Tau-Path Following Method for Solving the Riccati Equation with Fractional Order

Recently, many papers on fractional boundary value problems have been studied extensively. Several forms of them have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solutions. Several numerical techniques are used to solve such problems such as Laplace and Fourier transforms [1, 2], Adomian decomposition and variational iteration methods [3, 4], eigenvector expansion [5], differential transform and finite differences methods [6, 7], power series method [8], collocation method [9], and wavelet method [10, 11]. Many applications of fractional calculus on various branches of science such as engineering, physics, and economics can be found in [12, 13]. Considerable attention has been given to the theory of fractional ordinary differential equations and integral equations [14, 15]. Additionally, the existence of solutions of ordinary and fractional boundary value problems using monotone iterative sequences has been investigated by several authors [16–20]. We consider the Riccati equationwith fractional orders of the form


Introduction
Recently, many papers on fractional boundary value problems have been studied extensively.Several forms of them have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solutions.Several numerical techniques are used to solve such problems such as Laplace and Fourier transforms [1,2], Adomian decomposition and variational iteration methods [3,4], eigenvector expansion [5], differential transform and finite differences methods [6,7], power series method [8], collocation method [9], and wavelet method [10,11].Many applications of fractional calculus on various branches of science such as engineering, physics, and economics can be found in [12,13].Considerable attention has been given to the theory of fractional ordinary differential equations and integral equations [14,15].Additionally, the existence of solutions of ordinary and fractional boundary value problems using monotone iterative sequences has been investigated by several authors [16][17][18][19][20].
In this paper we study the Tau-path following method for solving the Riccati equation with fractional order.We organize this paper as follows.In Section 2, we present basic definitions and results of fractional derivatives.We extend basic results to path following method for the fractional case.In Section 3, we introduce the fractional-order Legendre Tau method with path following method for solving the Riccati equation with fractional order.In Section 4, we present some numerical results to illustrate the efficiency of the presented method.Finally, we conclude with some comments in the last section.

Preliminaries
In this section, we review the definition and some preliminary results of the Caputo fractional derivatives, as well as, the definition of the fractional-order Legendre functions and their properties.
The basic concept of this paper is the Legendre polynomials.For this reason, we study some of their properties.
Among the properties of the Legendre polynomials we list the following properties: (1) ∫ In order to use these polynomials on the interval [0, 1], we define the shifted Legendre polynomials by   () = L  (2 − 1).Using the change of variable  = 2 − 1,   () has the following properties: (1) ∫ The analytic closed form of the shifted Legendre polynomials of degree  is given by One of the common and efficient methods for solving fractional differential equations of order  > 0 is using the series expansion of the form ∑  =0     .For this reason, we define the fractional-order Legendre function by    () =   (  ).Using the properties of the shifted Legendre polynomials and the change of variable  =   , it is easy to show that )    (0) = (−1)  and    (1) = 1.In addition, {   () :  = 0, 1, 2, . ..} are orthogonal functions with respect to the weight function () =  −1 on (0, 1) with The closed form of    () is given by Using properties (4) and ( 5) of the Caputo fractional derivative, one can see that In the next theorem, we state one of the main properties of the fractional Legendre functions which will be used later in this paper.Theorem 4. For any nonnegative integers  and , where Proof.For any nonnegative integers  and , where For the proof of this case, see [25].Using the change of variables  = 2 − 1 and  =   , we obtain the result of the theorem.
Another important result which will facilitate applying the Tau method for fractional case is given in Theorem 5.

Fractional-Order Legendre
Tau-Path Following Method In this section, we present a numerical method for solving problem (1)- (2).We use the fractional-order Legendre Tau method to discretize problem (1)-( 2).The result is a nonlinear system.The initial guess that is used in the standard methods for solving the produced nonlinear system is one of the challenges.To overcome this problem, we use the path following method.Approximate the solution (), (), (), (), and () in terms of the fractional-order Legendre functions as follows: Thus,   () can be approximated by where  ()  is given by Theorem 6.For   , the residual is given by which can be written as where   = 0,   = 0,   = 0,   = 0 for  ≥  + 2, and  ()  = 0 for  ≥  + 1.Using Theorem 4, we can rewrite the residual as Since we are interested in the first  terms only, we ignore higher order terms and we rewrite the residual as Orthogonalize the residual with respect to the fractionalorder Legendre function as follows where () =  −1 .Therefore, (25) leads to the elementwise equation Therefore, we can write (26) as a system of  nonlinear equations in ( + 1) unknowns as where ]  , and  means the transpose of the matrix.
From the initial condition (2), one can see that or where From systems ( 27) and ( 29), we obtain a system of ( + 1) nonlinear equations in ( + 1) unknowns where The standard methods for solving system (30), such as secant method and Newton method, need a good initial guess which is not available.To overcome this problem, we look for another method which does not depend on the initial guess.The promise technique is the path following method which is described as follows.From (23), we can see that the function Ω is the sum of three terms.Two of them are linear systems which are Thus, we can rewrite Ω as where  1 and  2 are two  × ( + 1) matrices and Ω 1 : R +1 → R  is a nonlinear function of .Hence, system (30) can be written as where Thus,  satisfies the following properties (0, 0) = 0, It is worth to mention that  is a nonsingular matrix since  =  is the system produced by the following problem: which has a unique solution.Hence, it follows from the implicit function theorem that there is a smooth curve  :  → R +1 for some open interval  containing zero such that (0) = 0,   (0) ̸ = 0, rank(  (())) =  + 1, and for all  ∈ .Consider the solution () = ((), ()) (parametrized for convenience with respect to arc length) such that (0) = (0, 0).The solution curve  −1 ((0), (0)) should be either diffeomorphic to the circle or to the real line.Since the solution point (0, 0) is unique for  = 0, it follows that  cannot be closed, and hence, it is diffeomorphic to the real line.Since  is a smooth curve, () is bounded for () ∈ [0, 1].Moreover, the curve  reaches the level  = 1 after a finite arc length  1 .This means, ( 1 ) = (, 1), and hence,  is the zero of Ψ() − .Thus, we can take  = [0, 1].For more details about the proof, [29][30][31].To apply path following method numerically, we differentiate (36) to get which implies that   (0) is orthogonal to all rows of   ((0)).Thus, for all  ∈ [0, 1], (1) det (   (()) ()  ) > 0, (2) ‖  ()‖ = 1, where ‖ ⋅ ‖ is the Euclidean norm, (3)   (())  () = 0.
We use the predictor-corrector method to numerically trace the curve .The predictor step is the Euler-predictor which is given by where  is a point along the curve , and ℎ > 0 is a fixed step size.The corrector step is the Gauss-Newton corrector which is given by where   () + = (  ())  (  ()  ()  ) −1 .For more details, see [31].Thus, we will start from  0 = (0, 0) and then generate the sequence  1 ,  2 ,  3 , . . .We stop our procedure at   when the last component of   ≤ 1 and the last component of  +1 > 1.We can write   as   = (  ,   ).Thus,   will be the approximate solution to the system (33).

Numerical Results
In this section, we implement the Tau-Path following method to the nonlinear fractional Riccati differential equations.Two examples of nonlinear fractional Riccati differential equations are solved to show the efficiency of the presented method.
Example 7. Consider the following initial value problem [32]: The matrices in (33) are Then, the absolute maximum of the error for different values of  are given in Table 1 for  = 16.

Figure 1 :
Figure 1: The graphs of the approximate solution when  = 0.9999 and the graph of the exact solution when  = 1.

Example 8 .Figure 2 :
Figure 2: The graphs of the approximate solution when  = 0.9999 and the exact solution when  = 1.