NUMERICAL SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATION SYSTEM USING THE MÜNTZ-LEGENDRE POLYNOMIALS

Abstract: In the paper, we apply a numerical approach to approximate solution of a system of the fractional differential equations based on the Müntz-Legendre polynomials. Features of these polynomials make them so appropriate for solving the fractional differential equation system. In the proposed method, a system of the fractional differential equations is transformed into a system of the algebraic equations which is move easier to solve. The numerical examples show accuracy and convergence of the present method.


Introduction
Fractional calculus plays main role in physics, chemistry and engineering which has attracted the interest of many researchers in recent years. In comparison to the differential equations of accurate order, the fractional differential equations (FDEs) have an advantage that many phenomena in physics, chemistry and engineering can be described very successfully by them. Moreover, this allows one to apply the fractional differential equations to model these phenomena more accurately. Some of these models and applications are presented in [1,2,3,4,5]. Also many researchers tried to investigate and describe these kinds of the equations (see Ref. [6,7]). Odibat suggested an analytical examination for FDEs [8]. Jafari and Daftardar also proved existence and uniqueness solutions of the FDEs [9]. Obtaining an exact solution of these problems is extremely difficult thus there are a lot of numerical methods to approximate solutions of the FDEs such as, Homotopy method [10], finite differences method [11,12], Adomian decomposition method [13], Collocation method [14], and also Chen and et al. proposed a numerical method [15].
In this paper, we consider a system of the fractional differential equations, as follows: where D α i * f i (t) is derivative of order α i for f i (t) and 0 < α i ≤ 1. Thus for the system of Eq. (1), we consider the following initial condition: Note that y 1 , y 2 , . . . , y n are constant numbers. There are several kinds of the fractional derivatives of a function [16,17], but Rieman-Liouville and Caputo definitions are two of mostly used and famous for evaluating the fractional derivatives. In present paper, we use Caputo derivative. The Caputo fractional derivative of order 0 < α < 1 is defined in the following form: Recall that the above formula for α = 1 coincides with the usual derivative operator of the first order. We use The Müntz-Legendre polynomials [18,19,20] to approximate the solution of the Eq. (1). The main advantage of using the Muntz.Legendre polynomials is that their fractional derivative is still a Müntz-Legendre polynomial. Furthermore, approximate solutions of the fractional deferential equations with a collocation method based on the Müntz-Legendre polynomials show very good agreement with the exact solution. Another considerable advantage of the known polynomials in question is that the fractional derivatives of these polynomials can be expressed in terms of the same polynomials that are appropriate for solving the fractional differential equations.
A brief overview of the paper is as follows: first, the Jacobi polynomials are defined. Then, the definition of Müntz-Legendre polynomials and obtaining them via a stable recurrence relation are presented. After that, a method for evaluating the fractional derivatives of the Müntz-polynomials is introduced. Finally, the method of solving the fractional derivatives based on the Muntz.Legendre polynomials is explained and some numerical examples are given to demonstrate the effectiveness and accuracy of the proposed method.

Jacobi Polynomials
The Jacobi polynomials are extensively used for solving fractional differential equations. They are orthogonal on the interval [−1, 1] with respect to the weight function w (α,β) (t) = (1 − t) α (1 + t) β where α, β > −1. These polynomials can be obtained through the following recurrent relation: The initial derivative of the Jacobi polynomials can be obtained as follows:

Solution of the Fractional Differential Equations
In this section, we assume that α 1 = α 2 = · · · = α n = α then we present an approximate solution for the unknown fractional differential equations in Eq. (1) as follows: In addition, the fractional derivative of the unknown functions can be found by: where D α * L I,j (t : α) can be computed by the proposed Eq. (9) for evaluating the fractional derivative of the Müntz-Legendre polynomials and for the initial condition, we have: in this case, a system of Eq. (1) with boundary conditions is transformed into the following relations, (13) to findf i (t), we should obtain coefficients c ij ,for this purpose, we should satisfy the points θ k = R 2 − R 2 cos πk N , k = 1, 2, . . . , N into above system of Eq. (13), and we have,

Numerical Examples
Example 1. Consider the following system of the fractional deferential equations: The exact solution of these equations for α = 1 is given by f 1 (t) = e t sin(t) and f 2 (t) = e t cos(t). In Table 1, we list the maximum absolute errors obtained by f 1 (t), f 2 (t) with different values of N and α = 1. The approximate solutions obtained byf 1 (t) andf 2 (t) for different values of α and N = 30 are plotted in Figure 1.  Example 2. Consider the following system of the fractional deferential equations: The exact solution of these equations for α = 1 is given by f 1 (t) = e t/2 and f 2 (t) = te t . In Table 2, we list the maximum absolute errors obtained by f 1 (t), f 2 (t) with different values of N and α = 1. The approximate solutions obtained byf 1 (t) andf 2 (t) for different values of α and N = 30 are plotted in Figure 2.

Conclusion
In this paper, we presented a numerical method to obtain approximate solutions of the fractional deferential equations with a collocation method based on the Müntz-Legendre polynomials. The Müntz-Legendre polynomials are characterized for obtaining exact solutions of the fractional deferential equations. This is truly visible in results obtained through the numerical examples which their exact solutions are available.