Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.


Introduction
Fractional calculus, the calculus of derivative and integral of any order, is used as a powerful tool in science and engineering to study the behaviors of real world phenomena especially the ones that cannot be fully described by the classical methods and techniques [1].Differential equations with proportional delays are usually referred to as pantograph equations or generalized pantograph equations.The name pantograph was originated from the study work of Ockendon and Tayler [2].Many researchers have studied different applications of these equations in applied sciences such as biology, physics, economics, and electrodynamics [3][4][5].Solutions of pantograph equations were also studied by many authors numerically and analytically.Bhrawy et al. proposed a new generalized Laguerre-Gauss collocation method for numerical solution of generalized fractional pantograph equations [1].Tohidi et al. in [6] proposed a new collocation scheme based on Bernoulli operational matrix for numerical solution of generalized pantograph equation.Yusufoglu [7] proposed an efficient algorithm for solving generalized pantograph equations with linear functional argument.In [8], Yang and Huang presented a spectral-collocation method for fractional pantograph delay integrodifferential equations and in [9] Yüzbasi and Sezer presented an exponential approximation for solutions of generalized pantograph delay differential equations.Chebyshev and Bessel polynomials are, respectively, used in [10,11] to obtain the solutions of generalized pantograph equations.Operational matrices of fractional derivatives and integration have become very important tool in the field of numerical solution of fractional differential equations.In this paper, a member of Appell polynomials called Genocchi polynomials is used; although this polynomial is not based on orthogonal functions, it possesses operational matrices of derivatives with high accuracy.It is very important to note that this polynomial shares some great advantages with Bernoulli and Euler polynomials for approximating an arbitrary function over some classical orthogonal polynomials; we refer the reader to [6] for these advantages.On top of that, we International Journal of Differential Equations had successfully applied the operational matrix via Genocchi polynomials for solving integer-order delay differential equations [12] and fractional optimal control problems [13], and the numerical solutions obtained are comparable or even more accurate compared to some existing well-known methods.Motivated by these advantages, in this paper, we intend to extend the result for integer-order delay differential equations in [12] to fractional delay differential equations or so-called generalized fractional pantograph equations.To the best of our knowledge, this is the first time that the operational matrix based on Genocchi polynomials is applied to solve the fractional pantograph equations.On the other hand, some other types of polynomials were employed to solve some special type of fractional calculus problems; for example, Bessel polynomials were used for the solution of fractional-order logistic population model [14]; Bernstein polynomials were also used for the solution of Riccati type differential equations [15].
In this paper, we use the new operational matrix of fractional-order derivative via Genocchi polynomials to provide approximate solutions of the generalized fractional pantograph equations of the following form [1]: subject to the following conditions: where  , ,  , , and  , are real or complex coefficients;  − 1 <  < , 0 <  0 <  1 < ⋅ ⋅ ⋅ <  −1 < , while  , () and () are given continuous functions in the interval [0, 1].The rest of the paper is organized as follows: Section 2 introduces some mathematical preliminaries of fractional calculus.In Section 3, we discuss some important properties of Genocchi polynomials.In Section 4, we derive the Genocchi delay operational matrix and we apply the collocation method for solving fractional pantograph equation (1) using the Genocchi operational matrix of fractional derivative and the delay operational matrix in Section 5.In Section 6, the proposed method is applied to several examples and conclusion is given in Section 7.

Preliminaries
2.1.Fractional Derivative and Integration.We recall some basic definitions and properties of fractional calculus that we will use.There are various competing definitions for fractional derivatives [16,17].The Riemann-Liouville definition played a vital role in the development of the theory of fractional calculus.However, there are certain disadvantages of using this definition when modeling real world phenomena.To cope with these disadvantages, Caputo definition was introduced which is found to be more reliable in application.So we use this definition of fractional derivatives.We begin with the definition of Riemann-Liouville integral, in which the fractional integral operator  of a function () is defined as follows.
Definition 1.The Riemann-Liouville integral  of fractionalorder  of () is given by where Γ(⋅) is the Gamma function.The fractional derivative of order  > 0 due to Riemann-Liouville is defined by The following are important properties of Riemann-Liouville fractional integral   : () =  +  () ,  > 0,  > 0,     = Γ ( + 1) Γ ( +  + 1)  + . ( Definition 2. The Caputo fractional derivative   of a function () is defined as Some properties of Caputo fractional derivatives are as follows: International Journal of Differential Equations 3 where ⌈⌉ denotes the smallest integer greater than or equal to  and ⌊⌋ denotes the largest integer less than or equal to .
Similar to the integer-order differentiation, the Caputo fractional differential operator is a linear operator; that is, for  and  constants.

Genocchi Polynomials and Some Properties
Genocchi polynomials and numbers have been extensively studied in many different contexts in branches of mathematics such as elementary number theory, complex analytic number theory, homotopy theory (stable homotopy groups of spheres), differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), and quantum physics (quantum groups).The classical Genocchi polynomial   () is usually defined by means of the exponential generating functions [18][19][20].
where   () is the Genocchi polynomial of degree  and is given by − here is the Genocchi number.Some of the important properties of these polynomials include Before we move to the next level, we need the following linear independence on which the rest of theoretical results are based.
Proof.To show that  is the set of linearly independent elements of  2 [0, 1], it is enough to show that the Gram determinant is not zero.That is, where Now, to prove that this determinant is not equal to zero, we first reduce the Gram matrix to an upper triangular matrix by Gaussian elimination and it is not difficult to see that the elements of the diagonal of the reduced matrix are given by Clearly, one can see that, for any ∈ N, () ̸ = 0. Consequently, the determinant given by is not equal to zero.Therefore, the set  is the set of linearly independent sets.

Genocchi Operational Matrix
In this section, we derive the operational matrices for the delay and that of fractional derivative based on Genocchi polynomials for the solution of fractional pantograph equations.

4.1.
Genocchi Delay Operational Matrix.The Genocchi delay vector G( − ) can be expressed as where R is the  ×  operational delay matrix given by where Also, for any delay function ( − ), we can express it in terms of Genocchi polynomials as shown in (26): where C is given in (23).
The following lemma is also of great importance.
The proof of this lemma is obvious; one can use ( 7) and ( 8) on (10).

Genocchi Operational Matrix of Fractional Derivative.
If we consider the Genocchi vector G() given by G() = [ 1 (),  2 (), . . .,   ()], then the derivative of G() with the aid of ( 12) can be expressed in the matrix form by where Thus,  is  ×  operational matrix of derivative.It is not difficult to show inductively that the th derivative of G() can be given by In the following theorem, the operational matrix of fractional-order derivative for the Genocchi polynomials is given.
Theorem 5 (see [21]).Suppose that G() is the Genocchi vector given in (20) and let  > 0.Then, where   is  ×  operational matrix of fractional derivative of order  in Caputo sense and is defined as follows: where  ,, is given by − is the Genocchi number and   can be obtained from (23).

Upper Bound of the Error for the Operational Matrix of
Fractional Derivative   .We begin here by proving the upper bound of the error of arbitrary function approximation by Genocchi polynomials in the following Lemma.Lemma 6. Suppose that () ∈  +1 [0, 1] and  = Span{ 1 (),  2 (), . . .,   ()}; if C  G() is the best approximation of () out of , then where To see this, we set {1, , . . .,   } as a basis for the polynomial space of degree .
Since C  G() is the best approximation of () out of  and  1 () ∈ , from (18), one has Taking the square root of both sides, one has which is the desired error bound.
We use the following theorem from [22].
Theorem 7 (see [22]).Suppose that  is a Hilbert space and  is a closed subspace of  such that dim  < ∞ and  1 , The proof of this theorem obviously follows from Lemma 6.
The operational matrix error vector   is given by where Gram ( 1 () , . . .,   ()) ) By considering Theorem 8 and (43), we can conclude that by increasing the number of the Genocchi bases the vector    tends to zero.
For comparison purpose in Table 1, we show below the errors of operational matrix of fractional derivative based on Genocchi polynomials and shifted Legendre polynomials derived in [23,24] when  = 10 and  = 0.75 at different points on [0, 1].From this table, it is clear that the accuracy of Genocchi polynomials operational matrix of fractional derivative (GPOMFD) is better than the shifted Legendre polynomials operational matrix of fractional derivatives (SLPOMFD).We believe that this is the case for any value of  because the Genocchi polynomials have smaller coefficients of individual terms compared to shifted Legendre polynomials.

Collocation Method Based on Genocchi Operational Matrices
In this section, we use the collocation method based on Genocchi operational matrix of fractional derivatives and Genocchi delay operational matrix to solve numerically the generalized fractional pantograph equation.We now derive an algorithm for solving (1).To do this, let the solution of (1) be approximated by the first  terms Genocchi polynomials.Thus, we write where the Genocchi coefficient vector  and the Genocchi vector () are given by thus,     () and      (),  = 0, 1, . . .,  − 1, can be expressed, respectively, as follows: Substituting ( 44) and ( 46) in (1), we have where Also the initial condition will produce  other equations: To find the solution   () we collocate (47) at the collocation points   = /( − ),  = 1, 2, . . .,  − , to obtain for  = 1, 2, . . .,  − .Additionally, one can also use both the operational matrix of fractional derivative and delay operational matrix to solve problem (1).According to (44), we can approximate the delay function ( ,   +  , ) and its fractional derivative using the operational matrices P and R as follows: ( Hence, (49) or ( 52) is  −  nonlinear algebraic equation.Any of these equations together with (48) makes  algebraic equations which can be solved using Newton's iterative method.Consequently,   () given in (44) can be calculated.

Numerical Examples
In this section, some numerical examples are given to illustrate the applicability and accuracy of the proposed method.All the numerical computations have been done using Maple 18.
Example 1.Consider the following example solved in [25]: subject to The exact solution for this example is given by () =  2 .We solve the example when  = 0.6 and  = 0.3.In Table 2, we compare the errors obtained by our method with those obtained using FAM and new approach in [25].As reported in [25], the time required for the new method is 104.343750seconds and for the FAM the time taken is 215.031250 seconds for completing the same task, whereas in our method we only need 38.080 seconds to complete the computations.
Example 2 (see [1]).Consider the following generalized fractional pantograph equation: where The exact solution of this problem is known to be () =  3 .This problem is solved in [1] using generalized Laguerre-Gauss collocation scheme.We apply our technique with  = 4. Approximating (55) with Genocchi polynomials, we have and (59) gives Solving these equations, we have Thus, () = () is calculated and we have 0.9999938296 3 which is almost the exact solution.In Table 3, we compare the absolute errors obtained by our method (with only few terms  = 4) and the absolute errors obtained in [1] when  = 22 with different Laguerre parameters .
subject to The exact solution of this problem is known to be () =  2 .We solve (63) using our technique with  = 3 only.As in Example 2, we obtained the values of the coefficients to be Thus,   () = () is calculated to be  2 which is the exact solution and so there is nothing to compare for the error is zero.
The exact solution of this problem is known to be () =  2 .As in Example 3, we solve (66) using our technique with  = 3 and the values of the coefficients obtained are (68) Thus,   () = () is calculated and compared with the exact solution.This problem is solved using Taylor collocation method in [26] when  = 4, 5, and 6.In Table 4, we compare the absolute errors obtained by present method when  = 3 with the errors obtained when  = 4 in [26].

Conclusion
In this paper, a collocation method based on the Genocchi delay operational matrix and the operational matrix of fractional derivative for solving generalized fractional pantograph equations is presented.The comparison of the results shows that the present method is an excellent mathematical tool for finding the numerical solutions delay equation.The advantage of the method over others is that only few terms are needed and every operational matrix involves more numbers of zeroes; as such the method has less computational complexity and provides the solution at high accuracy.

Table 1 :
Comparison of the operational matrix errors for the GPOMFD and SLPOMFD.

Table 3 :
[1]parison of the absolute errors obtained by the present method and those obtained in[1]for Example 2.

Table 4 :
[26]arison of the absolute errors obtained by the present method and those in[26]for Example 4.