A New Technique for Solving Fractional Order Systems : Hermite Collocation Method

In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.


Introduction
As cited in [1] [2] [3], fractional order differential equations can be considered as a generalization of integer order ones and it is approved that the mathematical modelling on physical processes naturally leads to differential equations for fractional order.Consequently, applications of fractional differential equations appear very frequently in many fields, such as engineering, physics, finance, chemistry and bioengineering [1] [2] [4] [5] [6].Unfortunately, the resulting model equations are usually difficult to solve analytically.Therefore, it is vital to develop some numerical or approximate techniques.Nowadays, the studies on fractional order differential equations and their solutions have become very popular and attracted the attention of many researchers.So far many numerical or approximate schemes have been developed.Among them, finite difference approximation methods [7] [8] [9] [10], fractional linear multistep methods [11] [12] [13], quadrature formula approach [14], the Adomian decomposition method [15] [16] [17], variational iteration method [17] [18] [19], and differential transform method [20] [21] can be accounted.For some class of fractional order differential equations polynomial approximation methods were also given by Kumar and Agarwal and the references can be found in [22] [23] [24].
So far, a lot of works published on fractional order linear/non-linear differential equations but there are still works have to be done.In this work, we aim to extend the Hermite Collocation method (HCM) for obtaining solution to a system of fractional order differential equations with variable coefficients and specified initial conditions.
The technique constructs an analytical solution of the form of a truncated Hermite series with unknown coefficients.The orthogonal Hermite polynomials have the importance in the theory of light fluctuations and quantum states and, in particular, some problems of coastal hydrodynamics and meteorology [25].This method is the adaptation of Taylor collocation method with Hermite polynomials and first has been used to solve higher-order linear Fredholm integro differential equation in [26] and the development of the method can be found in [26].This paper is organized as follows.Section 2 involves some basic definitions and properties of fractional calculus.In Section 3, the theory and definitions of Hermite collocation method and the construction of this method for fractional order systems are presented.In Section 4, the matrix relations for initial conditions are defined and the Section 5 deals with the error estimate for the method.Section 6 involves some illustrative examples.Finally, the last section concludes with some remarks based on the reported research.

Preliminary and Notations
We first recall the following known definitions and preliminary facts of fractional derivatives and integrals which are used throughout this paper.
where Γ is the gamma function and , 0 x a α ≥ > .For consistency, we take 0 which is identity operator and holds ( ) ( ) Some properties of Caputo derivative are given as follows:

Establishing Hermite-Collocation Method for Fractional Order Systems with Variable Coefficients
In this section, we will consider the following system of fractional order differential equations (FDEs) with variable coefficients, ij P , , 1, 2, , , where , ij i P x g x are continuous functions on a x b ≤ ≤ .The initial conditions are defined as ( ) ( ) In Equation ( 5 where js a defines the unknown Hermite coefficients of the solution and N is a positive integer which is chosen sufficiently small for avoiding the laborious work such that N t ≥ .Therefore, the fundamental matrix relation of Equation ( 4) can be written as ( ) ( ) , .
Now, we need to define the Caputo derivatives of ( ) . By using Equation ( 6), therefore, we write (see [26]), ( ) ( ) where H and j A are defined as respectively.Now, we will describe the matrix representation of the truncated Hermite series in terms of rational power of the indepandant variable x, by using the following generalized formula: Now, in terms of N being odd or even, we denote the truncated series in matrix notation such as follows (see [26]): If N is an odd number: ( Hence, we have ( ) ( ) and letting, ( ) and then, substitution of Equation ( 11) into Equation (6) yields, ( ) ( ) T , 1, 2, , .
Now, the nαth order Caputo derivative of Equation ( 12) is written as or equivalently: Here, the matrix B is defined as follows ( ( ) [ ] ( ) ( ) Hence, if we substitute Equation ( 14) into Equation ( 13) we have: Therefore, the matrices in Equation ( 15), for 1, , j k =  , are clearly shown by where the each submatrix, T T , B F consisting of ( ) Consequently, the above matrix equation can be written as, where ( ) X x α appears as consisting of k rows and ( ) Equation ( 7) then, we have where n P and G are of the form: .


Apart from this, arranging Equation ( 16) for each collocation points then, we can write explicitly as, . ) Consequently, now we denote Equation ( 7) of the form: Then, by writing Equation (18) in Equation ( 19), the matrix form of the system of .
Now, Equation ( 21) constructs an algebraic system.To obtain the solution of the above system, the augmented matrix is written as follows: [ ] ( ) Solving the above system, as a result, we obtain the desired Hermite coefficients in the truncated Hermite series.Hence, writing j A in Equation ( 12) we evaluate the un- knowns ( ) j y x of the system of FDEs (Equation ( 4)).

Matrix Relations for Initial Conditions
In generally, we look for the solution of the system of FDEs under specified conditions.
However, preceding calculations do not involve these conditions.Therefore, we need to incorporate these conditions into the work.Then, we have to establish the new form of Equation ( 22) which involves initial conditions, Equation ( 5).Now, we start by writing Equation ( 5) explicitly for each 1, 2, , j k =  same as below: , , , k y y y  .For example, for 1 y we obtain t conditions such as follows: where and for 1, 2, , , , . Now writing Equation ( 16) into Equation ( 23) for , x a x b = = , we obtain Now, calling U as, then, the Hermite polynomial coefficients matrix which corresponds to the given initial conditions (Equation ( 5)), can be written as In Equation ( 25), U involves kt rows and ( ) × rows in Equation ( 21) and then replacing these rows by Equation ( 25), we obtain the whole augmented matrix of the system, , as follows: ; Hence, the system of algebraic equations of which unknowns are the hermite polynomial coefficients are shown by ( ) By the above theorem, the matrix of Hermite coefficients, A is uniquely determined by Equation (27).Finally, substitution of these coefficients into the truncated Hermite series gives the desired solution of the form:

Error Estimate for the Solution
The truncated Hermite series, Equation ( 29), is the approximate solution of Equation ( 4) with the given initial conditions, (Equation ( 5)).Since this solution should approximately satisfy the Equation ( 4) hence, the residuals

∑∑
give the error at the particular points Let us now denote the residuals by ( ) q E x as an error function.The error should be approximately zero or ( ) where i k is any positive constant.If the ( ) = is prescribed before then, the truncation limit for N is increased until ( ) q E x becomes smaller than 10 kα − (see [21] [26]).

Numerical Applications
The technique which we have developed to solve fractional order systems is quite feasible and accurate.To show the accuracy of the method the following system of FDEs with variable coefficients are solved.All the numerical calculations have been performed by using MatlabR2007b.

FDEs
, the system of FDEs is simply shown by [ ] or ; using the above relations, we obtain t-conditions for each unknown, 1 2

]
Figure1shows the HCM solution of the system, Equation(30).Figure2(a)) shows the Differential Transform solution and Figure 2(b)) Adomian Decomposition solution of the same system.

Figure 1 .
Figure 1.Approximate solution of the system in Example 1 by HCM.

Figure 2 .
Figure 2. (a)Approximate solution of the system in Example 1 by Differential Transform Method, (b) Adomian Decomposition Method.

Figure 3 .
Figure 3. Pollutant problem scheme of three lakes which connected by the channels.