An Efficient Numerical Method for Solving Volterra-Fredholm Integro- Differential Equations of Fractional Order by Using Shifted Jacobi-Spectral Collocation Method

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.


Introduction:
Fractional Calculus has been pulling in the consideration of researchers and specialists from long time ago, resulting in the improvement of numerous applications. Since the nineties of a century ago fractional calculus is being rediscovered and connected in an expanding number of fields, namely in several areas of Physics, Control Engineering, and Signal Processing, such as electromagnetism, communications, sciences, control, robotics, information and many other physical sciences and also in medical sciences (1,2,3,4,5).
Integral equations can be described as being a functional equation involving the unknown function under one or more integrals.
Differential equations as well as integral equations of fractional order belong to a wider class of equations in which the unknown object is a function (scalar function or vector function). Such kinds of equations are often encountered in mathematics and in various sciences that use the mathematical apparatus, and they are generally called functional equations.
Department of Accounting Al-Esra'a University College, Baghdad, Iraq E-mail: mohammed.ghazi@esraa.edu.iq An Integro-differential equation is an equation in which the unknown functions appear with derivatives, and either the unknown functions, or their derivatives, or both, appear under the sign of integration. This, however, is a purely formal classification, since we can easily pass from one type to the other. Numerous scientific models of physical wonders contain integro-differential equations; these equations emerge in numerous fields like physics, potential theory, astronomy, biological models, chemical kinetics and fluid dynamics.
As a special form of integro-differential equations of fractional order Volterra-Fredholm integro-differential equations of fractional order (36,37). That primary point of the Jacobi-collocation method over other techniques may be that Jacobi-collocation method provides a great finer rate for convergence (38,39).
In this article, we consider a Volterra-Fredholm integro-differential equation of fractional order as follows: …(1) Subject to the homogenous boundary conditions: u(a) = 0, u(b) = 0, a ≤ x ≤ b …(2) The organization of the rest of this article is as follows: in section 2 we present some essential definitions of the fractional calculus theory, in section 3 the Jacobi polynomial, and its properties are presented. While in section 4 we show how Jacobi polynomial with collocation technique may be used to replace problem (1)-(2) by an explicit system of linear algebraic equations. Moreover in section 5, we introduce some numerical cases to show the adequacy of the proposed method, concluding remarks are given in the last section.

Fractional Derivative and Integration
In this section, we might survey the essential definitions and properties of fractional integral and derivatives, which are utilized further in (3).

Definition (1):-
The left-sided and the right-sided Riemann-Liouville fractional integrals I a+ v f and I b− v f of order v ∈ ℂ (ℜ(α) > 0) are defined by:-

Definition (3):-
The Caputo fractional derivative operator of order v for the function f: [a, b] → ℝ, is given as follows: Where > 0, n is an integer and n − 1 < ≤ n.
The relation between Caputo fractional derivative and Riemann-Liouville: Where n is an integer and n − 1 < ≤ n. Also, for the Caputo fractional derivative we have for β ∈ N 0 and β ≥ ⌈ ⌉ or β ∉ N and β > ⌊ ⌋. ... (10) Where ⌈ ⌉ and ⌊ ⌋ be the ceiling and the floor function respectively.

…(26)
This is the end of the proof.

Function approximation
Consider linear Volterra-Fredholm integrodifferential equations of fractional order derivative of the form: Next, Eqs. (33)(34), after using (15), can be written as So, from using equations (32) with (35) and (36), we get (m + 1) linear algebraic equations which can be solved for the unknown coefficients .
We apply the proposed method for solving Eq. (37) the absolute error AE between our methods with the method given in (36) for different values of γ , δ with = 0.9 and m = 8 are shown in (Table 1), where are roots of the shifted Jacobi polynomial 8 (γ,δ) ( ). The diagrams of the exact and approximate solution for γ = δ = 0, γ = δ = −0.5 and for m = 4,16 together with = 0.75, = 0.9 are given in Fig. (1,2).   Next, after using (15), in Eq. (39), we can get ∑ (−1) By applying the suggested method for solving Eq. (38), the diagrams of the exact and approximate solution when = 0.9 and m = 16 are presented in Fig.3. Also the AE for different values of γ, δ with = 0.5 and m = 4 , m = 8 between our methods with the method given in (36) are shown in Tables 2 and 3.