New Solutions for System of Fractional Integro-Differential Equations and Abel ’ s Integral Equations by Chebyshev Spectral Method

Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations andAbel’s integral equations.


Introduction
In recent years, the topic of fractional calculus has attracted many scientists because of its several applications in many areas, such as physics, chemistry, biology, and engineering.For a detailed survey with collections of applications in various fields, see, for example, [1][2][3][4][5][6].
The numerical solution of differential equations of integer order has been a hot topic in numerical and computational mathematics for a long time.There are many different methods and different basis functions have been used to estimate the solution of fractional integro-differential equations or Abel's integral equations, such as Adomian decomposition method [7,8], fractional differential transform method [9,10], collocation method [11,12], homotopy perturbation method [13,14], homotopy analysis method [15,16], variational iteration method [17], discrete Galerkin method [18], and Haar wavelet method [19].
Spectral methods provide a computational approach that has achieved substantial popularity over the last four decades.They have gained new popularity in automatic computations for a wide class of physical problems in fluid and heat flow.Their fascinating merit is the high accuracy.So, they have been applied successfully to numerical simulations of many problems in science and engineering; see [20][21][22][23][24].
The operational matrix of fractional derivatives has been determined for some types of orthogonal polynomials, such as Chebyshev polynomials [25] and Legendre polynomials [26], and for integration has been determined for several types of orthogonal polynomials, such as Chebyshev polynomials [27], Laguerre series [28], and Legendre polynomials [29].Recently, the Bernstein operational matrix approach is developed for solving a system of high order linear Volterra-Fredholm integro-differential equations in [30].
And we use Abel's integral equation: where  = 0 or  = 1, () is a continuous function, and  is constant.

Basic Definitions
In this section, we summarize some basic definitions and properties of fractional calculus theory.
Definition 1.A real function (),  > 0, is said to be in the space   ,  ∈ R, if there exists a real number  > , such that Definition 3. The Riemann-Liouville fractional integral operator of order  ( ≥ 0), of a function  ∈   ,  ≥ −1, is defined as Definition 4. The Caputo fractional derivatives of order  are defined as where  − 1 <  ≤  and   is the classical differential operator of order .
For Caputo derivative, we have We use the ceiling function ⌈⌉ denoting the smallest integer greater than or equal to  and the floor function ⌊⌋ denoting the largest integer less than or equal to .Also N = {1, 2, 3, . ..} and N 0 = {0, 1, 2, . ..}. Recall that, for  ∈ N, the Caputo differential operator coincides with the usual differential operator of integer order.More properties of the fractional derivatives and the fractional integral can be found in [3,4].
In this form,  , () may be generated with the aid of the following recurrence formula: where  ,0 () = 1 and  ,1 () = 2/ − 1.The zeros of  , () are denoted by A function (), square integrable in (0, ), may be expressed in terms of the shifted Chebyshev polynomials as where the coefficients   are given by In practice, only the first ( + 1)-terms shifted Chebyshev polynomials are considered.Hence, if we write where the shifted Chebyshev coefficient vector  and the shifted Chebyshev vector () are given by then the derivative of the vector () can be expressed by where D (1) is the ( + 1) × ( + 1) operational matrix of derivative given by for example, for even , we have ) . (21)

The Shifted Chebyshev Operational Matrix (COM) Fractional Derivatives
The main objective of this subsection is to generalize the COM of derivatives for the fractional calculus.By using (19), it is clear that where  ∈ N and the superscript, in D (1) , denotes matrix powers.Thus Lemma 5. Let  , () be a shifted Chebyshev polynomial; then Theorem 6.Let () be the shifted Chebyshev vector defined in (18) and suppose  > 0; then where D () is the ( + 1) × ( + 1) COM of derivatives of order  in the Caputo sense and is defined as follows: where Note that, in D () , the first ⌈⌉ rows are all zero.
In our computations we used the Gaussian elimination method to solve the resulting linear system of algebraic equations and Newton's iteration method to solve the resulting nonlinear system of algebraic equations.Now, we can present the following problems.
Example 8. Consider the following system of fractional integro-differential equations [7,15]: with the initial conditions The exact solutions, when We use Chebyshev spectral method; we may write the approximate solution where Substituting (36) in ( 33) and ( 34), for  1 =  2 = 1, we get The roots of the shifted Chebyshev polynomial  1,8 () are given by (41) Thus using (36), we get Table 1 shows the comparison between the exact solution and the approximate solution with the absolute error at  = 8, and Table 2 shows the maximum of absolute error between exact solutions and approximate solutions for various choices of . Figure 1 shows the graph of the exact solution and the approximate solution at  = 8,  1 =  2 = 1. Figure 2 shows the graph of the exact solutions and the approximate solutions at  = 8,  1 =  2 = 0.9, 0.85, and 0.75.
Example 9. Consider the following nonlinear fractional system of integro-differential equations [15]: with the initial conditions The exact solutions, when We use Chebyshev spectral method; we may write the approximate solution Substituting ( 46) in ( 43) and (44), for  1 =  2 = 2, we get The roots of the shifted Chebyshev polynomial  1,2 () are Now, for calculating the shifted Chebyshev coefficient for  = 3, substitute (50) in (48), and solving the resulting nonlinear system of equations and (49), we get (52) Table 3 shows the comparison between the exact solution and the approximate solution with the absolute error at  = 3. Figure 3 shows the graph of the exact solution and the approximate solution at  = 3,  1 =  2 = 2. Figure 4 shows the graph of the exact solutions and the approximate solutions at  = 3,  1 =  2 = 1.9, 1.8, and 1.7.   Figure 4: The graphs of the exact solutions and the approximate solutions at  = 3,  1 =  2 = 1.9, 1.8, and 1.7.

Abel's Integral Equation
In order to use COM for Abel's integral equation of the form (3), we first approximate () by the shifted Chebyshev polynomials as By substituting (53) in (3), we get Then we have to collocate (54) at the ( + 1) shifted Chebyshev roots  ,+1, ,  = 0, 1, . . ., .These equations generate ( + 1) linear algebraic equations which can be solved for the unknown coefficients of the vector , using a suitable method.Consequently, () given in (53) can be calculated, which gives a solution of (3).
By applying the Chebyshev spectral method, we may write the approximate solution Substituting ( 56) in (55), we get The roots of the shifted Chebyshev polynomial  1,4 () are Now, calculating the shifted Chebyshev coefficient for  = 3 by substituting (58) in (57) and solving four equations yields , Therefore, we have which is the exact solution.
Example 11.Consider Abel's integral equation of the first kind [32] which has the exact solution () = √.Table 4 shows the comparison between the exact solution and the approximate solution with the absolute error at  = 20, and Table 5 shows the maximum of absolute error between exact solution and approximate solution for various choices of . Figure 5 shows the graph of the exact solution and the approximate solution at  = 20.
By applying the Chebyshev spectral method, we may write the approximate solution Substituting ( 63) in (62), we get The roots of the shifted Chebyshev polynomial  ,3 () are  Now, calculating the shifted Chebyshev coefficient for  = 2 by substituting (65) in (64) and solving three equations yields Therefore, we have which is the exact solution.
We use Chebyshev spectral method; we may write the approximate solution Substituting (69) in (68), we get The roots of the shifted Chebyshev polynomial  ,3 () are Now, calculating the shifted Chebyshev coefficient for  = 2 by substituting (71) in (70) and solving three equations yields Therefore, we have which is the exact solution.
Table 6 shows the comparison between the exact solution and the approximate solution with the absolute error at   7 shows the maximum of absolute error between exact solution and approximate solution for various choices of .(76)  Table 8 shows the comparison between the exact solution and the approximate solution with the absolute error at  = 20, and Table 9 shows the maximum of absolute error between exact solution and approximate solution for various choices of . Figure 7 shows the graph of the exact solution and the approximate solution at  = 20.

Conclusion
In this article, we develop the Chebyshev spectral method based on operational matrix for solving linear and nonlinear system of fractional integro-differential equations and Abel's integral equations.Our approach was based on the shifted Chebyshev collocation methods.It can be concluded that the Chebyshev spectral method is very powerful and efficient technique for finding exact solutions for wide classes of problems.

Figure 5 :
Figure 5: The graph of the exact solution and the approximate solution at  = 20.

Figure 6 :Figure 7 :
Figure 6: The graph of the exact solution and the approximate solution at  = 20.

Figure 6
shows the graph of the exact solution and the approximate solution at  = 20.Example 15.Consider the linear system of singular Volterra integral equations [10] are  () = ,  () = √ .

Table 9 𝑁
Maximum of absolute error of ()