Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients

Accepted Manuscript Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients Harendra Singh, H.M. Srivastava

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S0378-4371(19)30468-6 https://doi.org/10.1016/j.physa.2019.04.120 PHYSA 20884

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Physica A

Received date : 10 December 2018 Revised date : 16 February 2019 Please cite this article as: H. Singh and H.M. Srivastava, Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients, Physica A (2019), https://doi.org/10.1016/j.physa.2019.04.120 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

  

A computational method is proposed for the fractional Riccati differential equations. Convergence of proposed method is also established. Accuracy of proposed method is shown by comparing results from existing methods.

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Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients Harendra Singh1 and H. M. Srivastava2,3 1

Department of Mathematics, Post-Graduate College, Ghazipur 233001, Uttar Pradesh, India.

2

Department of Mathematics and Statistics, University of Victoria, British Colombia V8W 3R4, Canada. 3

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China. Abstract: This paper presents a computational method for the approximate solution of arbitrary-order non-linear fractional Riccati differential equations with variable coefficients. Proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the Jacobi polynomials. Convergence analysis of the proposed method is provided. Numerical results for different fractional orders of the Riccati differential equations are discussed. Figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of Jacobi polynomials such as the Legendre polynomials, the Chebyshev polynomials of the second kind, the Chebyshev polynomials of the third kind, the Chebyshev polynomial of the fourth kind, and the Gegenbauer (or ultraspherical) polynomials. Numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods. It is observed that the results from the proposed method are more accurate. Maximum absolute error and the root-mean square error tables are given for all five kinds of polynomials for comparison purposes. Keywords: Fractional-order Riccati differential equations, Jacobi polynomials, Collocation method, Operational matrix method, Convergence analysis, Error analysis.

E-mail addresses: Srivastava)

[email protected] (H. Singh), [email protected] (H.M.

1

1. Introduction Riccati differential equation is an initial value problem which play an important role in various area of science and engineering like diffusion problems, optimal control, robust stabilization and random processes [1, 2]. It also appears in econometric models, stochastic control, linear systems with Markovian jumps, river flows and dynamic games [3-5]. Furthermore, the solitary wave solution of nonlinear partial differential equations and one dimensional static Schrodinger equation which are important models of physics are also related to Riccati differential equation of integer order [6, 7]. The Riccati differential equation of arbitrary integer order with variable coefficients is given as: p(t ) D y(t )  q(t ) y(t )  r (t ) y 2 (t )  h(t ),

t [0,1]

(1)

with initial condition: D m y (0)   m ,

where, 

m  0,1, 2,....., M  1. M  

is a positive integer,

(2)

p (t ), q (t ), r (t ) and h(t ) are continuous function on

[0,1], p(t )  0  t  [0,1], and  m are constant.

Fractional calculus has many real life applications like as viscoelasticity [8, 9], chemical engineering [10], signal processing [11] and fluids mechanics [12]. Our physical problem is dependent on the initial conditions and it is known that fractional derivatives are non-local in nature. In addition, there are many classical and modern dynamical systems which are modelled by fractional order Riccati differential equation [13-15]. Therefore for handling the initial conditions of our physical problem and better understand of classical and modern dynamical systems it is essential requirement to replace integer order Riccati differential equation to fractional order Riccati differential equation. The fractional order Riccati differential equation is obtain by replacing integer order derivative by fractional order derivative and is given as: p(t ) D y(t )  q(t ) y(t )  r (t ) y 2 (t )  h(t ),

  0, t [0,1]

(3)

with initial condition:

Dm y(0)   m ,

m  0,1, 2,....., M 1. M    2

(4)

Some analytical and numerical techniques have been developed for the approximate solution of fractional Riccati differential equations. Analytical techniques are like homotopy perturbation method [16], optimal homotopy asymptotic method [17] and modified homotopy perturbation method [18]. In [19], authors obtained series solution using homotopy asymptotic method for Riccati differential equations. In [20], Dehghan and Lakestani used two numerical techniques based on operational matrix method for the approximate numerical solution of fractional Riccati differential equations. In [21], authors proposed iterative decomposition algorithm to solve Riccati differential equations with variable coefficients. A new technique which is based on the Taylor series expansion is used to solve these equations numerically [22]. Batiha et al. solved general Riccati differential equations using variational iteration method [23]. Generalized Riccati differential equations are solved by Rao using Adomian decomposition method (ADM) [24]. A comparative study of all developed methods for Riccati differential equations is given in [25]. In 2016, Kashkari and Syam [26] solved fractional Riccati differential equations using fractional operational matrix of Legendre polynomials. Recently in 2017, iterative reproducing kernel Hilbert spaces method is used to solve numerically Riccati differential equations [27]. In present article, we have proposed a computational method for the approximate solution of arbitrary order fractional Riccati differential equations with variable coefficients. Present method is a grouping of operational matrix method and collocation method. Jacobi polynomials are used as a basis for this method. Operational matrix method is a very powerful method for the solution of differential equations [28-35]. In this method first we take the finite dimensional approximation of unknown function by using Jacobi polynomials. Using this approximations along with operational matrix of integration, the residual function for the fractional Riccati differential equations is obtained. Now collocating the residual function at the required number of collocation points we obtained a system of nonlinear algebraic equations. On solving this system, we obtained unknowns in the approximations and finally the approximate solution for the fractional Riccati differential equations with variable coefficients. Some more numerical and analytical methods to solve various linear and nonlinear fractional model governed by differential equations can be found in [36-48]. Convergence analysis of proposed method is also established. Applicability and accuracy of the proposed method are shown by comparing the obtained results from existing analytical and numerical methods. Numerical results are shown through tables and figures for all five kinds polynomials namely Legendre polynomial, Chebyshev polynomial of second kind, 3

Chebyshev polynomial of third kind, Chebyshev polynomial of fourth kind, Gegenbauer polynomial. 2. Preliminaries and operational matrix of integration Lemma 2.1[49, 50]: If n  1    n, n  N , and y(t )  L2 [0,1] then D I  y (t )  y(t ) and n 1

I  D y (t )  y (t )   y ( k ) (0 ) k 0

tk , k!

t  0.

The shifted Jacobi polynomial of degree i on [0, 1] is given as, i

i(u ,v ) (t )   (1)i k k 0

(i  v  1)(i  k  u  v  1) tk . (k  v  1)(i  u  v  1)(i  k )!k !

(5)

which satisfies the following relation: 1



( u ,v ) n1

(t )n(2u ,v ) (t ) ( u ,v ) (t )dt  lnu1,v n1n2 ,

(6)

0

where  (u ,v ) (t )  t v (1  t )u ,  n1n2 is Kronecker delta function and lnu1,v n1n2 

(n1  u  1)(n1  v  1) . (2n1  u  v  1)(n1 )!(n1  u  v  1)

(7)

A function y (t )  L2 [0,1] , can be expanded as, n

y (t )  lim  d k k(u , v ) (t ), n 

(8)

k 0

where, d k 

1 lku ,v

1



( u ,v ) k

(t ) y (t ) ( u ,v ) (t )dt ,

k  0,1, 2,......

(9)

0

Truncating the series at n  m, we get, m

y   d kk( u ,v ) (t )  d T m (t ) ,

(10)

k 0

where, d and  m (t ) are ( m  1) 1 matrices given by

4

d  [d0 , d1 ,...., d m ]T and m (t )  [0(u ,v ) (t ),1(u ,v ) (t ),....,m(u ,v ) (t )]T .

(11)

Theorem 2.1: Let m (t )  [0(u ,v ) (t ),1(u ,v ) (t ),....,m( u ,v ) (t )]T , be the Jacobi vector consisting of the Jacobi polynomials and consider   0 , then

I i(u ,v ) (t )  I ( )m (t ), where, I

( )

(12)

   (i, j)  , is (m  1)  (m  1) Jacobi operational matrix of integration and its

entries are given by i

 (i, j, u, v)   ( 1)i  k k 0

j

(i  v  1)(i  k  u  v  1) (k  v  1)(i  u  v  1)(i  k )!(k    1)

  (1) j l l 0

( j  l  u  v  1)(u  1)(k  l    v  1)(2 j  u  v  1) j ! ( j  u  1)(l  v  1)( j  l )!(l )!(l  k  u  v    2)

(13)

0  i, j  m.

Proof: Pl. see [51-53]. 3. Method of solution In this section we present computational method for the approximate solution of arbitrary order fractional Riccati differential equation. Let us consider the following approximations for the arbitrary order fractional Riccati differential equation

D y (t )  w(t ).

(14)

Taking Riemann-Liouville integral of order  on both side of Eq. (14) and using Lemma 2.1, we get M 1

y(t )   D m y (0) m0

tm  I  w(t ). m!

(15)

From Eqs. (14) and (15), we can write,

y(t )  I  w(t )  a(t ), M 1

where,

 Dm y(0)

m0

(16)

tm  a(t ). m! 5

Using Eqs. (14) and (16) in Eq. (3), we get p(t ) w(t )  q(t )  I  w(t )  a(t )   r (t )  I  w(t )  a(t )   h(t ). 2

(17)

Eq. (17), can be written as p(t ) w(t )  q(t ) I  w(t )  r (t )  I  w(t )  a(t )   q(t )a(t )  h(t )  0. 2

(18)

Eqs. (3) and (18) are equivalent. Further taking the following approximations using Jacobi polynomials, w(t )  d T n (t ) ,

(19)

a (t )  AT n (t ),

(20)

q(t )a(t )  h(t )  BT n (t ).

(21)

Using operational matrix of integration and Eqs. (19), (20) and (21) in Eq. (18), we get following expression p(t )d T n (t )  q(t )d T I ( )n (t )  r (t )  d T I ( )n (t )  AT n (t )   BT n (t )  0, 2

(22)

where, I ( ) is Jacobi operational matrix of integration of order (n  1)  (n  1). The residual for Eq. (22), is defined as Rn (t )  p(t )d T n (t )  q(t )d T I ( )n (t )  r (t )  d T I ( )n (t )  AT n (t )   BT n (t ). 2

(23)

The unknown coefficients in the approximations can be obtained by solving a set of nonlinear algebraic equations which are obtained by collocating the residual functions at n  1 collocation points. So collocating the residual functions at n  1points, Rn (ti )  0,

ti 

i , n

i  0,1,....., n.

(24)

From Eq. (24), we obtain a set of n  1 nonlinear algebraic equations. On solving these equations we obtain unknown coefficients and hence approximate solution for arbitrary order nonlinear fractional Riccati differential equation with variable coefficients.

6

4. Convergence analysis Theorem 4.1. Let the function y :[0,1]  R and y  C ( m1) [0,1] and ym (t ) be the mth approximation of the function from qm(u ,v ) (t )  span {0(u ,v ) (t ),1(u ,v ) (t ),....,m(u ,v ) (t )} then em  y  ym

 ( u ,v )



M (1  u )(3  2m  v) , (m  1)! (4  2m  u  v)

(25)

where, M  max y ( m 1) (t ) .

(26)

t[0,1]

Proof: Pl. see [54-56]. Theorem 4.2. Let the error function of Riemann-Liouville fractional integral operator for Jacobi polynomials ei,m, :[ y0 ,1]  R is m -times continuously differentiable for y0  0 and

  m, then the error bound is given as follows: m

 ,

ei.m

 d( u ,v )

 d  i  v y0u     B m ! (  m  1)  y0   i 

 u  1, v  1 ,

(27)

where d  max  y0 ,1  y0 . Proof: Pl. see [57]. Theorem 4.3. If eI ,,m be the error vector for the operational matrix of integration of order  obtained by using ( m  1) Jacobi polynomials. Then

eI ,,m  I ( )m ( x)  I m ( x),

(28)

and the error vector in Eq. (28) tends to zero as m . Proof: Followed from Theorem 4.2. 2 Set of orthogonal polynomials on [0, 1] act as a basis for L [0, 1] .



dimensional subspace generated by i(u ,v ) on Wn as a linear combination of



 

Let Wn be the n -

2

0 i  n

( u ,v ) i 0 i  m

for L [0, 1] . So, we can write every functional

. The scalars in the linear combinations can be

7

chosen in such way that the functional minimizes. Let the minimum value of the functional on the space Wn be denoted by  n . From the construction of Wn and  n we can write,

Wn  W( n 1) and  n1   n . Theorem 4.4. Consider the functional L , then

lim  n (t )   (t )  inf L(t ) 2

(29)

tL [0, 1]

n 

Proof: Pl. see [58]. From Eq. (18), we can write L(t )  p(t ) w(t )  q(t ) I  w(t )  r (t )  I  w(t )  a (t )   q (t )a (t )  h(t ). 2

(30)

Using Eqs. (19), (20) and (21) in Eq. (30), we get L( e1 ) (t )  p(t )  d T n (t )  ed ,n   q(t )  d T I  n (t )  ed ,n 



 r (t )  d T I  n (t )  ed ,n   AT n (t )  eA,n



2

 BT n (t )  eB ,n ,

(31)

where,

ed ,n  d T  (t )  d T n (t ),

(32)

eA,n  AT  (t )  AT n (t ),

(33)

eB,n  BT  (t )  BT n (t ),

(34)

Using operational matrix of integration in Eq. (31), we obtain L( e1 ,e2 ) (t )  p(t )  d T n (t )  ed ,n   q(t )  d T I ( )n (t )  d T eI ,,n  ed ,n 



 r (t )  d T I ( )n (t )  d T eI ,,n  ed ,n   AT n (t )  eA,n



2

 BT n (t )  eB ,n ,

(35)

where,

eI ,,n  I n (t )  I ( )n (t ).

(36)

Now the residual function for Eq. (36), is given as 8

Rn( e1 ,e2 ) (t )  p(t )  d T n (t )  ed ,n   q(t )  d T I ( )n (t )  d T eI ,,n  ed ,n 



 r (t )  d T I ( )n (t )  d T eI ,,n  ed ,n   AT n (t )  eA,n



2

 BT n (t )  eB ,n ,

(37)

Now similar as in Eq. (23), collocating Eq. (37) at n  1points Rn( e1 ,e2 ) (ti )  0,

ti 

i , n

i  0,1,....., n.

(38)

From Eq. (38), we obtain a set of n  1 nonlinear algebraic equations. On solving these equations we obtain approximate solution for nonlinear fractional Riccati differential equations. Let this solution is denoted by  n* (t ) . Since p (t ), q (t ) and r (t ) are bounded continuous function on [0,1] . Using Theorem 4.1, 4.3 ( e ,e ) and taking n   , in Eq. (37), the functional L 1 2 (t ) in Eq. (35) comes close to the

functional L (t ) in Eq. (30). So for the large values of n ,

 n* (t )   n (t ) .

(39)

From Theorem 4.4 and Eq. (39), we conclude that

lim  n* (t )   (t ) n

Proof completed. 5. Numerical experiments and discussion In this section, we examine the accuracy and applicability of our proposed computational method by applying it on three test examples. We discuss Riccati differential equations with constant and variables coefficients. Accuracy of method is shown from absolute error and root mean square error tables for test examples. We have also compared the results from some existing numerical and analytical methods and it is observed that results from our method are more accurate. In figs. box  is denoted by alpha. In Table 1, given below we have listed the notations used for the different particular cases of Jacobi polynomials. Table 1 Example 1. Consider the following fractional Riccati differential equation [18, 21, 23, 24, 27]:

D y(t )  1  y 2 (t ),

t [0,1], 0    1,

(40) 9

y (0)  0.

(41)

The exact solution of Eq. (40), for integer order   1 , is given as:

y (t ) 

e2t  1 . e2t  1

(42)

The residual function of Eqs. (40) and (41), is given as Rn (t )  d T n (t )   d T I ( )n (t )   BT n (t ). 2

(43)

The collocation points for Eq. (43), is given by taking n  10. Rn (ti )  0,

ti 

i , n

i  0,1,.....,10.

(44)

The approximating coefficients obtained by solving Eqs. (43) and (44) at   1 , is given in Appendix A. We have discussed Example 1 using different orthogonal polynomials P1, P2, P3, P4 and P5. We have plotted the behaviour of approximate solution for each polynomials taking different value of   0.8, 0.9 and 1. Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Figs. 1-5, show the behaviour of approximate solution at   0.8, 0.9,1 and exact solution for P1, P2, P3, P4 and P5 respectively. From theses figs. it is observed that the behaviour of approximate solution is same for P1, P2, P3, P4 and P5. Further, it is also clear that the solution varies continuously for fractional order and coincide with exact solution at integer order derivative   1. In Fig. 6, given below we have plotted the absolute error of Example 1, for P1, P2, P3, P4 and P5. Fig. 6

10

In Table 2, given below we have compared the numerical results from our method to Adomian decomposition method (ADM), variational iteration method (VIM), iterative decomposition algorithm (IDA), modified homotopy perturbation method (MHPM), iterative reproducing kernel Hilbert spaces method (IRKHSM). Table 2 From Table 2, it is observed that the results from our proposed method is more accurate in compare to ADM [24], VIM [23], IDA [21], MHPM [18] and IRKHSM [27]. Example 2. Consider the following fractional Riccati differential equation [17, 18, 23, 27]:

D y(t )  1  2 y(t )  y 2 (t ),

t [0,1], 0    1,

(45)

y (0)  0.

(46)

The exact solution of Eq. (45), for integer order   1 , is given as:





 log 1  2 / 1  2 y (t )  1  2 tanh  2t   2 

  .  

(47)

The residual function of Eqs. (45) and (46), is given as Rn (t )  d T n (t )  2d T I ( )n (t )   d T I ( )n (t )   BT n (t ). 2

(48)

The collocation points for Eq. (48), is given by taking n  10. Rn (ti )  0,

ti 

i , n

i  0,1,.....,10.

(49)

The approximating coefficients obtained by solving Eqs. (48) and (49) at   1 , is given in Appendix B. We have discussed Example 2 using different orthogonal polynomials P1, P2, P3, P4 and P5. We have plotted the behaviour of approximate solution for each polynomials taking different values of   0.8, 0.9 and 1. Fig. 7 Fig. 8 Fig. 9 11

Fig. 10 Fig. 11 Figs. 7-11, show the behaviour of approximate solution at   0.8, 0.9,1 and exact solution for P1, P2, P3, P4 and P5 respectively. From theses figs. it is observed that the behaviour of approximate solution is same for P1, P2, P3, P4 and P5. Further, it is also clear that the solution varies continuously for fractional order and coincide with exact solution at integer order derivative

  1. In Fig. 12, given below we have plotted the absolute errors of

Example 2, for P1, P2, P3, P4 and P5. Fig. 12 In Table 2, given below we have compared the numerical results from our method to VIM, Optimal homotopy asymptotic method (OHAM), MHPM and IRKHSM. Table 3 From Table 3, it is observed that the results from our proposed method is more accurate in compare to VIM [23], OHAM [17], MHPM [18] and IRKHSM [27]. Example 3. Consider the following fractional Riccati differential equation with variable coefficients [24, 27]:

D y(t )  t 3 y 2 (t )  2t 4 y(t )  t 5  1,

t [0,1], 0    1,

y (0)  0.

(50) (51)

The exact solution of Eq. (50), for integer order   1 , is given as: y (t )  t.

(52)

The residual function of Eqs. (50) and (51), is given as Rn (t )  d T n (t )  t 3  d T I ( )n (t )   2t 4 d T I ( )n (t )  BT n (t ). 2

(53)

The collocation points for Eq. (53), is given by taking n  10. Rn (ti )  0,

ti 

i , n

i  0,1,.....,10.

(54)

From Eqs. (16) and (19), in this case we can write,

12

y (t )  d T I ( )n (t ).

(55)

We have discussed Example 3 using different orthogonal polynomials P1, P2, P3, P4 and P5. Case 1: Using P1 the approximating coefficients for integer order derivatives are given as d T  [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

1 1 d T I (1)  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]. 2 2

Using these values in Eq. (55), approximate solution is given as  1    2t  1   1 1   t, y (t )  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]  . 2 2    .   (0,0) (t )   10 

which is the exact solution. Case 2: Using P2 the approximating coefficients for integer order derivatives are given as d T  [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

1 1 d T I (1)  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]. 2 3

Using these values in Eq. (55), approximate solution is given as  1    3t  3  2   1 1   t, y (t )  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]  .   2 3  .   ( 1,1 )  102 2 (t )   

which is the exact solution. Case 3: Using P3 the approximating coefficients for integer order derivatives are given as d T  [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

1 1 d T I (1)  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]. 4 2

Using these values in Eq. (55), approximate solution is given as

13

 1     2t  1  2   1 1    t, y (t )  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0] .   4 2  .   ( 1 , 1 )  102 2 (t )   

which is the exact solution. Case 4: Using P4 the approximating coefficients for integer order derivatives are given as d T  [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

3 1 d T I (1)  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]. 4 2

Using these values in Eq. (55), approximate solution is given as  1     2t  3  2   3 1   t, y (t )  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]  .   4 2  .   ( 1, 1 )  10 2 2 (t )   

which is the exact solution. Case 5: Using P5 the approximating coefficients for integer order derivatives are given as d T  [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

1 1 d T I (1)  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]. 2 4

Using these values in Eq. (55), approximate solution is given as  1    4t  2   1 1 y (t )  [ , , 0, 0, 0, 0, 0, 0, 0, 0, 0]  .   t , 2 4    .   (1,1) (t )   10 

which is the exact solution.

14

We have plotted the behaviour of approximate solution for each polynomials taking different values of   0.8, 0.9 and 1. Fig. 13 Fig. 14 Fig. 15 Fig. 16 Fig. 17 Figs. 13-17, show the behaviour of approximate solution at   0.8, 0.9 and 1 for P1, P2, P3, P4 and P5 respectively. From theses figs. it is observed that the behaviour of approximate solution is same for P1, P2, P3, P4 and P5. Further, it is also clear that the solution varies continuously from fractional order to integer order. From the figures of absolute errors, it is clear that approximations from Legendre polynomial and Gegenbauer polynomial are more accurate in compare to other polynomials. Further, from the figures it is also clear that solution varies continuously for all polynomial and coincide to exact solution. In Table 4, given below we have listed maximum absolute error for all the test examples, using P1, P2, P3, P4 and P5. Table 4 From Table 4, it is observed that the results from P1 are more accurate in compare to P2, P3, P4 and P5. In Table 5, given below we have listed root mean square errors for all the test examples, using P1, P2, P3, P4 and P5. Table 5 From Table 5, it is observed that the results from P1 are more accurate in compare to P2, P3, P4 and P5. The computational order for the numerical results are given as [59, 60]:

E  Order= log 2  n  where En is maximum absolute error (max E ( xi )) for approximation 1i  N  E2 n  having n number of basis elements. In table 6, given below, we have listed the computational order for the proposed method (P1). [Table 6] 15

From table 6 it is clear that computational order of proposed method is excellent. In table 7 we have listed the CPU time taken for different polynomials. [Table 7] From table 7 it is observed that P1 and P5 are easier in compare to others for computational purpose. Conclusions: In this article, we have successfully applied our proposed method for the approximate solution of fractional Riccati differential equations with variable coefficients. Our proposed method is applicable for arbitrary order fractional Riccati differential equations. Convergence of solution is also shown. From Tables 2 and 3, it is clear that our proposed method is more accurate in compare to existing analytical and numerical methods like ADM, OHAM, VIM, IDA, MHPM and IRKHSM. In case, where exact solution is polynomial (Example 3), solution from our proposed method is same as the exact solution. A comparative study is made between five orthogonal polynomials. From Tables 4 and 5, it is clear that approximations from Legendre polynomial and Gegenbauer polynomial are more accurate in compare to Chebyshev second, third and four kind polynomials. This method can be applied in any bounded domain like [0, k ] instead of [0,1]. In future, we can use operational matrices of wavelets to achieve better accuracy. Acknowledgements The author is very grateful to the referees for their constructive comments and suggestions for the improvement of the paper. Appendix A: The unknown coefficients of Eq. (43) using P1 for n  10 and

  1 are given as

d T = [0.7615941535314235926553233384458, -0.3179025329047892164468393805186, -0.050648510134499671612666459983469, 0.02895957032384377986270819355185, -0.0011139537859852480434172405175924, -0.0010905010491249651985619146475105, 0.00016272488759292586835403790650545, 0.000020486762241883779558764358246292, -0.000007457743070547814010498558326188, 0.00000015301221713325441329525538058549, 0.00000021938892753016826306957873565706]. The unknown coefficients of Eq. (43) using P2 for n  10 and

  1 are given as

d T = [0.76794184395630229554466765444326, -0.216733405906167970434087643451, 16

-0.030185488717086587670056116736414, 0.016765655611806055427383856753897, -0.00065246266993065768233308376533954, -0.0005992153926349346404109114107316, 0.000089242569851047395304215440175302, 0.000010892381381017587501342502177053, -0.0000039971162628101562809807411129614, 0.000000081557197591046456503290111645905, 0.00000011552062833603707653230744691253]. The unknown coefficients of Eq. (43) using P3 for n  10 and

  1 are given as

d T = [0.90243810090799859577472062268073, -0.26899251355125303139670841681929, -0.074810125584208647872601946180281, 0.02940117266185572571543042506714, -0.000070026100951466661103937132466844,-0.0012271192389036638985315600201853, 0.00014024752606044218894492436572331, 0.000027448899432252051552347531561719, -0.0000074937268578443551214766535441335,-0.000000059712548666079647394425195304333, 0.00000022546381462428398140388832760526]. The unknown coefficients of Eq. (43) using P4 for n  10 and

  1 are given as

d T = [0.59514334096454946001274065995354, -0.34559700523437099679299419933528, -0.027329192002665407467298858997143, 0.027575945787480056347875491715711, -0.002015938543744152887770656586628,-0.00093500775988113712950542326150725, 0.00017842820298469416464174391835834, 0.000013667486920976938813685059220067, -0.0000072063775466534929523457636809959, 0.00000036035959000798797181837359784753, 0.00000021138550209233230062660408632695].

The unknown coefficients of Eq. (43) using P5 for n  10 and

  1 are given as

d T = [0.77172385558523971564378604524218,-0.16515688862890020269090194491006, -0.020011858797928866456754314959567, 0.01059054490609704082055636717514, -0.00040886986768516360109177065421815,-0.00035175775928037939783113413181323, 0.000051823919985966091097530612591002, 0.0000061097479913904794222782464568038, -0.0000022453532862534688042201415981949, 0.000000045327661911279294981002987825081, 0.000000063450736608155854290745468402055].

Appendix B: The unknown coefficients of Eq. (48) using P1 for n  10 and

17

  1 are given as

d T = [1.6894987212295046521974875309814, 0.3336708540917532377999842794373, -0.46770491775717638469567681136361, -0.080063155276222466300479003445362, 0.043186853507546861798976670427396, 0.0094448675063856723344834140240263, -0.0028266073931259795205671463731739, -0.00082393201205142061865448205894203, 0.00014787648339170258601915364300259, 0.00006716737223828963856128106196907, -0.0000064272565117271852718280994277941]. The unknown coefficients of Eq. (48) using P2 for n  10 and

  1 are given as

d T = [1.7473005300941204033615295518397, 0.23555238935455551076025544698307, -0.28862950986981831875970763934345,-0.047609410779683806042217951541881, 0.024575632103073432930585782561675, 0.0053254110747465107227405336080849, -0.0015540783298249010271284126444956,-0.00045435025545776031508291348664807, 0.000079684540019204732618116214178849, 0.000035368476856923150856047617176217, -0.0000034270762114946452196170857434528] The unknown coefficients of Eq. (48) using P3 for n  10 and

  1 are given as

d T = [1.4255158803329253873749520173968, 0.64356933100319879852717298562858, -0.38698790164366433233422996373357,-0.11287358849810106301055999527263, 0.03377930947084394675587407522194, 0.011618670627172190144431996831021, -0.0020243589920319995456791298291291,-0.00092809853441021484657742760908439, 0.000081252142863505534872050036029006, 0.000073343105626916168106708767126124, -0.0000064218761591272086400274980299745].

The unknown coefficients of Eq. (48) using P4 for n  10 and

  1 are given as

d T = [1.731245298766505731037663752619, -0.032110368403808770807057923614973, -0.51391931924984059049407333070151, -0.039444368038698400385934254000257, 0.050138851482224790713270925601557, 0.0065584301501617894774842737761283, -0.0034969173960706535286859634522143, -0.00065780158944968108269957314790857, 0.00020674137542887690360722138104032, 0.000060418050688321709187131361784964, -0.000007625788919179335769849952911218]. The unknown coefficients of Eq. (48) using P5 for n  10 and

  1 are given as

d T = [1.783039702291855331059927357039, 0.18399182241854252833145234498981, 18

-0.196679042705071266231805328903, -0.030740548501569720123185235491996, 0.015047914323294096592169532048462, 0.0031974324694437143552844748076458, -0.00090451712848365468694506573366991,-0.00026309050758302981366278131726339, 0.000044991960731987075925632055180665, 0.000019458740172173289767307948735428, -0.0000018713352082471293992328534512494].

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Figure Captions Fig. 1 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P1. Fig. 2 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P2. Fig. 3 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P3. Fig. 4 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P4. 23

Fig. 5 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P5. Fig. 6 Comparison of absolute errors for P1, P2, P3, P4 and P5 at   1. Fig. 7 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P1. Fig. 8 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P2. Fig. 9 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P3. Fig. 10 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P4. Fig. 11 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P5. Fig. 12 Comparison of absolute errors for P1, P2, P3, P4 and P5 at   1. Fig. 13 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P1. Fig. 14 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P2. Fig. 15 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P3. Fig. 16 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P4. Fig. 17 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P5.

24

Figure

Fig. 1 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P1.

Fig. 2 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P2.

Fig. 3 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P3.

Fig. 4 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P4.

Fig. 5 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P5.

Fig. 6 Comparison of absolute errors for P1, P2, P3, P4 and P5 at   1.

Fig. 7 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P1.

Fig. 8 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P2.

Fig. 9 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P3.

Fig. 10 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P4.

Fig. 11 Behaviour of exact solution and approximate solution for different values of   0.8, 0.9,1 using P5.

Fig. 12 Comparison of absolute errors for P1, P2, P3, P4 and P5 at   1.

Fig. 13 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P1.

Fig. 14 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P2.

Fig. 15 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P3.

Fig. 16 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P4.

Fig. 17 Behaviour of approximate solution for different values of   0.8, 0.9 and 1, using P5.

Table

Table 1 Particular cases of Jacobi polynomials and their notations are as follows:

u

v

0

Scheme 0

1

1

Legendre polynomial

Notation P1

Chebyshev polynomial of second kind

P2

Chebyshev polynomial of third kind

P3

Chebyshev polynomial of fourth kind

P4

Gegenbauer polynomial

P5

Table 2 Comparison of absolute error from our method (using P1) and from ADM, VIM, IDA, MHPM and IRKHSM.

t

Present method

ADM [24]

VIM [23]

IDA [21]

MHPM [18]

IRKHSM [27]

0.1

4.57E-9

8.82E-14

5.00E-11

1.00E-11

0.00E-0

9.05E-6

0.2

9.74E-10

1.78E-10

4.39E-9

0.00E-10

0.00E-0

1.72E-5

0.3

3.71E-9

1.51E-8

1.56E-7

2.50E-9

1.00E-6

2.38E-5

0.4

1.29E-9

3.49E-7

1.97E-6

5.61E-8

5.00E-6

2.85E-5

0.5

1.93E-9

3.92E-6

1.38E-5

6.03E-7

3.90E-5

3.11E-5

0.6

2.74E-9

2.80E-5

6.61E-5

4.09E-6

1.93E-4

3.17E-5

0.7

4.32E-9

1.46E-4

2.43E-4

2.01E-5

7.37E-4

3.07E-5

0.8

2.43E-9

6.04E-4

7.35E-4

7.78E-5

2.33E-3

2.81E-5

0.9

3.59E-10

2.09E-3

1.91E-3

2.50E-4

6.37E-3

2.32E-5

7.01E-9

6.30E-3

4.42E-3

6.99E-4

1.55E-2

1.19E-5

1

Table 3 Comparison of numerical results from our method (using P1) and results from VIM, OHAM, MHPM and IRKHSM.

t

Present method

VIM [23]

OHAM [17]

MHPM [18]

IRKHSM [27]

0.1

7.45E-7

1.98E-8

3.20E-5

1.00E-6

3.58E-5

0.2

8.51E-7

1.03E-6

2.90E-4

1.20E-5

7.58E-5

0.3

9.30E-7

8.85E-6

1.10E-3

1.00E-6

1.20E-4

0.4

1.08E-6

3.33E-5

2.50E-3

3.03E-4

1.66E-4

0.5

1.14E-6

7.26E-5

4.40E-3

1.55E-3

2.12E-4

0.6

1.14E-6

9.98E-5

5.50E-3

4.69E-3

2.52E-4

0.7

1.21E-6

8.84E-5

5.50E-3

1.05E-2

2.87E-4

0.8

1.04E-6

1.54E-5

3.80E-3

1.88E-2

3.40E-4

0.9

1.13E-6

4.99E-4

3.20E-3

2.80E-2

4.90E-4

1

4.84E-7

3.47E-3

3.40E-3

3.43E-2

9.22E-4

Table 4 Comparison of maximum absolute error.

Example no.

P1

P2

P3

P4

P5

1

7.0082E-9

1.2388E-8

2.2427E-8

3.2836E-8

2.1356E-8

2

1.2058E-6

1.1735E-6

1.2609E-6

3.8936E-6

1.1794E-6

3

0.0000

0.0000

0.0000

0.0000

0.0000

Table 5 Comparison of root mean square error.

Example no.

P1

P2

P3

P4

P5

1

5.6323E-10

7.3793E-10

4.6208E-10

1.8150E-9

9.1279E-10

2

2.2114E-7

2.1899E-7

2.2873E-7

2.7065E-7

2.2077E-7

3

0.0000

0.0000

0.0000

0.0000

0.0000

Table 6 Computational order of proposed method (P1) for different examples.

Example No.

1

2

Computational order 2

------

4

6.35476

8

8.14034

2

------

4

4.28494

8

9.10757

Table 7

CPU time for different examples using proposed method for

Example No.

P1 Time (m)

P2 Time (m)

P3 Time (m)

.

P4 Time (m)

P5 Time (m)

1

39.54

44.81

69.28

49.53

37.92

2

39.41

38.09

48.63

40.49

52.88

3

59.90

42.16

57.62

40.16

38.07