Numerical Solution of System of Fractional Delay Differential Equations Using Polynomial Spline Functions

The aim of this paper is to approximate the solution of system of fractional delay differential equations. Our technique relies on the use of suitable spline functions of polynomial form. We introduce the description of the proposed approximation method. The error analysis and stability of the method are theoretically investigated. Numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.


Introduction
Recently, the use of various types of spline function in the numerical treatment of ordinary differential equations [1]- [5] and delay differential equations [6]- [11] has been increasing.Many interesting applications in the area of mathematical biology, mathematical model of numerous engineering and physical phenomena have been studied [12] [13].The fractional differential equation of the form is studied by Kia Dithelm and N. J. Ford [14].In [15] [16], the Adams-Bashforth-Moulton method is used to approximate solutions of the initial value problem (1).An alternative is the backward differentiation formula presented in [17] where the idea of this method is based on discretizing the differential operator in the fractional differential Equation (1) by certain finite difference.Extrapolation principles in [7] are applied to improve the performance of the method presented in [17].Kia Dithelm in [18] studied that a fast algorithm for the numerical solution of initial value problems of the form (1) in the sense of Caputo identifies and discusses potential problems in the development of generally applicable schemes.More recently, Lagrange multiplier method and the homotopy perturbation method are used to solve numerically multi-order fractional differential equation see [19].
They assume that the functions ( ) , 1, 2 q i f i = and 0,1, 2, , q r = satisfy the Lipschitz condition of the form: , , , , , × × An extension of the spline functions form defined in [19] for approximating the solution of system of ordinary differential equations is investigated, namely, for the system (2) with unique solution [15] the solution of the first order delay differential equation of the form: , , , ( ) Ramadan, Z. in [21] discussed the system of the initial value problem where = , his method was presented which uses polynomial spline to approx- imate the solutions of the system.

Description of the Proposed Spline Approximation Method
Consider the system of first order delay differential equations: , .
The function g is called the delay function and it is assumed to be continuous on the interval [ ] , a b and to satisfy the inequality is continuous and satisfies Lipsechitz condition , , , , , , and there exists a constant 1 B such that , , , , , , ) [ ] ( ) ( ) is continuous and satisfies the Lipsechitz condition: , , , , , , and there exists a constant 2 B such that with These conditions assure the existence of unique solution y and z of system (4).
Let ∆ be a uniform partition to the interval [ ] , a b defined by the nodes Define the new form of system of fractional spline function ( ) S x and ( ) S x of polynomial form approximating the exact solution y and z by: ) ) ( ) , , , ( ) ( )

S x
∆ exist and are unique.

Error Estimation and Convergence Analysis
To estimate the error of the approximate solution, we write the exact solution ( ) y x and ( ) z x in the following Taylor form [11]: ) ) where ( ) ( ).
Moreover, we denote to the estimated error of ( ) y x and ( ) and at k x denote to the error , Define the modulus of continuity of Next lemma gives an upper bound to the error.

x e x e d e e h d e e i r d e e h h e h d e e i r d e h d e h h
( )  ( )

k d e h h h i r h e d e h d he h r h d h e d he h r
In the same manner we can prove that The lemma is proved.

Stability Analysis of the Proposed Method
For analyzing the stability properties of the given method, we make a small change of the starting values and study the changes in the numerical solution produced by the method.Now, we define the spline approximating function ( ) ( ) and W x W x as: ) ) where ( ) , , ( ) and use the notation , and , , and ,

e d e h d e h dh e d e h *
where, ( ) ( ) In the same manner we can prove that ( ) ( ) Thus the lemma is proved.

Numerical Example
Consider the system of fractional ordinary delay differential equations The exact solution is given by 2 2 and y x z x = = .The obtained numerical results are summarized in Table 1, Table 2 to illustrate the accuracy and the stability of the proposed spline method using spline function of polynomial form.The first column in each table, represents the different values of α, the second column represents the values of x.The third column gives the approximate solution at the corresponding points while the fourth column gives the absolute error between the exact solution and the obtained approximate numerical solution with the initial conditions ( ) ( ) . With small change in the initial conditions, ( ) ( ) 0 0.00001 and 0 0.00001 y z * * = = , the approximate solution is computed as    shown in the fifth column.To test the stability, the difference between the two approximate solutions is computed as shown in the six column From the obtained results in Table 1, Table 2 respectively, we can see that the proposed method gives acceptable accuracy and the method is shown to be stable.Moreover, the algorithm of the proposed method has recursive nature which makes it easy and simple to be programmed.

Conclusion
We adapt the spline functions with some additional assumptions and definitions for approximating the solution of system of ordinary delay differential equation with fractional order which studied in [7] [8].The error analysis and stability are theoretically investigated.A numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.The obtained numerical results reveal that the methods are stable and give high accuracy.

Table 1 .
The accuracy and stability of the proposed spline method using spline function of polynomial form (using h = 0.01).

Table 2 .
The accuracy and stability of the proposed spline method using spline function of polynomial form (using h = 0.01).