Advances in Mathematical Physics

Volume 2017 (2017), Article ID 3821870, 12 pages

https://doi.org/10.1155/2017/3821870

## New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations

Department of Mathematics and Statistics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, Johor, Malaysia

Correspondence should be addressed to Chang Phang

Received 14 September 2016; Revised 10 November 2016; Accepted 8 December 2016; Published 16 January 2017

Academic Editor: Luigi C. Berselli

Copyright © 2017 Jian Rong Loh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function . This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval . This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution . A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.

#### 1. Introduction

Fractional integro-differential equation (FIDE) is an equation that contains a fractional derivative term and an integral kernel operator term . An example is given as follows:Here, is the unknown function to be solved and , are the Volterra integral kernel and Fredholm integral kernel, respectively. is Caputo’s fractional derivative and , are coefficient functions.

In the past few years, various wavelet operational matrices and polynomial operational matrices for both fractional differentiation and fractional integration have been derived one after another to solve various kinds of fractional differential equations (FDEs) and fractional integro-differential equations (FIDEs). Examples of polynomial operational matrices that have been proposed in the past are Shifted Legendre polynomials [1], shifted Chebyshev polynomials [2], and shifted Jacobi polynomials [3] to name a few. The operational matrix method is usually combined with methods such as collocation method, spectral tau method, and tau method to solve FDE and FIDE. In [4], Zhu and Fan have proposed Chebyshev wavelet operational matrix of fractional integration and applied it to solve a certain type of nonlinear fractional integro-differential equations (FIDEs). Apart from the operational matrix approach, there exist various approaches such as Adomian decomposition method [5], spectral tau method [6], Taylor expansion method [7], and Bernoulli matrix method [8]. In [9], Alshbool et al. have proposed a new Bernstein function to solve fractional order differential equations.

In this paper, different from the existing methods in numerically solving FIDE, we explore the application of a relatively new polynomial, namely, Genocchi polynomials, in the numerical solution of FIDE. Genocchi polynomial belongs to a larger family class of polynomials which is the Appell polynomial family. Throughout this paper, we denote Genocchi polynomials by . We apply these polynomials to solve the FIDEs given the advantage of Genocchi polynomials having smaller coefficients of each individual term and relatively lesser terms compared to classical orthogonal polynomials. Thus, this is expected to provide us with smaller computational errors.

The rest of the paper is organized as follows: Section 2 introduces preliminary definitions and properties of the Caputo fractional derivative. Section 3 gives basic definitions and properties of , function approximation by , and the analytical expression of which is the integral of the product of two Genocchi polynomials. Section 4.1 derives the Genocchi polynomial operational matrix of Caputo’s fractional derivative. Section 4.2 shows the way to approximate the integral kernel in terms of Genocchi polynomials and derives the analytical expression of the kernel matrix. Section 5 presents the general approach of approximating (1) into a system of algebraic equations and using collocation methods to solve (1). Section 6 describes the general procedure of using Genocchi polynomials approximation to solve (1). Section 7 shows results of the proposed method. Section 8 sums up the findings of this paper.

#### 2. Preliminaries

##### 2.1. Fractional Calculus: Definitions

Fractional differentiation comes in different versions [11, 12]. In this paper, we consider Caputo’s fractional differentiation which provides a more realistic physical interpretation in real-life applications. Caputo’s fractional derivative operator of a function is defined as follows:

*Definition 1. *

Below are some properties of Caputo’s fractional derivatives: where denotes the smallest integer greater than or equal to and denotes the largest integer less than or equal to .

Caputo’s fractional differential operator is linear:where and are arbitrary constants.

#### 3. Genocchi Polynomials and Function Approximation

##### 3.1. Genocchi Polynomials: Definitions and Basic Properties

The Genocchi polynomials are defined as follows [13–15].

*Definition 2. *The Genocchi polynomial is defined by the generating function :where is the Genocchi number. are the Bernoulli number and Bernoulli polynomial, respectively.

Below are some of the important properties of Genocchi polynomials:

##### 3.2. Function Approximation by Genocchi Polynomials

We may approximate a function in terms of Genocchi polynomials by the following infinite series:where are the Genocchi polynomials and the Genocchi coefficients .

In practice, we truncate the infinite series up to number of Genocchi polynomials according to the desired accuracy of the problem into the following truncated Genocchi series:where in matrix notation,where is the Genocchi coefficient vector.

is the Genocchi vector.

In this case, the Genocchi coefficients may be computed as follows:

##### 3.3. Computing Genocchi Coefficients by Matrix Approach

Equation (10) fails to work for functions that are not differentiable at the points and . An example is shown below.

Let ; ; To avoid this problem that occurs in the case of differentiation, we may evaluate the Genocchi coefficients using Theorem 4. To prove Theorem 4, we first prove the following Theorem 3 which gives the analytical expression of the integral of the product of two Genocchi polynomials over an arbitrary interval , which will be used in the subsequent part of the paper.

Theorem 3. *Given any two Genocchi polynomials , , for where is the Genocchi number, , are the falling and rising factorial, respectively. In particular, we have the following for , *

*Proof. *Using the following expression: We obtain Continuing this relation recursively for times, we arrive at

Adopting the techniques used in [16] for function approximation, we prove the following theorem of function approximation using Genocchi polynomials.

Theorem 4. *Let and be the set of Genocchi polynomials up to order . Let . Since is a finite dimensional closed subspace of , then is the unique best approximation in Genocchi polynomials such that can be approximated by unique coefficients :The Genocchi coefficient matrix consisting of the unique coefficients is given by the following:where and as derived in Theorem 3.*

*Proof. *Let . Then, Thus, we have a system of equations which is written in matrix representation as follows: Therefore, the Genocchi coefficients matrix is where can be obtained from Theorem 3.

#### 4. Main Result

##### 4.1. Genocchi Polynomial Operational Matrix of Caputo’s Fractional Differentiation

In this section, we derive the analytical expression of the Genocchi polynomial operational matrix of Caputo’s fractional derivative, which is the matrix , where To derive , we first prove the following Lemmas 5 and 6.

Lemma 5. *Caputo’s fractional derivative of fractional order of a Genocchi polynomial of order is given by*

*Proof. *For , where . Substituting where is the Beta function for .

Lemma 6. *The matrix defined as is given by the following:where*

*Proof. *From Lemma 5,

Now, we can prove the following theorem that computes the operational matrix .

Theorem 7. *Given a set of , , of Genocchi polynomials, the Genocchi polynomial operational matrix of Caputo’s fractional derivative of order over the interval is the matrix and is given bywhere is given in Lemma 6 and with elements given in Theorem 3.*

*Proof. *From Lemma 5,

Here, we justify the better accuracy of the new operational matrix of Caputo’s fractional derivative due to Genocchi polynomials by comparing its errors and that obtained when using Shifted Legendre operational matrix. This may be due to the fact that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. We define the absolute error and relative error respectively, as follows: where is either the Genocchi polynomial or Shifted Legendre polynomial . In Figure 1, we compare the relative error obtained due to Genocchi polynomial operational matrix of fractional derivative (GPOMFD) and the Shifted Legendre operational matrix of fractional derivative (SLOMFD) derived in [17]. One clearly sees from the figure that the errors for GPOMFD become smaller than that of SLOMFD for and their difference gets larger as increases over the interval . Further, we also show the absolute error and relative error in Tables 1 and 2, respectively, for at different points.