [ Γ ( 3)] Γ (3 5) Γ (5 5) [ Γ ( 3)] Γ (3 5) Γ (5 7) 6 6 7 8 3 Γ ( 5) Γ (5 7) x 16[ Γ ( 5)] Γ (6 9) x [ Γ ( 3)] Γ ( 3) Γ (3 5) Γ (6 7) [ Γ ( 3)] [ Γ (3 5)] Γ (7 9) (31) (3) The, by the IDM, yx ( ) ca be approximated as For the particular case =1, x x x x yx ( )=1 Γ ( Γ ( 3) Γ ( Γ ( 3) 3 3 3 Γ ( x 8x Γ ( 3) x [ Γ ( ] Γ (3 Γ (3 3) Γ ( Γ ( 3) Γ (3 3) 3 Γ ( 5) x 8 Γ (3 3) x Γ Γ Γ [ Γ ( 3)] Γ (3 3) ( 1) ( 3) ( 3) 16 Γ (3 5) x 8 Γ ( 5) x Γ ( 3) Γ ( 3) Γ ( 5) [ Γ ( 3)] Γ ( 5) 5 5 16 Γ ( 5) x 8 Γ ( 5) Γ ( 5) x [ Γ ( 3)] Γ (5 5) Γ ( [ Γ ( 3)] Γ (3 5) Γ (5 5) 5 6 6 6 16 Γ ( 5) Γ ( 7) x 3 Γ ( 5) Γ (5 7) x [ Γ ( 3)] Γ (3 5) Γ (5 7) [ Γ ( 3)] Γ ( 3) Γ (3 5) Γ (6 7) 7 8 16[ Γ ( 5)] Γ (6 9) [ Γ ( 3)] [ Γ (3 5)] Γ (7 9) 3 5 6 7 8 9 11 1 15 x x x x x x 5x x x x yx ( )=1 x x 3 6 5 18 63 56 756 079 68 59535
America Joural of Computatioal ad Applied Mathematics 01, (3): 83-91 91 x Exact Approx. Solutio By IDM.1 0.11095 0.11065. 0.1976 0.156.3 0.39510 0.39335. 0.56781 0.563330.5 0.75601 0.775.6 0.953566 0.93997.7 1.1596 1.1336.8 1.36363 1.319708.9 1.56911 1.8835 1.0 1.68998 1.68571 5. Coclusios I this paper, the Iteratio Decompositio Method has bee successfully applied to fid approximate solutios of fractioal quadratic Riccati Differetial Equatios. The IDM is effective for Riccati differetial equatios, ad hold very great promise for its applicability to other oliear fractioal differetial equatios. The four examples used idicate the efficiecy ad accuracy of the method for fractioal quadratic Riccati differetial equatios. The results obtaied are i very acceptable agreemet with those obtaied by other kow methods. REFERENCES [1] Abbasbady, S., Homotopy Perturbatio Method for Quadratic Riccati Differetial Equatio ad Compariso with Adomia s Decompositio Method. Appl. Math. Comput., 17(1), 85-90, 006. [] Abbasbady, S. ad Ta, Y., Homotopy Aalysis Method for Quadratic Riccati Differetial Equatios, Comm. Noliear Sci. Num. Simul., 13(3):539-56, 008. [3] Amiikhah, H. ad Hemmatezhad, M., A Efficiet Method for Quadratic Riccati Differetial Equatios, Commu. Noliear Sci. Numer. Simul. 15:835-839, 010. [] Arikoglu, A. ad Ozkol, I., Solutio of Fractioal Differetial Equatios by Usig Differetial Trasform Method, Chaos, Solutos ad Fractab, 3(5):173-181, 007. [5] Baleau, O. Alipous, N. ad Jafari, H., The Berstei Operatioal Matrices for Solvig the Fractioal Quadratic Riccati Differetial Equatios with The Riema-Liouville Derivative, Abstract ad Applied Aalysis, 013. [6] Baleau, D. Diethelm, K., Scalas, E. Triyillo, J. J., Fractioal Calculus: Models ad Numerical methods, World Scietific, 01. [7] Cag, J, Ta, Y., Xu, H. ad Liao, S.J., Series Solutios of Noliear Riccati Differetial Equatios with Fractioal Order, Chaos, Solutos ad Fractals, 0(1), 1-9, 009. [8] Jafari, H. ad Tajadodi, H., he s Variatioal Iteratio Method for Solvig Fractioal Riccati Differetial Equatio, It. J. Diff. Equatios, 010. [9] Li, Y. ad Su, Numerical Solutio of Fractioal Differetial Equatios usig the Geeralized Block Pulse Operatioal Matrix, Compu. Math. Appl., 6(3):106-105. [10] Merda, M., O the solutios of Fractioal Riccati Differetial Equatios with Modified Riema-Liouville Derivative, It. J. Diff. Equatios, 01. [11] Momai, S. ad Shawagfeh, N., Decompositio Method for Solvig Fractioal Riccati Differetial Equatios, Appl. Math. Comput., 18(): 1083-109, 006. [1] Odibat, Z. ad Momai, S., Modified Homotopy Perturbatio Method: Applicatio to Quadratic Riccati Differetial Equatios of Fractioal Order, Chaos, Solutos ad Fractals, 36(1): 167-17, 008. [13] Odibat, Z. ad Momai, S., Applicatio of Variatioal Iteratio Method to Noliear Differetial Equatios of Fractioal Order, It. J. Noliear Sci. Num. Simulatio, 7(1): 7-3, 006. [1] Polluby, I., Fractioal Differetial Equatios, Vol.198 of Mathematics i Sciece ad Egieerig, Academic Press, sa Diego, Calif, USA, 1999. [15] Yuzbasi, S., Numerical Solutios of Fractioal Riccati type Differetial Equatios by Meas of the Berstei Polyomials, Applied Math. Comput., 19(11); 638-633, 013. [16] Reid, W. T., Riccati Differetial Equatios, Vol. 86, Academic Press, New York, NY, USA, 197. [17] Taiwo, O. A. ad Odetude, O. S., Approximatio of Multi-Order Fractioal Differetial Equatios by a Iterative Decompositio Method, Amer. J. Egr. Sci. Tech. Res., 1():1-9, 013. [18] Taiwo, O. A, Odetude, O. S. ad Adekule, Y. A, Numerical Approximatio of Oe Dimesioal Biharmoic Equatios by a Iterative Decompositio Methods, It. J. Math. Sci., 0(1): 37-, 009. [19] Yag, C. ad Hou, J., A Approximate Solutio of Noliear Fractioal Differetial Equatios by Laplace Trasform ad Adomia Polyomials, J. It. Comput. Sci., 10(1): 13-, 013.