On General Form of Fractional Delay Integro-Differential Equations

Abstract In this paper, a numerical solution of general form of fractional delay integro-differential equation (GFDIDE) is presented using spectral collocation method. The Chebyshev polynomials of the second kind are used as a basis function with the collocation scheme. The proposed equation represents a general form of intgro-differential equation with delayed argument, which has multi-terms of integer and fractional order derivatives for delayed or non-delayed terms. The operation matrices for all terms of GFDIDE are introduced according to fractional calculus. The reliability and efficiency of the proposed method are demonstrated by some numerical examples.

In this paper, a general form of fractional delay integro-differential equations (GFDIDEs) is presented.The proposed form of GFDIDEs is considered as a multi-term of fractional and integer order derivatives, where the terms contain delay argument are taken to be multi-term of fractional as well as integer order derivatives.In addition, matrices of derivatives for fractional order derivatives are presented for Chebyshev polynomials of the second kind, these matrices employed to deal with the GFDIDEs using spectral collocation method.The presented matrices of derivatives with the collocation method are used to deal with the proposed GFDIDEs as a matrix discretization method.The high accuracy of this method is obtained through some numerical test examples.The obtained numerical results show that the proposed method gives good accuracy comparing with other existing methods.Now we consider GFDIDEs as: under the conditons where 0 < e < 2, e 2 R and f i , l i are given constants, also Q i ðtÞ, Q Ã i ðtÞ, P i ðtÞ, P Ã i ðtÞ, g(t), b i ðtÞ, k i ðt, sÞ are well-defined functions.Therefor, m is equal to the greatest integer order derivative exist Maxðn 3 , n 4 Þ, or it takes the smallest integer greater than the highest fractional order derivative.Chebyshev polynomials of the second kind are used here to approximate the solution of proposed Eq.
(1), and the solution is expressed in the form: U r ðtÞ are Chebyshev polynomials of the second kind and A r are unknown Chebyshev coefficients and N is any positive integer such that N !m: The Chebyshev polynomials are orthogonal and defined on the interval [-1, 1], and Eq.(1) has two different arguments t, t À e must be define in [-1, 1], so t in Eq. ( 1) is 2 ½À1 þ e, 1: Also the integration limits of the integral part is in Chebyshev interval i.e. a i , b i 2 ½À1, 1:

Preliminars
In this section, definition and some properties for Caputo fractional derivative and the Chebyshev second kind polynomials are presented.

The Caputo fractional derivative
Definition 1.The Caputo fractional derivative operator D r of order r is defined in the following form: D r uðtÞ ¼ 1 Cðm À rÞ ð t 0 u ðmÞ ðsÞ ðt À sÞ rÀmþ1 ds, r > 0, (4) and where k and l are real constants.
for n 2 N 0 and n < dre; U n ðtÞ may written expressly in terms of t n in many forms as Ramadan, Raslan, El Danaf, and Abd El Salam (2017), one of them is: where From previous relation one can define that: for even n we find for odd n is we may write From above we can write U(t) as general matrix form as: where U(t) and X(t) matrices are in following form: UðtÞ ¼ U 0 ðtÞ U 1 ðtÞ ::: and L is a matrix given by In addition, L has size ðN þ 1Þ Â ðN þ 1Þ and the last row used for odd values of N ðN ¼ 2l þ 1Þ, where the previous last row will be the last row in even values of N ðN ¼ 2lÞ: Now, the j th order derivative of the matrix U(t) given from ( 6) as: U ðjÞ ðtÞ ¼ X ðjÞ ðtÞL T , j ¼ 0, 1, 2, ::::

Matrices of derivatives
In this section the generalized operational matrices for U(t), U ðjÞ ðtÞ, Uðt À eÞ, U ðsÞ ðt À eÞ, D i UðtÞ and D a i Uðt À eÞ are introduced according to the properties of Caputo fractional derivative.
Lemma 3.1.The ðjÞ th order derivative of the row vector U(t), is in the following form: where B is square matrix written as: Lemma 3.2.Uðt À eÞ, represents the row vector can be written as: where Corollary 3.1.The ðsÞ th integer order derivative for the row vector Uðt À eÞ, may written as: From previous lemmas with the fractional calculus properties we can introduce the following theorem:  : : Theorem 1.The th i fractional order derivative for U(t) takes the following form: X i ðtÞ ¼ 0, 0, :::0, t nÀ i , :::  (16) Proof.
Corollary 3.2.The a th i fractional order derivative for Uðt À eÞ takes the following form: Proof.By using (12) and by replacing t !ðt À eÞ we have: where X a i ðtÞ and B a i as the same as in Theorem 1. w

Fundamental relations
In this section we consider the GFDIDE (1) to find the matrix form of each term in this equation.We also convert the solution u(t) defined by a truncated Chebyshev series (3) and its derivative u ðkÞ ðtÞ also the fractional derivates D i u N ðtÞ with two arguments t, t À e can be written in the matrix form as: and where UðtÞ ¼ U 0 ðtÞ, U 1 ðtÞ, U 2 ðtÞ, :::, U N ðtÞ ½ , Therefore, by substituting from ( 8) into ( 21) we get the matrix relation Also, substituting ( 12) into ( 22) we can get: From ( 18) and ( 24) we have: Finally, form ( 11) and ( 23) we get: To find the solution for (1) and ( 2) using (3), the following collocation points are used:

The matrix representation for integral term
To find the matrix form for the integral term, we first assume that k i ðt, sÞ may expanded in univariate Chebyshev second kind series with respect to t as the following form: Therefore, the matrix representation for k i ðt, sÞ is given by where H i ðtÞ ¼ ½h i0 ðtÞ, h i1 ðtÞ, ::::, h iN ðtÞ, and h ij ðtÞ well defined functions.
Replacing the relations ( 20) and ( 31) in the integral part of (1), then we have: where Thus, the integral term of (1) has the following matrix representation: In the end, we get the matrix form for the conditions ( 2) with (20) as the following form: we can write (34) in this form: where , v i1 , ::::, v iN ½ :

The collocation scheme
According to the typical collocation method, substitute (25-28) and (33), into (1) and then substituting the collocation points t j (29).Hence, the fundamental matrix equation takes this form: or in short where , , Equation (37) represents system of algebraic equations, which contains ðN þ 1Þ Chebyshev second kind coefficients unknowns, and shortly may written as: where p, q 2 f0, 1, 2, :::, Ng: By replacing the last m rows in (38) by the rows of ( 35), then we may construct the following augmented matrix to get the solution of ( 1 if the rank of the matrix M is equal to the rank of the augmented matrix ½ M; G then the solution of the algebric system exists, and if the two ranks equal to N þ 1 then the solution is unique.Therefor the matrix inverse method is used to get the solution as: (40)

Numerical results
In this section, the above results are illustrated by introduce some numerical examples for GFDIDEs.
Mathematica 7 program is used to obtain the introduced numerical results for five examples.

:
After we make all calculations for our problem, we can get the solution as: Thus, the solution for the problem (41) is.
which is represents the exact solution for the proposed problem (41).
Example 2. Consider the second order linear FIDE (G€ ulsu et al., 2010) with the subjected conditions uð0Þ ¼ 1, u 0 ð0Þ ¼ 0, e ¼ 1 so t 2 ½0, 1, and the exact solution is uðtÞ ¼ cos ðtÞ at a ¼ 2, gðtÞ ¼ 2t 2 þ tð sin ðtÞ þ cos ðtÞÞ À cos ðtÞ þ sin ðt À 1Þþ cos ðt À 1Þ þ 4t sin ð1Þ: Thus, for N ¼ 9 with (3) and ( 29), and the fundamental matrix equation of ( 45) is: After we make all calculations for our problem, we can get the solution as: where A comparison between numerical results with the exact solution at different a, N ¼ 9 is mentioned in Table 1. Figure 1 shows the behavior of the numerical results with exact solution at N ¼ 9. We note that Eq. ( 45) found in G€ ulsu et al. (2010) in the ordinary case (a ¼ 2), we don't list their results because the authors in this reference obtained the numerical solution using the interval ½À1, 0 and it is incorrect.
Example 4. Consider the following linear FDIDE (Mohammed, 2014): subject to uð0Þ ¼ 0 with the exact solution uðtÞ ¼ t 2 À t at e ¼ 0: By using ( 3) and ( 29) with Q 0 ðtÞ ¼ 1, at N ¼ 8 the fundamental matrix equation of ( 52) is: After we make all calculations for our problem (when e ¼ 0), we can get the solution as:  Then the solution is of Eq. ( 52) which is the exact solution of the problem (52).A comparison between numerical results at different e with the exact solution (e ¼ 0), N ¼ 8 is listed in The initial conditions are uð1Þ ¼ 1, u 0 ð1Þ ¼ À2, and the exact solution is uðtÞ We apply the suggested method with Equation ( 57) and condition are presents linear system of ðN þ 1Þ algebraic equations in the coeffcients c i .The solution of this system gives the Chebyshev coefficients: A 3 ¼ 1:73868 Â 10 À16 , A 4 ¼ À6:61509 Â 10 À18 : Thus, the approximate solution of this problem becomes.3. Table 4 shows the comparison of the numerical solution of the present method at N ¼ 4 and results in G€ urb€ uz et al. ( 2014) with the exact solution.Figure 3 shows the comparison of the present method numerical results of u(t) at different values of for N ¼ 4 with exact solution (v ¼ 2).

Conclusion
In this paper, a numerical solution of general form of linear fractional delay integro-differential equation (GFDIDE) is presented using spectral collocation method.The proposed equation represents a general form of intgro-differential equations with delayed argument, which has multi-terms of delayed or nondelayed terms with integer and fractional order derivatives for these terms.The Chebyshev polynomials of the second kind are used as a basis function with the collocation method for GFDIDEs.The collocation scheme reduces the proposed equation to system of algebraic equation.The operational matrices for all terms of GFDIDE according to the fractional calculus are introduced.The accuracy and competence of the suggested scheme have been explained by some numerical examples.

Figure 2 .
Figure 2. Comparison of the numerical results with exact solution at N ¼ 8 for example 4.

Table 1 .
Comparison between numerical results with exact solution at different a, N ¼ 9.
t Exact solution Present method Present method Present method a ¼ 2 a ¼ 2 a ¼ 1:9 a ¼ 1:8 Figure 1.Comparison of the numerical results with exact solution at N ¼ 9.

Table 2 .
Comparison between numerical results with the exact solution at different values of e, N ¼ 8 for example 4.

Table 2 .
Figure2shows the behavior of the numerical results with exact solution (e ¼ 0) at N ¼ 8.

Table 3 .
Comparison of the absolute errors for example 5 for different N values at ¼ 2.

Table 4 .
Numerical solution of example 5 for different N values.