ISSN 2347-1921 Volume 14 Number 1 Journal of Advance in Mathematics 7416 | Page January 2018 www.cirworld.com Hermite collocation method for solving Hammerstein integral equations Y. A. Amer 1 , A. M. S. Mahdy 1, 2 and H. A. R. Namoos 3 1 Department of Mathematics, Faculty of Science, ZagazigUniversity, Zagazig, Egypt yaser31270@yahoo.com 2 Department of Mathematics, Faculty of Science, Taif University, Saudi Arabia amr_mahdy85@yahoo.com 3 Department of Mathematics, Faculty of Education Ibn-Al-Haitham, Baghdad University, Baghdad- Iraq abdulah123s@yahoo.com ABSTRACT In this paper, we are presenting Hermite collocation method to solve numerically the Fredholm-Volterra-Hammerstein integral equations. We have clearly presented a theory to find ordinary derivatives. This method is based on replacement of the unknown function by truncated series of well knownHermite expansion of functions. The proposed method converts the equation to matrix equation which corresponding to system of algebraic equations with Hermite coefficients. Thus, by solving the matrix equation, Hermite coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique. Keywords Fredholm-Hammerstein integral equations, Hermite polynomials, Volterra integral equation. 1. INTRODUCTION The linear and non linearFredholm and Volterra integral equation have been a growing interest inrecent years ([21], [22]). This are an important branch of modern mathematics and arise frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy and biology ([1], [5]). There are several methods for approximating the solution of linear, non-linear integral equations ([6]-[10], [13]). and solving fractional integro-differential equations ([11],[12]). We consider the Hammerstein integral equations in the forms ([21],[22]):- = + 1 1 1 0 , + 2 2 0 , .(1) Where , 1 , and 2 , are given functions, 0 ≤, ≤ 1, and 1 , 2 are arbitrary constants. Hermite polynomials are widely used in numerical computation. One of the advantages of using Hermite polynomials as a tool for expansion functions is the good representation of smooth functions by finite Hermite expansion provided that the function is infinitely differentiable ([2], [15], [18], [20]). The Hermite collocation method in [16] solving convection diffusion equation, in [17] solving linear differential equations and in [19] solving linear complex differential equation. The paper is organized as follows: Section 2, we will study some properties of the Hermite polynomials. In Section 3, we take his idea about an approximate formula of the integral derivative. In Section 4, procedure solution using the proposed numerical method. In Section 5, we give numerical implementation. In Section 6, the paper ends with a brief conclusion. 2. Some properties of the Hermite polynomials Definition:-The Hermite polynomials are given by ([2], [20]):- = −1 2 − 2 . Some main properties of these polynomials are : The Hermite polynomials evaluated at zero argument 0 and are called Hermite number as follows ([2], [20]):- 0 = 0, , −1 2 2 2 − 1! , (2) Where − 1! is the factorial. The polynomials are orthogonal with respect to the weight function = − 2 with the following condition [2]:-