Numerical study on factional differential-algebraic systems by means of Chebyshev Pseudo spectral method

A numerical treatment to a system of Caputo fractional order differential- algebraic equations (SFDAEs) is presented throughout this article. The suggested method based upon the shifted Chebyshev pesedu- spectral method (SCPSM). The shifted Chebyshev polynomials (SCPs) are handled to reduce the SFDAEs into the solution of linear/ nonlinear systems of algebraic equations. By using some tested applications, the effectiveness and the accuracy of the suggested approach are demonstrated graphically. Also numerical comparisons between the proposed technique with other numerical methods in the existing literature are held. The numerical results show that the proposed technique is computationally efficient, accurate and easy to implement.

Many physical phenomena's are obviously designated by a system of differential-algebraic equations (SDAEs). These types of systems follow in the displaying of the mechanical systems subject to constraints, power systems, electrical networks, optimal control, chemical process and in other numerous applications [17]. SFDAEs have currently confirmed to be a suitable devise in the displaying of the numerous engineering and physical problems such as non-integer order optimal controller design, electrochemical processes, complex biochemical [18].
The approximate and numerical solutions of these types of systems have been a focus of several researchers especially the nonlinear systems, because most of these systems don't have exact solutions. Numerical methods to solve SDAEs have been given such as, the numerical algorithms for computing the matrix Green's operator [19], implicit Runge-Kutta method [20], Padé approximation method [21,22], homotopy perturbation method [23], Adomain decomposition method [24] and variation iteration method (VIM) [25].
For FDEs, the spectral collocation method (also called pseudo-spectral method) is more applicable and commonly applied to numerically solve different types of the fractional differential equations [15,32]. In collocation technique, expansion coefficients are determined by constructing the approximate solution to satisfy the differential equation at some applicably selected points from the domain identified as collocation points. Recently, various types of orthogonal polynomials and collocation points are used in spectral collocation approximations [15,32].
CPs have many useful properties. These polynomials present, among others, very good properties in the approximation of functions. This encourages many researchers for using these polynomials for solving different types of differential equations and FDEs [32][33][34][35].
The proposed technique used the properties of Chebyshev polynomials (CPs) to reduce the SFDAEs into a system of algebraic equation which is greatly simplifying the problem. To the best of our knowledge, the numerical treatment of SFDAEs has not been established by using SCPSM yet.
To check the accuracy of the suggested method, five numerical applications including comparisons between our obtained results with those achieved by using other existing methods are presented. This article is prescribed as follows: The basic definition of the Caputo fractional derivative and the main properties of the CPs are summarized in Section 2. In Section 3, the necessary theorems of the upper bound of errors and the convergence analysis of the fractional derivatives of the SCPs are explained. Section 4 contains the procedures for the implementation of the suggested method to nonlinear FDAEs. Some applications are discussed in Section 5. Finally, a brief conclusion finishes the paper in Section 6.

Definition 2.1:
A real function f (t), t > 0, is assumed to be in the space C μ, μ ∈ , if there exists a real number p > μ such that f (t) = t p g(t) where g(t) ∈ C(0, ∞), and is assumed to be in space C m μ if and only if f (m) ∈ C μ , m ∈ N.

Main properties of the CPs
The CPs, T n (z) of degree n are determined by the following recurrence relation [32][33][34] T n+1 (z) = 2zT n (z) − T n−1 (z), T 0 (z) = 1, The analytic form of T n (z) is defined by where n 2 is the integer part of n 2 . The SCPs T * n (t) of degree n defined in the interval [0, L], constructed by offering the change of variable z = 2 L t − 1 and defined by The square integrable function f (t) in [0, L], can be approximated using the first (m + 1) terms of the SCPs as Where the coefficients c i are given by

The approximation of the fractional derivatives of the SCPs and its convergence analysis
The approximate formulation of the non-integer derivatives, the truncating error and the convergence analysis of the SCPs are considered in the following theorems.

Theorem 3.1: (Chebyshev truncation theorem) [32]
The error in approximating x(t) by the sum of its m-terms is bounded by the sum of the absolute values of all neglected coefficient. If then Theorem 3.2: Let f (t) be approximated by SCPs as in (8) and suppose that α > 0, then Where ψ (α) i,k is given by For Proof, see [32].

Theorem 3.3: [32,35]
The Caputo fractional derivative of order α for the SCPs can be expressed in terms of the SCPs themselves as in the following form where Theorem 3.4 [32,35] The error

Solution to the SFDAEs
In this section we will explain the main steps of the procedure of applying the SCPSM for solving the following SFDAEs: y 1 , y 2 , · · · y n , y 1 , y 2 , · · · , y n ), i = 1, 2, · · · , l − 1, t ≥ 0, 0 < α i ≤ 1, Subject to the initial conditions Step 1: Approximate y i (t) by using the SCPs as: Step 2: Use Eq. (12) to approximate the Caputo fractional derivatives, then the SFDAEs (17) is reduced to Step 3: Approximate the initial conditions (18) by using SCPs: Step 4: For a suitable collocation points, use the of the SCPs roots, T * m i +1− α i (t).
Step 5: The obtained equations of the previous step 4 with Eq. (21) represent a system of linear/nonlinear algebraic equations which contain Step 6: Solve the algebraic system by using the Newton iteration method to obtain the unknowns.
Step 7: The approximated solutions will be

Numerical applications
In this section five numerical applications of SFDAEs are solved by the proposed technique; the applications include variable and constant coefficient linear nonlinear SFDAEs Application 5.1: Consider the following variable coefficient linear SFDAE [27][28][29] (22) With the initial conditions For the special case at α = 1, system (22) has exact solution: By implementing the proposed technique using m 1 = And the approximated equations for the initial conditions will be 5 j=0 c j T * j (0) = 1 and The equations obtained by collocating system (25) at the first five roots of the SCPs T * 5 (t) with Eqs. (26) represent a system of linear algebraic equations which contains twelve equations for twelve unknowns; c j and b j , j = 0, 1, . . . , 5. These unknowns are obtained by using Newton iteration method.
For α = 1. The estimated solutions are     Table 1. Executing time of this problem is measured using Mathematica 10 software on CPU Intel(R) Core(TM) i3, it's apparent that the solution doesn't require much CPU time. It is remarkable that the proposed technique is very effective even with using few terms of SCPs and the overall errors can be made smaller by adding more terms of SCPs.
In Table 2 the approximate numerical solutions for x(t) for α = 0.75 and 1 are compared with the solutions given by VIM [27], HAM [28] and TF [29]. These numerical results demonstrate the harmony between our method and the other numerical methods used in the comparisons.

Application 5.2:
Consider the following nonlinear SFADE [27][28][29] ⎧ ⎪ ⎪ ⎨ With 0 < α ≤ 1 and initial conditions For the special case when α = 1, the exact solution is By employing the proposed technique using m 1 j=0 a j T * j (t) We obtain a system of nonlinear algebraic equations which contains eighteen equations for eighteen unknowns; c j , b j and a j , j = 0, 1, . . . , 5. These unknowns are obtained by using Newton iteration method.
For α = 1. The estimated solutions will be x(t) = 0.9999 + t(3.0001 + t(2.4967 + t(0.85017   y(t) = 1. + t(6.00725 + t(13.647 + t (19.38 + t(4.417 + 11.9554t)))) The numerical solutions of Application 5.2 are shown graphically through Figure 2 and tabulated in Table 3 (a-c) and Table 4(4a and 4b). From Figure 2, it is easy to conclude that the obtained solutions are continuously depend on the fractional derivative and the estimated solutions are in good agreement with the exact solution at α = 1. Table 3(a-c) show the effectiveness of the suggested technique even with using few terms of SCPs and the accuracy of the method is increased by adding more terms of the SCPs. Also the solutions do not need much CPU time to produce very accurate numerical solutions. Table 4(a,b) demonstrate the accuracy of the proposed technique when compared with the numerical methods in [27][28][29].  where 0 < α ≤ 1 and initial conditions:

Application 5.3: Consider the following variable coefficient nonlinear SFDAEs [29]
At α = 1. The exact solution is: By applying our proposed technique using m 1 = m 2 = m 3 = 5, y(t) = 5 j=0 c j T * j (t), z(t) = 5 j=0 b j T * j (t) and w(t) = 5 j=0 a j T * j (t) to the fractional system (5-8), we obtain a system of linear algebraic equations with unknowns; c j , b j and a j , j = 0, 1, . . . , 5. These unknowns are attained by using Newton iteration method. For α = 1, the estimated solutions will be y(t) = −1.56 × 10 −17 + t(−0.0002 + t(1.004 The numerical solutions of Application 5.3 are shown graphically through Figure 3 and the absolute errors between our approximate solutions and the exact solutions for different values of m with their CPU time are given in Table 5(a-c) Application 5.4: Consider the following nonlinear SFDAEs [26] ⎧ ⎪ ⎪ ⎨ with the initial conditions At α = 1. System (31) has exact solution: x(t) = t 2 , y(t) = t 4 , z(t) = 2t 3 + t + 1. By using our proposed technique using m 1 = m 2 = m 3 = 5, x(t) = 5 j=0 c j T * j (t), y(t) = 5 j=0 b j T * j (t) and z(t) = 5 j=0 a j T * j (t) to the fractional system (5-6), we obtain a system of linear algebraic equations with unknowns; c j , b j and a j , j = 0, 1, . . . , 5. These unknowns are gained by using Newton iteration method.  Table 6(a-c).
The exact solution of this problem is y 1 (t) = t 5 2 , y 2 (t) = t 2 , y 3 (t) = sin(t) By using our proposed technique using m 1 = m 2 = m 3 = 5, y 1 (t) = 5 j=0 c j T * j (t), y 2 (t) = 5 j=0 b j T * j (t) and y 3 (t) = 5 j=0 a j T * j (t) to the fractional system (33), we obtain a system of linear algebraic equations with unknowns; c j , b j and a j , j = 0, 1, . . . , 5. These unknowns are obtained by solving the algebraic system. Then the estimated solutions are The numerical results of Application 5.5 are graphically illustrated in Figure 5. A numerical comparison between our attained solutions with the results in [31] is tabulated in Table 7(a-c). The numerical results demonstrate the effectiveness and the accuracy of the proposed technique even by using a few terms of SCPs, and our obtained results are quite similar to the results given by SGOM [31].

Conclusion
Through this paper, The SCPSM method has been extended to solve the linear and nonlinear SFDAEs. The numerical results of some tested applications were a good evidence for the applicability and efficiency of the suggested method. A specific advantage of the suggested implementation is that it transfers the fractional differential equations into a system of algebraic equations which is easier to solve. Also, satisfactory results are obtained by using a few terms of the SCPs, and the efficiency of the anticipated method is increased by using more terms of SCPs. The obtained solutions are continuously depended on the fractional derivative and this note confirms the physical meaning of the behaviour of the solution for the proposed real problems. Also the solutions do not require much CPU time.