Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series

In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

In this work, we introduce a general form of nonlinear fractional integro-differential equations (GNFIDEs) with linear functional arguments, which is a more general form of nonlinear fractional pantograph and Fredholm-Volterra integro-differential equations with linear functional arguments [65][66][67][68][69]. The spectral collocation method is used with Chebyshev polynomials of the first kind as a matrix discretization method to treat the proposed equations. An operational matrix for derivatives is presented. The introduced operational matrix of derivatives includes fractional order derivatives and the operational matrix of ordinary derivative as a special case. No other studies have discussed this point.
The proposed GNFIDEs with linear functional arguments are presented as follows: where η i ∈ [a, b], and m is the greatest integer order derivative, or the highest integer order greater than the fractional derivative. The general form (1) contains at least three different arguments, then the following corollary defines the interval that the independent variable x belongs to. Chebyshev polynomials of the first kind are used in this work to approximate the solution of suggested equation (1). The Chebyshev polynomials are characterized on [-1, 1].

Corollary 1.1
The independent variable x of (1) belongs to [a, b], which is the intersection of the intervals of the different arguments and

General notations
In this section, some definitions and properties for the fractional derivative and Chebyshev polynomials are listed [63,64,70,71].

The Caputo fractional derivative
The Caputo fractional derivative operator D γ t of order γ is characterized in the following form: where x > 0, n -1 < γ ≤ n, n ∈ N 0 , and where λ i and γ are constants. • The Caputo fractional differentiation of a constant is zero.
where γ denotes to the smallest integer greater than or equal to γ .

Chebyshev polynomials
The Chebyshev polynomials T n (x) of the first kind are defined as follows: orthogonal polynomials in x of degree n are defined on [-1, 1] such that T n (x) = cos nθ , where x = cos θ and θ ∈ [0, π]. The polynomials T n (x) are generated by using the following recurrence relations:

Corollary 2.1
The Chebyshev polynomials T n (x) are explicitly expressed in terms of x n in the following form: where w (n) k = (-1) k 2 n-2k-1 n nk nk k , 2k ≤ n.

Procedure solution using the collocation method
The solution y(x) of (1) may be expanded by Chebyshev polynomial series of the first kind as follows [64]: By truncating series (5) to N < ∞, the approximate solution is expressed in the following form: where T(x) and C are matrices given by Now, using (4), relation (6) may written in the following form: where W is a square lower triangle matrix with size (N + 1) × (N + 1) given by For example, Then, by substituting from (6) in (1), we get We can write (9) as follows: The collocation points are defined in the following form: where h = ba N , l = 0, 1, 2, . . . , N.
By substituting the collocation points (11) in (10), we get In the following theorem we introduce a general form of operational matrix of the row vector T(x) in the representation as (7), such that the process includes the fractional order derivatives, and ordinary operational matrix given as a special case when α i → α i . (7), then the fractional order derivative of the vector D α i T(x) is
and B α i as in (15), where the proposed operational matrix represents a kind of unification of ordinary and fractional case. Now, we give the matrix representation for all terms in (12) as representation (13). * The first term in (12) can be written as follows: wherē In addition, H p i is a square diagonal matrix of the coefficients for the linear argument, and the elements of H p i can be written as follows: Moreover, E ξ i is a square upper triangle matrix for the shift of the linear argument, and the form of E ξ i is . * The second term in (12) can be written as follows: wherē The matrix representation for the variable coefficients takes the form . * Matrix representation for integral terms: Now, we try to find the matrix form corresponding to the integral term. Assume that K d (x, t) can be expanded to univariate Chebyshev series with respect to t as follows: Then the matrix representation of the kernel function K d (x, t) is given by where Substituting relations (13) and (27) in the present integral part, we obtain where So, the present integral term can be written as: where . * Matrix representation for integral terms: Now, we try to find the matrix form corresponding to the integral term. By the same way V c (x, t) can be expanded as (26) Then the matrix representation of the kernel function V c (x, t) is given by where G c (x) = g c,0 (x) g c,1 (x) · · · g c,N (x) .
Substituting relations (13) and (31) in the present integral part, we obtain where So, the present integral term can be written as follows: Now, by substituting equations (24), (25), and (29) into (12), we have the fundamental matrix equation We can write (34) in the form where Corollary 3.1 Suppose k ≥ 2, then the matrix representation for the terms free of derivatives in (1), by using (6), we obtain We can achieve the matrix form of (37) by using the collocation points as follows: (38) * We can achieve the matrix form for conditions (2) by using (6) on the form or where System (34), together with conditions, gives (N + 1) nonlinear algebraic equations which can be solved for the unknown c n , n = 0, 1, 2, . . . , N . Consequently, y(x) given as equation (6) can be calculated.

Numerical examples
In this section, several numerical examples are given to illustrate the accuracy and the effectiveness of the method.

Error estimation
if the exact solution of the proposed problem is known, then the absolute error will be estimated from the following: where y Exact (x) is the exact solution and y Approximate (x) is the achieved solution at some N . The calculation of L 2 error norm also can obtained as follows: where h is the step size along the given interval. We can easily check the accuracy of the suggested method by the residual error. When the solution y Approximate (x) and its derivatives are substituted in (1), the resulting equation must be satisfied approximately, that is, for x ∈ [a, b], l = 0, 1, 2, . . .
In Table 5, we contribute the numerical results y(x l ), for N = 9, of our proposed scheme together with numerical results y(x l ), for N = 10, of the Legendre collocation method (LCM) [69] and [68]. It is observed that the proposed scheme reaches the same results of [69] with lower degree of approximation. Moreover, the proposed scheme has superior results with regard to the ADM [66] as shown in [69]. In addition, the numerical results associated with our presented method LCM and generalized differential transform method (GDTM) [67] for N = 10 and ν 4 = 3.75 are given in Table 6. As shown in Table 4 of [69], the ADM has very weak approximations with regard to GDTM and LCM. Therefore, we do not consider ADM in Table 6. From this table, we can find that our achieved results  Table 7. The residual error E 10 is given in Table 8 for different values of ν 4 as follows: 3.75, 3.5.
Finally, since problem (56) defines on [0, 1] the proposed method applied with the Chebyshev nodes (zeros of Chebyshev polynomials) as collocation points. Table 9 compares the absolute errors for different values of N at ν 4 = 4 using Chebyshev nodes collocation points, namely 1 2 (1 + cos iπ N ), i = 0, 1, . . . , N . Also, the comparison of the L 2 error norm according to (42) using both equally spaced (11) and Chebyshev nodes collocation points is given in Table 10. Comparing Table 7 with Table 9 and the L 2 results in Table 10, one finds that the nodes of Chebyshev fall on [-1, 1] and they are chosen with the collocation method as collocation points if the problem is also defined in the same interval, and better results will be obtained than any choice of other form of collo-     cation points, and any modification in the nodes to fit the interval of the problem does not give the good results as expected than the original zeros of the Chebyshev polynomials.

Conclusion
A numerical study for a generalized form of nonlinear arbitrary order integro-differential equations (GNFIDEs) with linear functional arguments is introduced using Chebyshev series. The suggested equation with its linear functional argument represents a general form of delay, proportional delay, and advanced nonlinear fractional order Fredholm-Volterra integro-differential equations. Additionally, we have presented a general form of the operational matrix of derivatives. The fractional and ordinary order derivatives have been obtained and presented in one general operational matrix. Therefore, the proposed operational matrix represents a kind of unification of ordinary and fractional case. To the best of authors knowledge, there is no other work discussing this point. We have presented many numerical examples that greatly illustrate the accuracy of the presented study to the proposed equation and also show how that the propose scheme is very competent and acceptable.