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\markboth{\small{N. H. Sweilam, M. M. Khader, W. Y.
Kota}}{\small{Numerical Solution of Hammerstein IEs using Legendre
Approximation}}
\date{}
\begin{document}
{\scriptsize
\emph{\textbf{International Journal of Applied Mathematical Research}, xx (xx) (20xx) xxx-xxx}
\emph{\copyright Science Publishing Corporation}
\emph{www.sciencepubco.com/index.php/IJAMR}
\emph{}}
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\centerline {\Large{\bf On the Numerical Solution of Hammerstein
Integral }}
\centerline{}
\centerline{\Large{\bf Equations using Legendre Approximation}}
\centerline{}
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\centerline{\bf {N. H. Sweilam, M. M. Khader, and W. Y. Kota}}
\centerline{}
\centerline{Department of Mathematics, Faculty of Science, Cairo
University, Giza, Egypt }
\centerline{Email:\,\,n$_{-}$sweilam@yahoo.com}
\centerline{Department of Mathematics, Faculty of Science, Benha
University, Benha, Egypt}
\centerline{Email:\,\,mohamed.khader@fsc.bu.edu.eg}
\centerline{Department of Mathematics, Faculty of Science, Mansoura
University, Damietta, Egypt}
\centerline{Email:\,\,wafaa$_{-}$kota@yahoo.com}
\centerline{}
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\begin{abstract}
In this study, Legendre collocation method is presented to solve
numerically the Fredholm-Hammerstein integral equations. This method
is based on replacement of the unknown function by truncated series
of well known Legendre expansion of functions. The proposed method
converts the equation to matrix equation, by means of collocation
points on the interval $[-1,1]$ which corresponding to system of
algebraic equations with Legendre coefficients. Thus, by solving the
matrix equation, Legendre coefficients are obtained. Some numerical
examples are included to demonstrate the validity and applicability
of the proposed technique.
\end{abstract}
{\bf Keywords:} \emph{Fredholm-Hammerstein integral equations,
Integral equation, Legendre collocation matrix method, Legendre
polynomials, Volterra integral equation.}
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\section{Introduction}
%=============================
In recent years, there has been a growing interest in the Fredholm
and Volterra integral equations. This is an important branch of
modern mathematics and arise frequently in many applied areas which
include engineering, mechanics, physics, chemistry, astronomy,
biology~\cite{1}-\cite{5}. There are several methods for
approximating the solution of linear and non-linear integral equations~\cite{10}-\cite{14}.\\
We consider the Hammerstein integral equations in the forms
\begin{equation}\label{1}
x(t)=f(t)+\lambda_1\int_0^1 K_1(t,s) F(x(s))ds + \lambda_2\int_0^t
K_2(t,s) G(x(s))ds,
\end{equation}
where $f(t)$, $K_1(t,s)$ and $K_2(t,s)$ are given functions,
$0\leq{t,s}\leq1,$ and $\lambda_1,\lambda_2$ are arbitrary
constants.
Orthogonal polynomials are widely used in applications in
mathematics, mathematical physics, engineering and computer science.
One of the most common set of the Legendre polynomials
${P_0(t),P_1(t),...,P_N(t)}$ which are orthogonal on $[-1,1]$ with
respect to the weight function $w(t)=1.$ The Legendre polynomials
$P_n(t)$ satisfy the Legendre differential equation
\begin{center}
$(1-t^2)u^{\prime\prime}(t)-2tu^{\prime}(t)+n(n+1)u(t)=0,\,\,\,\,\,-1