SHIFTED GEGENBAUER OPERATIONAL MATRIX AND ITS APPLICATIONS FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS

SHIFTED GEGENBAUER OPERATIONAL MATRIX AND ITS APPLICATIONS FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS T.M. El-Gindy, H.F. Ahmed and Marina B. Melad Assuit University, Faculty of Science, Department of Mathematics, Assuit, Egypt. Minia University, Faculty of Science, Department of Mathematics, Minia, Egypt. Assuit University Branch of NewVally, Faculty of Science, Department of Mathematics, Assuit, Egypt.


Introduction
Recently, the recurrent appearance of ordinary and partial fractional differential equations have attracted the attentions of numerous studies in different fields like fluid mechanics, viscoelasticity, biology, physics and engineering,etc [1]; this is because the fractional-order models are more accurate than integer-order models [2][3][4].It is difficult to find exact solutions for several FDEs, so approximate and numerical mechanisms are introduced [5].Different analytical and numerical methods such as Adomian decomposition method [6][7][8], variational iteration method [9][10][11], homotopy perturbation method [12][13][14], homotopy analysis method [15,16], collocation method [17], Galerkin method [18], spectral methods [17,18] and other methods are investegated.For more decades, spectral methods have obtained a great interest in solving differential equations.These methods are characterized by their precision for any number of unknowns.There are three main spectral images, they are the Galerkin, collocation and Tau methods [19].In the spectral methods, the explicit formula for operational matrices of fractional integration and fractional differentiation for classical orthogonal polynomials are needed [19].The mechanism of the proposed method depends on transforming the FDEs into a system of AEs which is very easy to solve.Recently, some types of orthogonal polynomials have been introduced as basis functions of the operational matrices of fractional derivatives and integrals which are used to solve ordinary and partial fractional differential equations [21,22].In this paper we investigate the operational matrix of the fractional integral of the shifted Gegenbauer polynomials and use it with the Tau method to present numerical solutions to MOFDEs and systems of fractional differential equations.The Gegenbauer polynomials have many useful properties.The most important characteristic of them is achieving rapid rates of convergence, for more details see [23][24][25][26].The paper is organized as follows.In section 2, we review necessary definitions and properties of fractional calculus and ultraspherical (Gegenbauer) polynomials.In section 3, the SGOM of fractional integration is proved.In section 4, the proposed mechanism of applying SGOM of fractional integration for solving linear MOFDEs and systems of fractional differential equations is discussed.In section 5, Some applications of the proposed method are given.Finally a conclusion is given in section 6.

Fractional Calculus Definitions
Definition 1.One of the popular definition of fractionl integeral is the RL, which get from the relation For more properties of I ν see [27], we just recall the next property D ν is the RL fractional derivative of order ν is defined by where m is the smallest integer order greater than ν.Lemma 1.

Shifted ultraspherical (Gegenbauer) polynomials and some properties
The ultraspherical (Gegenbauer) polynomials C (α) j (x), of degree j ∈ Z + , and associated with the parameter α > −1 2 are a sequence of real polynomials in the finite domain [−1, 1].They are a family of orthogonal polynomials which has many applications.Definition 1.The Gegenbauer polynomials are the Jacobi polynomials, P • There are useful relations between the Chebyshev polynomial of the first and second kind and the Legender polynomial with the Gegenbaure polynomials as follows • The Gegenbauer polynomials can be created from the next recurrence equation • The orthogonality relation of the Gegenbauer polynomials is given by the weighted inner product where ω (α) (x) is the weight function, it is an even function given from relation and is the normalization factor and δ i,j is the Kronecker delta function.
• The shifted Gegenbauer polynomials are formed by replacing the variable x with 2x L − 1, 0 ≤ x ≤ L. So, we can write shifted Gegenbauer polynomials as • The analytical form of the shifted Gegenbauer polynomial is given from S,j (0) = (−1) j Γ(j + 2α) Γ(2α)j! . (2.5) • The orthogonal relation of shifted Gegenbauer polynomials is getting from where ω (α) S (x) is the weight function, it is even function given from the relation • This polynomial recover the shifted Chebyshev polynomial of the first kind T S,j (x) ≡ C (0) S,j (x), the shifted Legendre polynomial L S,j (x) ≡ C S,j (x), and the shifted Chebyshev polynomial of the second kind C S,j (x) ≡ 1 j+1 U S,j (x).
• Consider f (x), g(x) be integrable functions in [0, L].These functions can be approximated by shifted Gegenbauer polynomials as: S,N,j (x) = G T φ(x), where fS,N,i , gS,N,j are obtained from relation (2.9).The approximation of the product f (x)g(x) can be obtained from the following relation where Υ T = [υ S,N,0 , υS,N,1 , . . ., υS,N,k ] is an unknown vector, its elements are calculated by υS,N,k = 1 (2.13) • The q times repeated integration of Gegenbauer vector is obtained by where P (q) is called the operational matrix (OM) of the integration of φ(x).
At this section, the shifted Gegenbauer operational matrix (SGOM) of fractional integration in [0, 1] will be proved.
Theorem(1) Let φ(x) be the shifted Gegenbauer vector and ν > 0 then where P (ν) is called OM of fractional integration of order ν in the RL sense, it is a square matrix of order (N + 1) × (N + 1) and is written as where ξ i,j,k is given by:
Writing the last equation in a vector form gives which finished our proof.
4 Procedure Solution of SGOM for Solving Fractional Differential Equations

Multi-order fractional differential equations
At this section, we apply the SGOM method to a MOFDE with RL fractional derivative.So, let ν be the highest fractional order of FDE.By using the properties of fractional integral, the proposed FDE is transformed into an integral equation which can be approximated by using SGOM.Consider the following MOFDE: with the initial conditions where γ i (i = 1, . . ., k + 1) are real constant coefficients, m − 1 < ν ≤ m, and 0 < β 1 < β 2 < . . .< β k < ν.Moreover D ν y(x) ≡ y (ν) (x) refers to the RL fractional derivative of order ν.
The existence, uniqueness and continuous dependence of the solution of the problem (4.1-4.2) are discussed in [28].By applying the RL integral of order ν on Eq.(4.1) and use Eq.(2.4), we get where where The functions y(x) and g(x) are approximated by shifted Gegenbauer polynomials as where the vector G = [g 0 , g 1 , . . ., g N ] T is given and T is an unknown vector.By using theorem (3) the RL integral of order ν and (ν − β j ) of Eq. (4.5), can be written as and where P (ν) is the (N + 1) × (N + 1) square matrix of fractional integration of order ν.Using Eqs.(4.5-4.8) the residual R N (x) for Eq.(4.4) can be written as By using Tau method, we generate (N − m + 1) linear algebraic equations from S,N,j (x) By using Eq.(4.5) in Eq. (4.2), we get
The RL integral of orders (ν) and (ν − D r ) of Eq. (4.18), can be written as and where P (ν) is the (N + 1) × (N + 1) square matrix of fractional integration of order (ν).By using Eq.(4.18-4.21) the residual R N,i (x) for system (4.17) can be written as By applying the Tau method, we generate (N − m + 1)n linear algebraic equations From Eqs.(4.23) and (4.24), (N − m + 1)n and mn set of linear algebraic equations are generated.This linear system can be solved easily for the unknowns coefficient of the vector Y i , i = 1, . . ., n.So from Eq. (4.18) we can calculate y N,i (t) which is the solution of problem (4.12).

Systems of fractional non-linear differential equations with initial conditions
Consider the following system of non-linear FDEs With the initial conditions y where F i (i = 1, 2, . . ., r) are non-linear functions.To solve this system by using SGOM, we will follow the same procedure as discussed in the subsection (4.2) with the help of the Eqs.(2.12)-(2.13).

Illustrative Problems
Now, some problems are given to clarify the applicability and accuracy of the proposed mechanism.
From Eq. (4.5) the approximate solution with N = 3, is written as 2 ) φ(x).From theorem (3), we have Unfortunately, to the best of our knowledge, there is no available data in literature to compare this running time with other authors.For this purpose, we developed a mathematica code to run the same problem using the method shifted Jacobi operational matrix(SJOM) [19] on our computer, the run time found to be 4.73 second which is less than our method.The approximate and exact solutions of Problem ( 1) are displayed in Figure 1 while the absolute errors between the exact and approximate solution at N=3 are listed in Table 1.
Then the solution will be written as Executing this problem takes 2.26 second using Mathematica 9 software on CPU Intel(R) Core(TM) i3 at ν = 1, N = 3.Unfortunately, to the best of our knowledge, there is no available data in literature to compare this running time with other authors.For this purpose, we developed a mathematica code to run the same problem using the method shifted Jacobi operational matrix(SJOM) [19] on our computer, the run time found to be 2.18 second which is less than our method.Figure 2 show the approximate and exact solutions of Problem (2).The absolute errors between the exact and approximate solution at N=3 are tabulated in Table 2. x Absolute error 0 0 0.1 1.33511 × 10 − 14 0.2 5.09853 × 10 − 14 0.3 1.09118 × 10 − 13 0.4 1.8395 × 10 − 13 0.5 2.71838 × 10 − 13 0.6 3.6901 × 10 − 13 0.7 4.71567 × 10 − 13 0.8 5.75984 × 10 − 13 0.9 6.78457 × 10 − 13 1 7.75158 × 10 − 13 Table 2: Absolute error of proposed method for Problem (2).
So, the solution is The executing of this problem takes 3.18 second using Mathematica 9 software on CPU Intel(R) Core(TM) i3 at ν = 1, N = 2. Figure (3), displays a comparison between the approximate solution and exact solution.Table (3) illustrates the The second initial condition is for ν > 1 only.The exact solution of this problem is [29] y .

Conclusions
A general formulation of the SGOM of RL fractional integral has been developed.This method has been used to approximate the solutions of a set of MOFDEs and systems of FDEs.The proposed mechanism has been depended on the shifted Gegenbauer polynomials and the Tau method.The applicability, accuracy and rapidity by using few terms of the shifted Gegenbaur polynomials of the proposed mechanism are illustrated by numerical problems.
α) S,N,j (0) = d i , i = 0, 1, . . ., m − 1. (4.11) From Eqs.(4.10) and (4.11), (N − m + 1) and m set of linear algebraic equations are generated.This linear system can be solved easily to get the unknown coefficients of the vector Y.So from Eq.(4.5) we can calculate y N (x) which is the solution of problem (4.1).

Table 1 :
Absolute error of proposed method for Problem (1).

Table 6 :
1 exact y 1 approx.Absolute error of y 1 y 2 exact y 2 approx.Absolute Error of y 2 Absolute error of y 1 and y 2 for N=5 of Problem (5).
[31]lem(6): HIV Model Find the solution of the following non-linear system of FDEs[31]