An Accurate Spectral Galerkin Method for Solving Multiterm Fractional Differential Equations

This paper reports a new formula expressing the Caputo fractional derivatives for any order of shifted generalized Jacobi polynomials of any degree in terms of shifted generalized Jacobi polynomials themselves. A direct solution technique is presented for solving multiterm fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using spectral shifted generalized Jacobi Galerkin method. The homogeneous initial conditions are satisfied exactly by using a class of shifted generalized Jacobi polynomials as a polynomial basis of the truncated expansion for the approximate solution. The approximation of the spatial Caputo fractional order derivatives is expanded in terms of a class of shifted generalized Jacobi polynomials with , and is the polynomial degree. Several numerical examples with comparisons with the exact solutions are given to confirm the reliability of the proposed method for multiterm FDEs.

Spectral method is one of the principal methods of discretization for the numerical solution of most types of differential equations. The three most widely used spectral versions are the Galerkin, Tau, and collocation methods (see, for instance [26][27][28][29][30][31][32]). Recently, spectral method is a class of important tools for obtaining the numerical solutions of fractional differential equations. They have excellent error properties and they offer exponential rates of convergence for smooth problems. In the present paper we intend to extend the application of Galerkin method based on generalized Jacobi polynomials form solving linear problems to solve multiterm FDEs. To the best of our knowledge, there are not so many results on using this technique to solve such problems arising in mathematical physics. This partially motivated our interest in such a method.
Spectral Galerkin method for the numerical solution of fractional differential equations is characterized by expanding the solution by a truncated series of the trial functions. The unknown coefficients of this expansion will be determined by minimizing the error between the exact and numerical solutions in appropriate weighted space. This method provides exponential rates of convergence. An explicit expression for the derivatives of an infinitely differentiable function of any degree and for any fractional order in terms of the function itself is needed. Doha et al. [16] have obtained such a relation in the case of the basis functions of expansion that are shifted Jacobi polynomials. Another formula for shifted Legendre coefficients is obtained by Bhrawy et al. [17]. Moreover, in [33] the authors expressed explicitly the Caputo fractional derivatives of generalized Laguerre polynomials of any degree in terms of the generalized Laguerre polynomials themselves to solve fractional initial value problems on the half line.
An explicit expression for any Caputo fractional order derivative of the shifted generalized Jacobi polynomials of any degree in terms of the shifted generalized Jacobi polynomials themselves is the first goal of this paper. The fundamental goal of this paper is to develop a direct solution technique based on shifted generalized Jacobi-Galerkin method (SGJG) for solving multiterm FDEs with homogeneous and nonhomogeneous initial conditions. Finally, we present some numerical results exhibiting the accuracy and efficiency of our numerical algorithm.
The next section of this paper is for fractional preliminaries. Section 3 is devoted to proving a formula that expresses the Caputo fractional order derivative of the shifted generalized Jacobi polynomials. In Section 4, we construct and develop algorithms for solving linear FDEs by using shifted generalized Jacobi Galerkin spectral method. In Section 5, several examples are presented. Finally, some concluding remarks are given in the last section.

Preliminaries and Notations
In this section, we present some basic knowledge of fractional calculus, orthogonal shifted Jacobi polynomials, and generalized Jacobi polynomials these are most relevant to spectral approximations.
Similar to the integer-order differentiation, the Caputo's fractional differentiation is a linear operation; that is, where and are constants.
It is convenient to standardize the Jacobi polynomials so that where ( ) = Γ( + )/Γ( ). In this form the polynomials may be generated using the standard recurrence relation of Jacobi polynomials starting from , 0 ( ) = 1 and where = + + 1.

Generalized Jacobi Polynomials.
Recently, Guo et al. [36] presented and developed the generalized Jacobi approximation, in which the parameters and considered in the generalized Jacobi polynomialŝ, ( ) might be any real numbers. In this section, we give some properties of such polynomials. Let̂= We denote the set of integers by Z. For any , ∈ Z, the generalized Jacobi polynomials are defined by (see [36,37]) For our present purposes it is convenient to use the shifted Jacobi polynomials , ( ); let = (0, 1) and , ( ) = (1 − ) . We define the shifted GJPs and separate them into four cases as follows.
Proof. Firstly, Secondly, Thirdly, And lastly, A function ( ), square integrable in (0, 1), can be expressed in terms of shifted generalized Jacobi polynomials as where the coefficients are given by Proof. The analytic form of the shifted generalized Jacobi polynomials ,− ( ) of degree − is given by (26). Using Now, approximating + −] by terms of shifted generalized Jacobi series, we have where , is given from (28) where ] ( , , , ) is given as in (30), and this proves the theorem.
Mathematical Problems in Engineering 5

Shifted Generalized Jacobi Galerkin Method for FDEs
In this section, we are interested in employing the SGJG method to solve the linear multiterm FDE subject to the homogeneous initial conditions where ( = 1, . . . , ) and 0 < 1 < 2 < ⋅ ⋅ ⋅ < −1 < ], − 1 < ] ≤ are constants, ] ( ) ≡ (]) ( ) denotes the Caputo fractional derivative of order ] for ( ), and ( ) is a given source function. Let us first introduce some basic notation that will be used in the upcoming sections. We set where V ( ) ( ) denotes th-order differentiation of V( ) with respect to . Then the shifted generalized Jacobi-Galerkin approximation to (36) is to find ∈ such that By virtue of (31) and making use of the orthogonality relation of shifted generalized Jacobi polynomials (21), and after some rather lengthy calculation, we get Thereby, we can write (41) in the following matrix system form

Illustrative Examples
Several test examples are carried out in this section. The results obtained by the present methods reveal that the present method is very effective and convenient for linear FDEs.
whose exact solution is given by ( ) = 13 .
In Table 2, we present the maximum absolute errors, using SGJG method with various choices of ] and .
In Table 3, we present the maximum absolute errors, using SGJG method with various choices of ] and .
In Table 4, we exhibit maximum pointwise error using SGJG method with two choices of the shifted generalized Jacobi parameters , and = 8, 12, 16, 20, 24. We observe from this table that the suggested algorithm provides accurate and stable numerical results. This numerical experiment demonstrates the utility of the method.

Conclusion
We have derived a new formula expressing explicitly the Caputo fractional derivatives for any fractional-order of shifted generalized Jacobi polynomials of any degree in terms of shifted generalized Jacobi polynomials themselves. We have derived a Galerkin method, involving a specified class of the shifted generalized Jacobi polynomials, which permits us to numerically solve an important class of FDEs. Indeed, in Section 5, we demonstrated that for all parameter shifted generalized Jacobi considered, the method results in rather small errors with relatively few modes are considered. Since the method is rather robust, it is likely that it may be applied to other types of FDEs. For instance, one-and two-dimensional time-dependent FDEs Copyright of Mathematical Problems in Engineering is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.