OPEN ACCESS Numerical Solution of Linear Volterra Integral Equation with Delay using

CORRESPONDENCE: adhraa.mathkoor@uomisan.edu.iq ABSTRACT Bernstein polynomial is one of the most valuable and attractive method used to develop numerical solution for several complex models because of its robustness to demonstrate approximation for anonymous equations. In this paper, Bernstein polynomial is proposed to present effective solution for the 2 nd kind linear Volterra integral equations with delay. To evaluate proposed method, the experiments are contacted using two examples and it is obtained the validity and applicability of the


INTRODUCTION
Approximate methods for solving numerically various classes of integral equations (Shihab & Mohammed Ali, 2015) are very rare.
Several methods have been proposed for numerical solution of these equations (Mustafa & AL-Zubaidy, 2011), Bhatta and Bhatti (2006) presented numerical solution Kdv equation using linear and non-linear differential equation both partial and ordinary by modified Bernstein polynomials. Bhattacharya and Mandal (2008) used of Bernstein polynomials is numerical solution of Volterra integral equations. AL-Zawi (2011) used Bernstein polynomials for solving Volterra integral equation of the second kind. Alturk (2016) presented application of the Bernstein polynomial for solving Volterra integral equations with convolution kernels as well. Mohamadi et al. (2017) introduced Bernstein multiscaling polynomial and application by solving Volterra integral equations. A solution for Volterra integral equation of the first kind based on Bernstein polynomials. Maleknejad et al. (2012) demostrated Analytical and numerical solution of volterra integral equation of the second kined.
Many researchers have used Volterra integral equation with delay. Mustafa and Latiff Ibrahem (2008) proposed numerical solution of Volterra integral equation with delay using Block methods. Nouri and Maleknejad (2016) used the numerical solution of delay integral by using Block-pulse functions.
In this work, a robust proposed approach is explored for recruiting Bernstein polynomial to solve linear Volterra integral equation of 2 nd kind with delay. Competitive results are obtained after benchmarks evaluation.

BERNSTEIN POLYNOMIALS AND PROPERTIES
It is worth to mention that, the Bernstein polynomials are useful polynomial formula defined on [0,1]. Its degree n form a basis for the power polynomials of degree n. Bernstein polynomials area set of polynomials (Shihab and Mohammed Ali, 2015) is defined by.
Also Bernstein Polynomials Properties are described below (Nouri and Maleknjad 2016): The polynomials form a partition of unity that is ∑ , ( ) = 1 =0 and can be used for approximating of any function in [a,b]. Moreover, using binomial expansion of (1 − ) − , it can be defined (Maleknejad et al. 2012).

THE SOLUTION OF LINEAR VOLTERRA INTEGRAL EQUATION WITH DELAY FOR SECOND KIND
We consider the integral of the 2 nd kind given by where u(x) is an unknown function to be determined, k(x, t) is a continuous kernel function, f(x) represents a known function. To determine approximated solution in the Bernstein polynomials basis on [a, b] as (Mustafa & AL-Zubaidy, 2011), the following formula is applied i.e. u(x) = ( ) = 0 0, ( ) + ⋯ + , ( ) where ( = 0,1, … , )are unknown constants to be determined by substituting equation (3) in equation (2) we obtain: where Choosing ( = 0,1, … , ) As described above we obtain the linear system where , = 0,1, … , , and = ( ) The system described in equation (6) is solved to obtain the unknown constants ( = 0,1, … , ) which are used to obtain unknown function u(x) in equation (3). Example (1): Consider the following Linear Volterra Integral Equation with delay of the second kind (Muhammad, 2017):

NUMERICAL EXAMPLES
where the exact solution is represented by u(x) = .  (Mustafa and Latiff Ibrahem, 2008): with the exact solution u(x) = . Table 3 represents the exact solution and absolute error by using Bernstein polynomial with n=11. Table 4 contains the absolute error by using Bernstein polynomial with n=11 and =0.99086.

CONCLUSION AND RECOMMENDATIONS
In this paper, Bernstein polynomial method for solving Volterra integral equations with delay of the second kind is proposed. For this method we used different values of because of its effect on Bernstein polynomials for each example above. Thus, we have noticed a difference in the coroner. In general, the results illustrate efficiency and accuracy of the method. The mean absolute error of the numerical examples at the point x in Tables 1-4 for n=11 are computed. According to the numerical results obtained from the illustrative example, we conclude that: