Generalized Spline Approach for Solving System of Linear Fractional Volterra Integro-Differential Equations

In this paper generalized spline method is used for solving linear system of fractional integro-differential equation approximately. The suggested method reduces the system to system of linear algebraic equations. Different orders of fractional derivative for test example is given in this paper to show the accuracy and applicability of the presented method.


Introduction
The concept of fractional or non-integer order derivative and integration can be traced back to the genesis of integer order calculus itself.Almost every mathematical theory applicable to the study of non-integer order calculus was developed through the end of 19 th century.However, it is in the past hundred year that the most intriguing leaps in engineering and scientific application have been found.The calculation technique has to change in order to meet the requirement of physical reality in some cases [1].The use of fractional differentiation for the mathematical modeling of real world physical problems has been wide spread in recent years,e.g. the modeling of earthquake , the fluid dynamic of viscoelastic material properties, etc, [2].There are several approaches to the generalization of the notation of differentiation of fractional orders, e.g.Riemann-Liouville, Grunwald-Letnikov, Caputo and generalized functions approach [3].In this paper, we present numerical solution of the system of integro-differential equation with fractional derivative.
With initial conditions:

2-
, where 2n [wi-1,wi] class of all functions defined on [wi-1,wi] has derivative of order 2n.Definition (2), [5]: Let : , → is an interpolating generalized spline function of f associated with the partition  and the operator L, if in addition to the conditions of definition (2.1), on  and for i = 0,…, N and g = 1,2, …, n 1 .
Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operatin .
Where, and µ are constants, For the Caputo's derivative we have [6]: We use the ceiling function ┌ ┐to denote the smallest integer greater than or equal to v , and 1,2, … .Recall that for ∈ , the Caputo differential operator coincides with the usual differential operator of linear order.

Solution of system of linear fractional integro-differential equation
In this paper, the generalized spline function is applied to study the approximate solution of system of fractional integro-differential of equations are given in equation ( 1) Be the generalized spline function to approximate the solution of equation ( 1) Where , 1, … ,2 be the basis function of generalized spline and 2n is order of * 0 and be the coefficients, 1, … , , ∈ , 1, … ,2 .Now, substituting equation (4) in equation ( 1), Adding the initial conditions of equation ( 1) as a new raw in the following matrices: or in the system form: and F are constant matrices with dimensions (N+ )×2n and (N+ )×1 respectively .
By solving the above system, we obtain the values of the unknown coefficients and the approximate solution of equation (1).
To demonstrate the accuracy and applicability of the presented method illustrative example for solving linear system of fractional integro-differential equations with different values of fractional derivative is provided.
Then by solving the system: Where is constant matrix and: Finally , Gauss elimination method may be used to solve system ( 21 To show the implementation of the method figure (1) is given the approximate solution of the system of fractional integro-differential equation.

Conclusions
In this paper, the application of generalized spline functions investigated to obtain approximate solution of system of linear fractional integro-differential equations and we give illustrative example with different α to show this approximation.As a comparison with the exact solution, table (1) and figure (1) showed the result.
∈ .Then the Caputo fractional derivative of the exponential function has the form: Г 1 , where , is the two-parameter function of Mittag-leffler type.Proof:To prove the theorem, the relation between Caputo and Riemann-Liouville fractional derivative: