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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{}}

\centerline{}
\centerline{}
\centerline{}

\centerline {\Large{\bf On the Numerical Solution of Hammerstein
Integral }}

\centerline{}

\centerline{\Large{\bf  Equations using Legendre Approximation}}

\centerline{}

%% My definition
\newcommand{\mvec}[1]{\mbox{\bfseries\itshape #1}}

\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{}

\newtheorem{Theorem}{\quad Theorem}[section]

\newtheorem{Definition}[Theorem]{\quad Definition}

\newtheorem{Corollary}[Theorem]{\quad Corollary}

\newtheorem{Lemma}[Theorem]{\quad Lemma}

\newtheorem{Example}[Theorem]{\quad Example}

\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.}

%=============================
\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}
$(1-t^2)u^{\prime\prime}(t)-2tu^{\prime}(t)+n(n+1)u(t)=0,\,\,\,\,\,-1<t<1,\,\,\,n\geq0,$\end{center}
and are given by the following relation
\begin{equation}\label{2}
P_n(t)=\frac{1}{2^n}\sum_{k=0}^{[n/2]} (-1)^k
\frac{(2n-2k)!}{(n-k)!(n-2k)k!}
t^{n-2k},\,\,\,\,\,\,\,\,\,\,n=0,1,2,\ldots.
\end{equation}
Also, the recurrence formula associated with Legendre polynomials is
given with the relations,
\begin{equation}\label{3}
P_0(t)=1,\,P_1(t)=\frac{1}{2}(3t^2-1),\,
(n+1)P_{n+1}(t)={(2n+1)tP_n(t)-nP_{n-1}(t)},\,\,n\geq1.
\end{equation}
%In this paper, we reduce the problem (\ref{1}) for solving system of
%non-linear algebraic equations by truncated approximating series
%\begin{equation}\label{5}
 %   x(t)\simeq x_N(t) = \sum_{i=0}^{N-1} c_i \Phi_i (t),
%\end{equation}
%where the elements $\Phi_0(t),\Phi_1(t),...,\Phi_{N-1}(t).$ are the
%orthogonal basis functions defined on a certain interval $[a,b]$.
%Here we choose $\Phi_0(s)$ as Legendre functions.
Legendre polynomials occur in the solution of Laplace equation of
the potential, $\nabla^2\Phi(x)=0,$ in a charge-free region of
space, using the method of separation of variables, where the
boundary conditions have axial symmetry, the solution for the
potential will be
$$\Phi(r,\theta)=\sum_{l=0}^\infty[A_lr^l+B_lr^{-(l+1)}]P_l(\hbox{cos}{\theta}),$$
$A_l$ and $B_l$ are to be determined according to the boundary
condition of each problem.
 They also appear when solving Schr\"{o}dinger equation in three dimensions for a central force.

%==========================================================================================
The Legendre and Chebyshev polynomials are mostly used to solve
several problems of differential equations or integral equations,
for example, the Legendre pesudo-spectral method is used to solve
the delay and the diffusion differential
equations~(\cite{6},~\cite{7}). Chebyshev polynomials are used to
introduce an efficient modification of homotopy perturbation
method~\cite{8}. Also, the polynomial approximation is used to solve
high-order linear Fredholm integro-differential equations with
constant coefficient~\cite{11} and others
(\cite{9},~\cite{15}-\cite{17}).
%==============================
\section{Procedure Solution using the Proposed
\\Numerical Method}
%==============================

We consider the Fredholm-Volterra integral equation (\ref{1}). The
function $x(t)$ may be expanded by infinite series of Legendre
polynomials as follows
\begin{equation}\label{4}
x(t) = \sum_{n=0}^{\infty} c_n P_n (t),
\end{equation}
where $c_n =(x(t),P_n(t))$. If we consider truncated series in
Eq.(\ref{4}), we obtain
\begin{equation}\label{5}
x(t) \simeq \sum_{n=0}^{N} c_n P_n (t)=C^T P(t),
\end{equation}
such that $C$ and $P$ are matrices given by
%\begin{equation}\label{6}
%C=\begin{bmatrix}
%          c_0 & c_1 & \ldots & c_N \\
%        \end{bmatrix},\,\,\,\,\,
%    P(t)=\begin{bmatrix}
%           P_0(t) & P_1(t) & \ldots & P_N(t) \\
%         \end{bmatrix}^T.
%\end{equation}
\begin{equation}\label{6}
C=[c_0\,\,\,\,  c_1\,\,\,\,  ...\,\,\,\, c_N],\quad
P(t)=[P_0(t)\,\,\,\, P_1(t)\,\, ... \,\,\,\,P_N(t)]^{T}.
\end{equation}
%where $P_n(t)$, are Legendre polynomials of degree $n$ which
 %are orthogonal on the interval $[-1,1]$.
%Let us assume that the function $x(t)$ defined by Eq.(\ref{1}) can
%be expanded to the truncated Legendre series in the form
%Eq.(\ref{6}). Then the solution expressed by (\ref{6}) can be
%written in the form Eq.(\ref{7}).
Then we substitute the approximation Eq.(\ref{5}) into Eq.(\ref{1})
we get
\begin{equation}\label{7}
C^T P(t)=f(t)+\lambda_1\int_0^1 K_1(t,s) F(C^T P(s))ds +
\lambda_2\int_0^t K_2(t,s) G(C^T P(s))ds.
\end{equation}
Now, to use the Legendre collocation method which is a matrix method
based on the Legendre collocation points depended by
\begin{equation}\label{8}
t_i=-1+\frac{2}{N}i,\,\,\,\,\,\,\,\,\,\,\,\,\,i=0,1,...,N,
\end{equation}
we collocate Eq.(\ref{7}) with the points (\ref{8}) to obtain
\begin{equation}\label{9}
C^T P(t_j)=f(t_j)+\lambda_1\int_0^1 K_1(t_j,s) F(C^T P(s))ds +
\lambda_2 \int_0^{t_j} K_2(t_j,s) G(C^T P(s))ds.
\end{equation}
The integral terms in Eq.(\ref{9}) can be found using composite
trapezoidal integration technique as:
\begin{equation}\label{10}
\int_0^1 K_1(t_j,s) F(C^T
P(s))ds\cong\frac{h}{2}(\Omega_1(s_0)+\Omega_1(s_m)
+2\sum_{k=1}^{m-1}\Omega_1(s_k)),
\end{equation}
where $\Omega_1(s)=K_1(t_j,s) F(C^T P(s))$, $h=\frac{1}{m}$, for an
arbitrary integer $m,$ $s_i=ih,\,\,\,i=0,1,...,m$ and
\begin{equation}\label{11}
\int_0^{t_j} K_2(t_j,s) G(C^T
P(s))ds\cong\frac{h_j}{2}(\Omega_2(\bar{s}_0)+\Omega_2(\bar{s}_m)
+2\sum_{k=1}^{m-1}\Omega_2(\bar{s}_k)),
\end{equation}
where $\Omega_2(s)=K_2(t_j,s)G(C^T P(s))$, $h_j=\frac{t_j}{m}$, for
an arbitrary integer
$m,$ $\bar{s}_i=ih.$ \\
 Eq.(\ref{9}) gives $(N+1)$ system of linear or non-linear algebraic equations, which can be solved for
$c_k,\,\,\,k=0,1,...,N$. So the unknown function $x(t)$ can be
found.
%
%==============================
\section{Numerical Implementation}
%==============================

In this section, to achive the validity, the accuracy and support
our theoretical discussion of the proposed method, we give some
computational results of numerical examples.\\ %Where we implement the
%ADM to obtain the approximate solution of the same examples.\\
\\
{\bf Example 1. } \\Consider Eq.(\ref{1}) with the following
functions and coefficients
$$f(t)=t^3-(6-2e)e^t,\,\,\,\,\lambda_1=1,\,\,\,\,\lambda_2=1,$$
$$K_1(t,s)=e^{(t+s)},\,\,\,\,\,K_2(t,s)=0,\,\,\,\,\,F(x(s))=x(s),\,\,G(x(s))=0.$$
Eq.(\ref{1}) takes the form
\begin{equation}\label{12}
x(t)=t^3-(6-2e)e^t+\int_0^1 e^{(s+t)}x(s)ds.
\end{equation}
We apply the suggested method with $N=4$, and approximate the
solution $x(t)$ as follows
\begin{equation}\label{13}
x_N(t)=\sum_{i=0}^4 c_i P_i(t)=C^T P(t).
\end{equation}
By the same procedure in the previous section and using Eq.(\ref{9})
we have
\begin{equation}\label{14}\begin{split}
\sum_{i=0}^{4} c_i P_i(t_j)-(t_j^3-(6-2e)e^{t_j})-
\frac{h}{2}(\Omega(s_0)+\Omega(s_m)+2\sum_{k=1}^{m-1} \Omega(s_k))
=0,\,\,j=0,1,2,3,4,
\end{split}
\end{equation}
where $\Omega(s)=e^{(s+t_j)}\sum_{i=0}^{4} c_i P_i(s)$ and the nodes
$s_{l+1}=s_{l}+h,\,\,l=0,1,...,m$, $s_0=0$ and $h=\frac{1}{m}.$\\
Eq.(\ref{14}) represents linear system of $N+1$ algebraic equations
in the coefficients $c_i$, by solving it using the conjugate gradient method, % By solving it
%using the well known Newton iteration method with suitable solution
we obtain
\begin{center}
$c_0=-0.0048,\,\,\,\,c_1=0.5955,\,\,\,\,c_2=-0.0015,\,\,\,\,
c_3=0.3998,\,\,\,\,c_4=-0.0001.$
\end{center} Therefore, the approximate solution of this example
using Eq.(\ref{13}) is given by
\begin{center}
$x(t)=-0.0048P_0(t)+0.5955P_1(t)-0.0015P_2(t)+0.3998P_3(t)-0.0001P_4(t).$
\end{center}
The exact solution of this example is $x(t)=t^3$.
\begin{center}
\includegraphics{f1}\vspace{5cm}\end{center}
\begin{center}{Fig.1. The behavior of the exact solution and
the approximate solution at $N=4.$} \end{center}
 %Figure behavior of the exact and the approximate solutions in some points using the
%presented method
The behavior of the approximate solution using the proposed method
with $N=4$ and the exact solution are presented in Fig.1. %, where
%we compared our solution with the solution using ADM.
From this Fig., it is clear that the proposed
method can be considered as an efficient method to solve the linear integral equations.\\
%========================================================================================================
\\
{\bf Example 2. }
\\Consider Eq.(\ref{1}) with the following functions and coefficients
$$f(t)=2te^t-e^t+1,\,\,\,\lambda_1=1,\,\,\,\lambda_2=-1,\,\,\,\,\,K_1(t,s)=0,$$
$$K_2(t,s)=(s+t),\,\,\,\,\,F(x(s))=0,\,\,\,\,\,G(x(s))=e^{x(s)}.$$
Eq.(\ref{1}) takes the following form
\begin{equation}\label{15}
x(t)=2te^t-e^t+1-\int_0^t(s+t)e^{x(s)}ds.
\end{equation}
We apply the suggested method with $N=4$, and approximate the
solution $x(t)$ as follows
\begin{equation}\label{16}
x_N(t)=\sum_{i=0}^4 c_i P_i(t)=C^T P(t).
\end{equation}
By the same procedure in the previous section and using Eq.(\ref{9})
we have
\begin{equation}\label{17}
\sum_{i=0}^{4} c_i P_i(t_j)-f(t_j)+
\frac{h_j}{2}(\Omega(s_0)+\Omega(s_m)+2\sum_{k=1}^{m-1}\Omega(s_k))
=0,\,\,\,\,\,\,\,j=0,1,2,3,4,
\end{equation}
where the nodes $s_{l+1}=s_{l}+h,\,\,l=0,1,...,m$, $s_0=0$ and
$h_j=\frac{t_j}{m},\,\,\,$$\Omega(s)=(s+t_j)e^{C^TP(s)}.$\\
Eq.(\ref{17}) presents non-linear system of $N+1$ algebraic
equations in the coefficients $c_i$. By solving it by using the
Newton iteration method with suitable initial solution we obtain
\begin{center}
$c_0=0.0002,\,\,\,\,\,\,c_1=0.9895,\,\,\,\,\,\,\,c_2=0.0022,\,\,\,\,\,\,
c_3=-0.0088,\,\,\,\,\,\,c_4=0.0023.$
\end{center} Therefore, the approximate solution of this example
can be found using (\ref{16}) as follows
\begin{center}
$x(t)=0.0002P_0(t)+0.9895P_1(t)+0.0022P_2(t)-0.0088P_3(t)+0.0023P_4(t).$
\end{center}
The exact solution of this problem is $x(t)=t$.\\
The behavior of the approximate solution using the proposed method
with $N=4$ and the exact solution are presented in Fig.2. From this
Fig., it is clear that the proposed method can be considered as an
efficient method to solve the nonlinear integral equations.
\begin{center}
\includegraphics{f2}\vspace{3cm}\end{center}
\begin{center}{Fig.2. The behavior of the exact solution and
the approximate solution at $N=4$.
}\end{center}%\newpage\hspace{-0.75cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\{\bf Example 3.}
\\Consider Eq.(\ref{1}) with the following functions and coefficients
$$f(t)=te+1,\,\,\,\,\,\lambda_1=-1,\,\,\,\,\,\lambda_2=1,\,\,\,\,\,K_1(t,s)=s+t,$$
$$K_2(t,s)=0,\,\,\,\,\,\,F(x(s))=e^{x(s)},\,\,\,\,\,G(x(s))=0.$$
Eq.(\ref{1}) takes the following form
\begin{equation}\label{18}
x(t)=te+1-\int_0^1(s+t)e^{x(s)}ds.
\end{equation}
The exact solution of this problem is $x(t)=t$.\\
We apply the suggested method with $N=3$, and approximate the
solution $x(t)$ as in (\ref{16}), and by the same procedure in the
previous section with using Eq.(\ref{9}) we have
\begin{equation}\label{19}
\sum_{i=0}^{3} c_i P_i(t_j)-f(t_j)+
\frac{h}{2}(\Omega(s_0)+\Omega(s_m)+2\sum_{k=1}^{m-1}\Omega(s_k))
=0,\,\,\,\,\,\,\,j=0,1,2,3,
\end{equation}
where $s_{l+1}=s_{l}+h,\,\,l=0,1,...,m$, $s_0=0$ and $h=\frac{1}{m}$
and $\Omega(s)=(s+t_j).\\e^{(\sum_{i=0}^{3} c_i
P_i(s))}.$\\
Eq.(\ref{19}) presents non-linear system of algebraic equations. By
solving it using the well known Newton iteration method with
suitable initial solution we obtain
\begin{center}
$c_0=-0.0023,\,\,\,\,\,\,c_1=1.0013,\,\,\,\,\,\,c_2=0.0\,\,\,\,\,
c_3=0.0\,.$
\end{center} Therefore, the approximate solution of this example
can be found using (\ref{16})
\begin{center}
$x(t)=-0.0023P_0(t)+1.0013P_1(t)+0.0P_2(t)+0.0P_3(t).$
\end{center}
\begin{center}
\includegraphics{f3}\vspace{3cm}\end{center}
\begin{center}{Fig.3. The behavior of the exact solution and
the approximate solution
at $N=3.$ %and comparison with ADM.
}\end{center} The behavior of the approximate solution using the
proposed method with $N=3$ and the exact solution are presented in
Fig.3.\\
%========================================================================================
\\{\bf Example 4.}
\\ Consider Eq.(\ref{1}) with
the following functions and coefficients
$$f(t)=\frac{t}{2}-\frac{t^4}{12}-\frac{1}{3},\,\,\,\lambda_1=1,\,\,\,\lambda_2=-1,\,\,\,K_1(t,s)=s+t,$$
$$K_2(t,s)=s-t,\,\,\,F(x(s))=x(s),\,\,\,G(x(s))=x^2(s).$$
Eq.(\ref{1}) takes the following form
\begin{equation}\label{20}
x(t)=\frac{t}{2}-\frac{t^4}{12}-\frac{1}{3}+\int_0^1(s+t)x(s)ds+\int_0^t(s-t)x^2(s)ds.
\end{equation}
We apply the suggested method with $N=4$, and approximate the
solution $x(t)$ as follows:
\begin{equation}\label{21}
x_N(t)=\sum_{i=0}^4 c_i P_i(t)=C^T P(t).
\end{equation}
By the same procedure in the previous section and using Eq.(\ref{9})
we have
\begin{equation}
\begin{split}
\sum_{i=0}^{4} c_i P_i(t_j)-f(t_j)-
\frac{h}{2}(\Omega_1(\bar{s}_0)+\Omega_1(\bar{s}_m)&+2\sum_{k=1}^{m-1}\Omega_1(\bar{s}_k))-
\frac{h_j}{2}(\Omega_2(s_0)+\Omega_2(s_m)\\&+2\sum_{k=1}^{m-1}\Omega_2(s_k))
=0,\end{split}\label{22}\end{equation} where
$\bar{s}_{l+1}=\bar{s}_{l}+h,\,\,\,\,s_{l+1}=s_{l}+h_j,l=0,1,...,m,\,\,\,\,s_{0}=\bar{s}_0=0,\,\,\,\,$
$h=\frac{1}{m},\,\,\,\,$ $h_j=\frac{t_j}{m},$ and
$\Omega_1(s)=(s+t_j)(\sum_{i=0}^{4} c_i
P_i(s)),\,\,\,\,\Omega_2(s)=(t_j-s){(C^TP(s))}^2.$\\
Eq.(\ref{22}) presents non-linear system of $N+1$ algebraic
equations. By solving it using Newton iteration method we obtain
\begin{center}
$c_0=-0.0012,\,\,\,\,\,\,c_1=0.9987,\,\,\,\,\,\,c_2=-0.0039,\,\,\,\,\,\,\,
c_3=0.0007,\,\,\,\,\,\,c_4=-0.0017.$
\end{center} Therefore, the approximate solution of this example
can be found using (\ref{21}):
$$x(t)=-0.0012P_0(t)+0.9987P_1(t)-0.0039P_2(t)+0.0007P_3(t)-0.0017P_4(t).$$
%\newpage\hspace{-0.75cm}
The exact solution of this problem is $x(t)=t$.\\
The behavior of the approximate solution using the proposed method
using $N=4$ and the exact solution are presented in Fig.4. From
Fig.4, it is clear that the proposed method can be considered as an
efficient method to solve the nonlinear Hammerstein integral
equations.
\begin{center}
\includegraphics{f4} \vspace{3cm}\end{center}

\begin{center}{Fig.4. The behavior of the exact solution and
the approximate solution at $N=4.$ }\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%==========================
\section{Conclusion}
%==========================
An approximate method for the solution of linear and non-linear
Fredholm-Volterra integral equations in the most general form has
been proposed and investigated. For this purpose, the presented
method which is based on the Legendre polynomials is proposed to
find the approximate solution. A comparison of the exact solution
reveals that the presented method is very effective and convenient.
The numerical results show that the accuracy improves with
increasing $N$, hence for better results, using number $N$ is
recommended. Also, from the obtained approximate solution, we can
conclude that the proposed method gives the solution in an excellent
agreement with the exact solution. All computations are done using Matlab programming.\\

{\bf ACKNOWLEDGEMENTS}

The authors are very grateful to the editor and referees for
carefully reading the paper and for their comments and suggestions
which have improved the paper.
%===================================================================================
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\end{document}