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\markboth{P. K. Sahu and S. Saha Ray}{Nonlinear Hammerstein Fuzzy Integral Equations}
\title[Nonlinear Hammerstein Fuzzy Integral Equations]{The Comparison for Accurate Solutions of Nonlinear Hammerstein Fuzzy Integral Equations}\thanks 

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%\author[D. Juki\'c]{Dragan Juki\'c\corrauth}
%\address{Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia}
%\email{{\tt jukicd@mathos.hr} (D. Juki\'c)}


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\author[P .K. SAHU and S. SAHA RAY]{P. K. SAHU\affil{1} and S. SAHA RAY\affil{2} \corrauth}
\address{\affilnum{1}\comma \affilnum{2}\ Department of Mathematics, National Institute of Technology Rourkela, Odisha-769008, India}
\emails{{\tt santanusaharay@yahoo.com} (S. Saha Ray)} 
%\\   \affilnum{2}\ Department of Mathematics, National Institute of Technology Rourkela, Odisha-769008, India}
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\begin{abstract}
In this paper, efficient numerical techniques have been proposed to solve nonlinear Hammerstein fuzzy integral equations. The proposed methods are based on Bernstein polynomials and Legendre wavelets approximation. Usually, nonlinear fuzzy integral equations are very difficult to solve analytically as well as numerically. The present methods applied to the integral equations, reduces to solve system of nonlinear algebraic equations. Again this system has been solved by Newton's method. The numerical results obtained by present methods have been compared with that of by homotopy analysis method. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented methods.
\end{abstract}
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%%%%% Keywords %%%%%%%%%%%
\keywords{Bernstein polynomial, Legendre wavelets, Hammerstein integral equation, Fuzzy integral equation, Fuzzy calculus}

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\ams{45G10, 65R20}

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\section{Introduction}

The application of fuzzy integral equations has been developed in recent years. The concept of fuzzy sets was originally introduced by Zadeh \cite{Zadeh-1975}, led to the definition of fuzzy numbers and its implementations in fuzzy control \cite{Chang-1972} and approximate reasoning problems \cite{Zadeh-1983}. The study of fuzzy integral equations has been invented by investigations of Kaleva \cite{Kaleva-1987} and Seikkala \cite{Seikkala-1987} for fuzzy Volterra integral equations. In the past few years, many works have been done by several authors in the theory of fuzzy integral equations. The approximate-analytical methods like Adomian decomposition method \cite{Babolian-2005}, homotopy analysis method \cite{Molabahrami-2011}, and homotopy perturbation method \cite{Allahviranloo-2008} have been used to solve fuzzy integral equations. Numerical techniques are available in the literature to solve fuzzy integral equations. Fuzzy Fredholm integral equations have been solved by Nystrom method \cite{Abbasbandy-2007}, sinc function \cite{Keyanpour-2011}, residual minimization method \cite{Jahantigh-2008}, fuzzy transforms method \cite{Ezzati-2012} and others. \\

Previously, many numerical methods have been used to solve nonlinear Hammerstein integral equations. The nonlinear Hammerstein fuzzy Fredholm integral equations and Hammerstein Volterra delay integral equations have been solved by the learned researchers Bica et al. \cite{Bica-2014,Bica-2013}. Recently, the nonlinear Hammerstein Fredholm integral equation have been solved by B-spline wavelet method and variational iteration method \cite{Sahu-cmes-2013}, and fuzzy integro-differential equations have been solved by two dimensional Legendre wavelet method \cite{Sahu-2014}. Bernstein polynomial approximation has been applied to solve integral equations by many researchers. Fuzzy Volterra integral equations \cite{Mosleh-2013} and fuzzy Fredholm integral equations \cite{Fard-2011} have been solved by Bernstein polynomials. Bernstein polynomials have been applied to solve systems of nonlinear Hammerstein Fredholm integral equations in \cite{Sahu-mma-2013}.\\

In this paper, nonlinear Hammerstein fuzzy Fredholm integral equations have been solved by Bernstein polynomial collocation method (BPCM) and Legendre wavelet method (LWM). This proposed methods are used to solve Hammerstein integral equation by reducing to a system of nonlinear algebraic equations. Again this system has been solved by Newton's method. Also the results obtained by presented methods have been compared with the results obtained by homotopy analysis method (HAM). Some illustrative examples are introduced to justify the accuracy and applicability of the present methods. The comparison between the present methods establishes that Bernstein polynomial collocation method provides more accurate solutions than Legendre wavelet method solutions.\\

This paper has been organized as follows: in section 2, we present some preliminaries and notations of nonlinear Hammerstein fuzzy Fredholm integral equations. In section 3, we discuss the properties of Bernstein polynomial and function approximation. In section 4, we discuss the properties of Legendre wavelets and function approximation. In section 5, we establish the function approximation of nonlinear Hammerstein fuzzy Fredholm integral equation by two dimensional Bernstein polynomial and two dimensional Legendre wavelets. In section 6, we discuss the convergence analysis of present method. Section 7 deals with the illustrative examples which show the efficiency and accuracy of these presented methods with regard to HAM.



\section{Preliminaries of fuzzy integral equation}\label{sec:2}
In this section the most basic notations used in fuzzy calculus are introduced. We start with defining a fuzzy number.

\noindent
\textbf{Definition 2.1:} (See ref. \cite{Goetschel-1986}) A fuzzy number $u$ is represented by an ordered pair of functions $\left( \underline{u}(r),\overline{u}(r)\right)$; $0\leq r\leq 1$ which satisfying the following properties.
\begin{enumerate}
\item $\underline{u}(r)$ is a bounded monotonic increasing left continuous function over $[0,1]$.
\item $\overline{u}(r)$ is a bounded monotonic decreasing left continuous function over $[0,1]$.
\item $\underline{u}(r)\leq \overline{u}(r), \quad 0\leq r\leq 1$.
\end{enumerate}
For arbitrary $u(r)=\left( \underline{u}(r),\overline{u}(r)\right) $, $v(r)=\left( \underline{v}(r),\overline{v}(r)\right) $ and $k>0$, we define addition $(u+v)$ and scalar multiplication by $k$ as
\begin{enumerate}
\item[i.] $\left( \underline{u+v}\right)(r)=\underline{u}(r)+\underline{v}(r)$
\item[ii.] $\left( \overline{u+v}\right)(r)=\overline{u}(r)+\overline{v}(r)$
\item[iii.] $\left( \underline{ku}\right)(r)=k\underline{u}(r)$, $\left( \overline{ku}\right)(r)=k\overline{u}(r)$
\end{enumerate}

\noindent
\textbf{Definition 2.2:} (See ref. \cite{Wu-2001}) For arbitrary fuzzy numbers $u,v\in E$ (the space $E$ is the set of all fuzzy numbers), we use the distance 
\begin{equation*}
D(u,v)=\underset{0\leq r\leq 1}{\sup}\left[ \max \left\lbrace \vert \overline{u}(r)-\overline{v}(r)\vert ,  \vert \underline{u}(r)-\underline{v}(r)\vert \right\rbrace \right]
\end{equation*}
and it is shown that $(E,D)$ is a complete metric space. 

\noindent
\textbf{Remark 2.3:} If the fuzzy function $f(t)$ is continuous in the metric $D$ (see ref. \cite{Babolian-2005}), its definite integral exists. Also 
\begin{align*}
\left( \underline{\int_{a}^{b}f(t;r)dt}\right) =\int_{a}^{b}\underline{f}(t;r)dt,\\
 \left( \overline{\int_{a}^{b}f(t;r)dt}\right) =\int_{a}^{b}\overline{f}(t;r)dt
\end{align*}
We have followed ref. \cite{Goetschel-1986} and defined the integral of a fuzzy function using the Riemann integral concept.\\
\noindent
\textbf{Definition 2.4:} (See ref. \cite{Bede-2004}) If $f:R\longrightarrow E$ be a fuzzy function (where $E$ is a subset of a Banach space) and $t_{0}\in R$, Then the derivative $f'(t_{0})$ of $f$ at a point $t_{0}$ is defined by
\begin{equation}\label{2.1}
f'(t_{0})=\underset{h\rightarrow 0^{+}}{\lim}\frac{f(t_{0}+h)-f(t_{0})}{h},
\end{equation}
provided that this limit taken with respect to the metric $D$, exists.\\

%\noindent
%The elements $f(t_{0}+h)$ and $f(t_{0})$ in eq. \eqref{2.1} are in Banach space $B=C[0,1]\times C[0,1]$. Thus, if $f(t_{0}+h)=(\underline{a},\overline{a})$ and $f(t_{0})=(\underline{b},\overline{b})$, then $f(t_{0}+h)-f(t_{0})=(\underline{a}-\underline{b},\overline{a}-\overline{b})$.\\
%
%\noindent
%Clearly, $[f(t_{0}+h)-f(t_{0})]/h$ may not be a fuzzy number for all $h$. However, if it approaches $f'(t_{0})$ (in $B$) and $f'(t_{0})$ is also a fuzzy number (in $E$), this number is the fuzzy derivative of $f(t)$ at $t_{0}$. In this case, if $f=(\underline{f},\overline{f})$, then $f'(t_{0})=(\underline{f'}(t_{0}),\overline{f'}(t_{0}))$, where $\underline{f'}$, $\overline{f'}$ are the classic derivatives of $\underline{f}$, $\overline{f}$, respectively and $t_{0}\in R$.\\


\noindent
In this paper, we consider the nonlinear Hammerstein fuzzy Fredholm integral equation of the form
\begin{align}\label{2.2}
u(t)=g(t)+ \int_{0}^{1}H(t,s)F(u(s))ds, 
~~H(t,s)\in C([0,1]\times [0,1]),~~t\in [0,1],
\end{align}
where $u$, $g$ and $F$ are fuzzy functions, and $H(t,s)$ is positive in $[0,1]$. Let
\begin{align*}
&u(t,r)=\left( \underline{u}(t,r),\overline{u}(t,r)\right)\\
&g(t,r)=\left( \underline{g}(t,r),\overline{g}(t,r)\right)\\
&F(u(t,r))=\left( \underline{F}(u(t,r)),\overline{F}(u(t,r))\right)
\end{align*}
Eq. \eqref{2.2}, in crisp sense, converted into a system as
\begin{align}\label{2.3}
\underline{u}(t,r)=\underline{g}(t,r)+\int_{0}^{1}H(t,s)\underline{F}\left(u(s,r)\right)ds
\end{align}
\begin{align}\label{2.4}
\overline{u}(t,r)=\overline{g}(t,r)+\int_{0}^{1}H(t,s)\overline{F}\left(u(s,r)\right)ds
\end{align}

\section{Bernstein polynomials and function approximation}\label{sec:3}
The general form of $n$-th degree Bernstein polynomials on the interval $[0,1]$ defined in \cite{Sahu-mma-2013,Bhatti-2007} as
 \begin{align}\label{3.1}
 B_{i,n}(x)=\binom ni x^{i}(1-x)^{n-i},\quad i=0,1,...,n,
 \end{align}
 where $\binom ni=\frac{n!}{i!(n-i)!}$.\\
 
\noindent
Explicitly, if $f$ be a continuous function on $[0,1]$, then $B_{n}(f)$ is called $n$-th Bernstein polynomial for $f$ and defined as (see ref. \cite{Qian-2011})
\begin{equation*}
B_{n}(f)(x)=\sum_{i=0}^{n}f(\frac{i}{n})B_{i,n}(x).
\end{equation*}
 
 Note that each of these $n+1$ polynomials having degree $n$ satisfies the following properties:
\begin{align*}
&B_{i,n}(x)=0,\quad if\quad i<0\quad or \quad i>n\\
&B_{i,n}(0)=B_{i,n}(1)=0,\quad for\quad 1\leq i\leq n-1\\
&\sum_{i=0}^{n}B_{i,n}(x)=1
\end{align*}

Bernstein polynomials defined above form a complete basis \cite{Sahu-mma-2013,Bhatti-2007} over the interval $[0,1]$. A function $u(x)$ defined over $[0,1]$ can be approximated by Bernstein polynomials basis functions of degree $n$ as
\begin{equation}\label{3.2}
u(x)\approx \sum_{i=0}^{n}c_{i}B_{i,n}(x)
\end{equation}
Similarly, in two dimensional case, any function $u(x,t)\in [0,1]\times [0,1]$ can be approximated as
\begin{equation}\label{3.3}
u(x,t)\approx \sum_{i=0}^{n_{1}}\sum_{j=0}^{n_{2}}c_{i,j}B_{i,n_{1}}(x)B_{j,n_{2}}(t)
\end{equation}

\noindent
We suppose $\Vert . \Vert$ to be the maximum norm on $[0,1]$, then the error bound
\begin{equation}\label{3.4}
\vert (B_{n}f)(x)-f(x)\vert \leq \frac{1}{2n}x(1-x)\Vert f''\Vert.
\end{equation}
given in Chapter 10 of \cite{Devore-1993}, shows that the rate of convergence is at least $\frac{1}{n}$ for $f\in C^{2}[0,1]$. Namely, for each function $f:[0,1]\rightarrow R$ and a point of continuity $x$ of $f$ we have
\begin{equation}\label{3.5}
\lim _{n\rightarrow \infty }(B_{n}f)(x)=f(x).
\end{equation}
and the convergence is uniform if $f$ is continuous. Voronovskaya \cite{Voronovskaya-1932} derived that if the second derivative $f''(x)$ of the function $f$ exists, then
\begin{equation}\label{3.6}
\lim _{n\rightarrow \infty }n((B_{n}f)(x)-f(x))=\frac{1}{2}x(1-x)f''(x).
\end{equation}
Therefore, $(B_{n}f)(x)=f(x)+O(1/n)$.\\

\section{Legendre wavelets and function approximation}\label{sec:4}
Wavelets constitute a family of functions constructed from dilation and translation of a single function called mother wavelet. When the dilation parameter $a$ and the translation parameter $b$ vary continuously, we have the following family of continuous wavelets as 
\begin{equation} \label{4.1}
\Psi_{a,b}(t)=\vert a\vert^{-\frac{1}{2}}\Psi \left( \dfrac{t-b}{a}\right),~a,b\in R,~a\neq 0.
\end{equation}
If we restrict the parameters $a$ and $b$ to discrete values as $ a=a_{0}^{-k} $, $ b=nb_{0}a_{0}^{-k} $, $ a_{0}>1 $, $ b_{0}>0 $ and $n$, and $k$ are positive integers, we have the following family of discrete wavelets:
$$\psi _{k,n}(t)=\vert a_{0}\vert^{\frac{k}{2}}\psi (a_{0}^{k}t-nb_{0}),$$

where $\psi _{k,n}(t)$ form a wavelet basis for $L^{2}(R)$. In particular, when $a_{0}=2$ and $b_{0}=1$, then $\psi _{k,n}(t)$ form an orthonormal basis.

Legendre Wavelets $\psi _{n,m}(t)=\psi (k,n,m,t)$ have four arguments. $n=1,2,...,2^{k-1},\;k\in Z^{+}$, $m$ is the order of Legendre polynomials and $t$ is normalized time. They are defined on the interval $[0,1)$ as \cite{Venkatesh-2013,Yousefi-2005}
\begin{align} \label{4.2}
\psi _{n,m}(t)= 
\begin{cases}
\sqrt{m+\frac{1}{2}}2^{\frac{k}{2}}P_{m}(2^{k}t-2n+1), & \frac{n-1}{2^{k-1}}\leq t< \frac{n}{2^{k-1}} \\
\quad  0, & \text{otherwise}
\end{cases}
\end{align}
where, $m=0,1,...,M-1$ and $n=1,2,...,2^{k-1}$. 

The coefficient $\sqrt{m+\frac{1}{2}}$ is for orthonormality, the dilation parameter is $a=2^{-k}$ and translation parameter is $b=(2n-1)2^{-k}$.

Here $P_{m}(t)$ are the well-known Legendre polynomials of order $m$, which are defined on the interval $[-1,1]$, and can be determined with aid of the following recurrence formulae:
\begin{flushleft}
\begin{align*}
&P_{0}(t)=1,\\ 
&P_{1}(t)=t,\\ 
&P_{m+1}(t)=\left(\frac{2m+1}{m+1}\right)tP_{m}(t)-\left(\frac{m}{m+1}\right)P_{m-1}(t),~~m=1,2,3,...
\end{align*}
\end{flushleft}
The two-dimensional Legendre wavelets are defined as \cite{Banifatemi-2007,Sahu-2014}
\begin{align}\label{4.3}
&\psi _{n_{1},m_{1},n_{2},m_{2}}(x,t) \nonumber \\
&=\begin{cases}
AP_{m_{1}}(2^{k_{1}}x-2n_{1}+1)P_{m_{2}}(2^{k_{2}}t-2n_{2}+1),&
 \frac{n_{1}-1}{2^{k_{1}-1}}\leq x<\frac{n_{1}}{2^{k_{1}-1}},\frac{n_{2}-1}{2^{k_{2}-1}}\leq t<\frac{n_{2}}{2^{k_{2}-1}} \\
0,&  \text{Otherwise} \end{cases}
\end{align}
where $A=\sqrt{\left( m_{1}+\frac{1}{2}\right) \left( m_{2}+\frac{1}{2}\right)}2^{\frac{k_{1}+k_{2}}{2}}$, and $n_{1}=1,2,...,2^{k_{1}-1}$, $n_{2}=1,2,...,2^{k_{2}-1}$, $k_{1},k_{2} \in Z^{+}$, and $m_{1}$, $m_{2}$ are the order of Legendre polynomials. $\psi _{n_{1},m_{1},n_{2},m_{2}}(x,t)$ forms a basis for $L^{2}([0,1)\times [0,1))$.\\


A function $f(x,t)$ defined over $[0,1)\times [0,1)$ can be expanded in terms of Legendre wavelets as \cite{Banifatemi-2007,Sahu-2014}
\begin{align}\label{4.4}
f(x,t)=\sum _{n_{1}=1}^{\infty}\sum _{m_{1}=0}^{\infty}\sum _{n_{2}=1}^{\infty}\sum _{m_{2}=0}^{\infty}  c_{n_{1},m_{1},n_{2},m_{2}}\psi _{n_{1},m_{1},n_{2},m_{2}}(x,t)
\end{align}
If the infinite series in Eq. \eqref{4.4} is truncated, then it can be written as

\begin{flushleft}
\begin{align}\label{4.5}
f(x,t)=\sum _{n_{1}=1}^{2^{k_{1}-1}}\sum _{m_{1}=0}^{M_{1}-1}\sum _{n_{2}=1}^{2^{k_{2}-1}}\sum _{m_{2}=0}^{M_{2}-1} 
c_{n_{1},m_{1},n_{2},m_{2}}\psi _{n_{1},m_{1},n_{2},m_{2}}(x,t)
= C^{T}\Psi (x,t)
\end{align}
\end{flushleft}
where $C$ and $\Psi (x,t)$ are $\left( 2^{k_{1}-1}2^{k_{2}-1}M_{1}M_{2}\times 1\right) $ vectors given by
\begin{align}\label{4.6}
C=[ c_{1,0,1,0},...,c_{1,0,1,M_{2}-1},...,c_{1,0,2^{k_{2}-1},M_{2}-1},...,c_{2^{k_{1}-1},M_{1}-1,2^{k_{2}-1},M_{2}-1}] ^{T}
\end{align}
\begin{align}\label{4.7}
& \Psi (x,t)=[ \psi_{1,0,1,0}(x,t),...,\psi_{1,0,1,M_{2}-1}(x,t),...,\psi_{1,0,2^{k_{2}-1},M_{2}-1}(x,t),..., \nonumber \\
& \quad \quad \quad \quad \quad ...,\psi_{2^{k_{1}-1},M_{1}-1,2^{k_{2}-1},M_{2}-1}(x,t)] ^{T}
\end{align}


\section{Analysis of present methods}\label{sec:5}
\subsection{Method I: Bernstein polynomial collocation method}

\noindent
Consider the eq. \eqref{2.3} for solving by Bernstein polynomial collocation method, first approximate the unknown function $\underline{u}(t,r)$ by using the eq. \eqref{3.3} as
\begin{equation}\label{5.1}
\underline{u}(t,r)\approx \sum_{i=0}^{n_{1}}\sum_{j=0}^{n_{2}}c_{i,j}B_{i,n_{1}}(t)B_{j,n_{2}}(r)
\end{equation}
Eq. \eqref{2.3} can be reduced as 
\begin{equation}\label{5.2}
\sum_{i=0}^{n_{1}}\sum_{j=0}^{n_{2}}c_{i,j}B_{i,n_{1}}(t)B_{j,n_{2}}(r)=\underline{g}(t,r)+\int_{0}^{1}H(t,s)\underline{F}\left( \sum_{i=0}^{n_{1}}\sum_{j=0}^{n_{2}}c_{i,j}B_{i,n_{1}}(s)B_{j,n_{2}}(r)\right)ds
\end{equation}
Utilizing eq. \eqref{5.2} with the collocation points $t_{l}$ and $r_{m}$ defined as 
\begin{align*}
&t_{l}=t_{0}+lh_{1},\quad t_{0}=0,~~h_{1}=\frac{1}{n_{1}},~~~l=0,1,...,n_{1},\\
&r_{m}=r_{0}+mh_{2},\quad r_{0}=0,~~h_{2}=\frac{1}{n_{2}},~~~m=0,1,...,n_{2},
\end{align*}
eq. \eqref{5.2} reduces to a system of $(n_{1}+1)(n_{2}+1)$ number of  nonlinear algebraic equations with same number of unknowns as $c_{i,j}$, $i=0,1,...,n_{1}$, $j=0,1,...,n_{2}$. This algebraic system has been solved by Newton's method to obtain the unknowns $c_{i,j}$, $i=0,1,...,n_{1}$, $j=0,1,...,n_{2}$. Hence we get the solution $\underline{u}(t,r)$ from eq. \eqref{5.1} and same algorithm can be applied to obtain the approximate solution of $\overline{u}(t,r)$.\\

\subsection{Method II: Legendre wavelet method}
\noindent
Consider the eq. \eqref{2.3} for solving by Legendre wavelet method, first approximate the unknown function $\underline{u}(t,r)$ by using the eq. \eqref{4.5} as
\begin{equation}\label{5.3}
\underline{u}(t,r)=C^{T}\Psi(t,r)
\end{equation}
Now, eq. \eqref{2.3} can be reduced as
\begin{equation}\label{5.4}
C^{T}\Psi(t,r)=\underline{g}(t,r)+\int_{0}^{1}H(t,s)\underline{F}\left( C^{T}\Psi(s,r)\right) ds
\end{equation}
In order to use the Gauss-Legendre integration formula for eq. \eqref{5.4}, we transfer the interval $[0,1]$ to $[-1,1]$ by means of the transformation $\tau =2s-1$.\\

\noindent
Therefore, eq. \eqref{5.4} can be written as 
\begin{equation}\label{5.5}
C^{T}\Psi(t,r)=\underline{g}(t,r)+\frac{1}{2}\int_{-1}^{1} H\left(t,\frac{\tau +1}{2}\right)\underline{F} \left( C^{T}\Psi\left(\frac{\tau +1}{2},r\right)\right)d\tau
\end{equation}
By using the Gauss-Legendre integration formula, we get
\begin{equation}\label{5.6}
C^{T}\Psi(t,r)=\underline{g}(t,r)+\frac{1}{2}\sum_{j=1}^{M}w_{j}H\left(t,\frac{\tau_{j} +1}{2}\right)\underline{F} \left( C^{T}\Psi\left(\frac{\tau_{j} +1}{2},r\right)\right)
\end{equation}
wher $\tau_{j}$ are $m$ zeros of Legendre polynomials $P_{m+1}$ and $w_{j}$ are the corresponding
weights.
Now we collocate the eq. \eqref{5.6} at $t_{p}=\frac{2p-1}{2^{k_{1}}M_{1}},~~~~p=1,2,...,2^{k_{1}-1}M_{1}$ and $r_{q}=\frac{2q-1}{2^{k_{2}}M_{2}},~~~~q=1,2,...,2^{k_{2}-1}M_{2}$, we have
\begin{equation}\label{5.7}
C^{T}\Psi(t_{p},r_{q})=\underline{g}(t_{p},r_{q})+\frac{1}{2}\sum_{j=1}^{M}w_{j}H\left(t_{p},\frac{\tau_{j} +1}{2}\right)\underline{F} \left( C^{T}\Psi\left(\frac{\tau_{j} +1}{2},r_{q}\right)\right)
\end{equation}
Eq. \eqref{5.7} gives a system of $2^{k_{1}-1}M_{1}\times 2^{k_{2}-1}M_{2}$ number of algebraic equations with same number of unknowns for $C^{T}$. Again solving this system numerically by Newton's method, we can find the value for unknowns for $C^{T}$ and hence obtain the approximate solution for $\underline{u}(t,r)$. Same algorithm can be applied to obtain the approximate solution of $\overline{u}(t,r)$.

\section{Convergence analysis}
In this section, we have discussed about the convergence of Bernstein polynomial collocation methods. \\
\noindent
\textbf{Theorem 1:}  Let $u(t)\in H^{k}(0,1)$ (Sobolov space) and $u_{n}(t)=\sum_{j=0}^{n}b_{j}\tilde{U}_{j}(t)$ be the best approximation polynomial of $u(t)$ in $L_{w}^{2}$-norm, then the truncation error is $$\Vert u(t)-u_{n}(t)\Vert _{L_{w}^{2}[0,1]}\leq C_{0}n^{-k}\Vert u(t)\Vert _{H^{k}(0,1)},$$ where $C_{0}$ is independent of $n$ and $u(t)$.\\

\noindent
\textbf{Proof:} See ref. \cite{Canuto-1988}.\\

\noindent
\textbf{Theorem 2:}\\
Let $u(t)\in H^{(k)}(0,1)$ (Sobolov space) and $u_{n}(t)=\sum_{i=0}^{n}c_{i}B_{i,n}(t)$ be the approximate solution of $u(t)$ where, $B_{i,n}(t)$ are Bernstein polynomials, $i=0,1,...,n$, then the error term $\Vert e_{n}(t)\Vert $ is bounded and convergence to zero.\\

\noindent
\textbf{Proof:}\\
The Bernstein polynomials are not orthogonal. These polynomials can be expressed in terms of second kind shifted Chebyshev polynomials which are orthogonal with respect to the weight function $\tilde{w}(t)=w(2t-1)$ and $w(t)=\sqrt{1-t^{2}}$.
$$B_{i,n}(t)=\frac{1}{2^{n}}\binom {n}{i}\sum_{s=0}^{n}d_{s}^{i,n}\frac{1}{2^{s}}\sum_{m=0}^{[s/2]}\lbrace \binom {s}{m}-\binom {s}{m+1}\rbrace\tilde{U}_{s-2m}(t) $$
where, $\tilde{U}(t)=U(2t-1)$ is shifted Chebyshev polynomials and $d_{s}^{i,n}=\sum_{k}(-1)^{s-k}\binom {i}{k}\binom {n-i}{s-k}$. The summation over $s$ being choosen as follows:\\
For $i<n-i$, 
\begin{enumerate}
\item $k=0$ to $s$, for $s\leq i$
\item $k=0$ to $i$, for $i<s\leq n-i$
\item $k=s-(n-i)$ to $n-i$, for $n-i<s\leq n$
\end{enumerate}
For $i=n-i$ ($n$ is even integer), 
\begin{enumerate}
\item $k=0$ to $s$, for $s\leq i$
\item $k=s-i$ to $i$, for $i<s\leq n$
\end{enumerate}
For $i>n-i$, $i$ and $n-i$ above are interchanged.\\

We can approximate $u(t)\approx \sum_{i=0}^{n}c_{i}B_{i,n}(t)$ which can be expressed as $u(t)\approx \sum_{j=0}^{n}b_{j}\tilde{U}_{j}(t)$, where $b_{j}$'s are the scalar multiplication of $c_{j}$'s.\\

If $u(t)$ is approximated by $u_{n}(t)=\sum_{j=0}^{n}b_{j}\tilde{U}_{j}(t)$, then we find $\overline{b}_{j}$ as approximation of $b_{j}$ and $\overline{u}_{n}(t)=\sum_{j=0}^{n}\overline{b}_{j}\tilde{U}_{j}(t)$.\\
We have \begin{align*}
\Vert u(t)-\overline{u}_{n}(t)\Vert _{2}&=\Vert u(t)-u_{n}(t)+u_{n}(t)-\overline{u}_{n}(t)\Vert _{2}\\
& \leq \Vert u(t)-u_{n}(t) \Vert _{2} +\Vert u_{n}(t)-\overline{u}_{n}(t)\Vert _{2}
\end{align*}
Now,\begin{align*}
\Vert u_{n}(t)-\overline{u}_{n}(t)\Vert _{2}&=\left[ \int_{0}^{1}(u_{n}(t)-\overline{u}_{n}(t))^{2}\tilde{w}(t)dt \right]^{1/2}\\
&=\left[ \int_{0}^{1}\left( \sum_{j=0}^{n}(b_{j}-\overline{b}_{j})\tilde{U}_{j}(t)\right)^{2}\tilde{w}(t)dt \right]^{1/2}\\
&= \left( \sum_{j=0}^{n}\frac{\pi}{4}(b_{j}-\overline{b}_{j})^{2}\right) ^{1/2}
\end{align*}
Using $(n+1)$-point Gauss-Chebyshev rule, we have (see ref. \cite{Delves-1985})
$$\vert b_{j}-\overline{b}_{j}\vert \leq C_{1}n^{-k+1}$$
and $$\Vert u_{n}-\overline{u}_{n} \Vert _{2}\leq C_{2}(n+1)^{1/2}n^{-k+1}.$$
Thus, $$\Vert u(t)-\overline{u}_{n}(t) \Vert _{2}\leq C_{0}n^{-k}\Vert u(t)\Vert +C_{2}(n+1)^{1/2}n^{-k+1}.$$

Hence the error term $\Vert e_{n}(t) \Vert=\Vert u(t)-\overline{u}_{n}(t)\Vert \longrightarrow 0$ as $n,k\longrightarrow \infty$.


\section{Illustrative examples}\label{sec:6}
\label{sec:5}
\textbf{Example 1}\\
Consider the following nonlinear Hammerstein fuzzy integral equation
\begin{equation}\label{6.1}
u(t)=g(t)+\int_{0}^{1}H(t,s)e^{-s}(u(s))^{2}ds,\quad t\in [0,1]
\end{equation}
with
$$H(t,s)=\begin{cases} \frac{1}{6}s^{2}(1-t)^{2}(3t-s-2ts),&0\leq s\leq t\leq 1\\
\frac{1}{6}t^{2}(1-s)^{2}(3s-t-2ts),&0\leq t\leq s\leq 1 \end{cases}$$
and
$$g(t,r)=g_{1}(t,r)+g_{2}(t,r),\quad t,r\in [0,1],$$
where 
\begin{align*}
&g_{1}(t,r)=\left[ (1-t)^{2}(3t+1)\left(1-\frac{1-r}{10}\right),(1-t)^{2}(3t+1)\left(1+\frac{1-r}{10}\right)\right]\\
&g_{2}(t,r)=\left[t^{2}(2-t)\left( e-\frac{1-r}{10}\right),t^{2}(2-t)\left( e+\frac{1-r}{10}\right)\right]
\end{align*}
Eq. \eqref{6.1}, in crisp sense, converted into a system as
\begin{align}\label{6.2}
\underline{u}(t,r)=\underline{g}(t,r)+\int_{0}^{1}H(t,s)e^{-s}\left(\underline{u}(s,r)\right)^{2}ds
\end{align}
\begin{align}\label{6.3}
\overline{u}(t,r)=\overline{g}(t,r)+\int_{0}^{1}H(t,s)e^{-s}\left(\overline{u}(s,r)\right)^{2}ds
\end{align}
where $$g(t,r)=(\underline{g}(t,r),\overline{g}(t,r))=(\underline{g_{1}}(t,r)+\underline{g_{2}}(t,r),\overline{g_{1}}(t,r)+\overline{g_{2}}(t,r))$$

The exact solution of this problem is not known. This problem has been solved by Bernstein polynomial collocation method and Legendre wavelet method, and compare with the approximate-analytical method like homotopy analysis method \cite{Saharay-2011,Abbasbandy-2012}. Since, exact solution of this problem is unknown, the solution in HAM has been considered as standard solution.  \\

\begin{itemize}
\item \textbf{Comparison with HAM solution}
\end{itemize}

\noindent
In homotopy analysis method \cite{Saharay-2011,Abbasbandy-2012}, the $m^{th}$ order deformation equation approximating $\underline{u}(t,r)$ is given by
\begin{align*}
L\left[ \underline{u}_{m}(t,r)-\chi_{m}\underline{u}_{m-1}(t,r)\right] 
=\hbar \Re_{m}\left( \underline{u}_{0},\underline{u}_{1},...,\underline{u}_{m-1}\right) 
=\hbar\frac{1}{(m-1)!}\frac{\partial ^{m-1}}{\partial q^{m-1}}N(\varphi (t,r;q))\vert _{q=0}
\end{align*}
where 
\begin{align*}
N(\varphi (t,r;q))=\varphi (t,r;q)-\underline{g}(t,r)-\int_{0}^{1}H(t,s)e^{-s}(\varphi (s,r;q))^{2}ds
\end{align*}
and $\chi _{m}=\begin{cases}0,& m\leq 1\\ 1,&m>1 \end{cases}$.\\
\noindent
Here auxiliary parameter \cite{Abbasbandy-2012} $\hbar =-1$  belongs to the convergence region of HAM series solution.\\

Now using $m^{th}$ order deformation equation of $\underline{u}(t,r)$, we recursively obtain
\begin{align*}
 &\underline{u}_{0}(t,r)=0\\
 &\underline{u}_{1}(t,r)=\frac{1}{10}(9+r+9t+rt-47t^{2}+20et^{2}-3rt^{2}+28t^{3}-10et^{3}+2rt^{3})\\
 &\underline{u}_{2}(t,r)=\frac{1}{25}e^{-t}r^{2}t^{6}-\frac{2}{5}e^{1-t}rt^{6}+...\\
 & \text{and so on}.
 \end{align*} 
Thus the HAM solution 
\begin{equation*}
\underline{U}_{m}(t,r)=\sum_{i=0}^{m}\underline{u}_{i}(t,r)
\end{equation*}

Similarly, using $m^{th}$ order deformation equation of $\overline{u}(t,r)$, we recursively obtain
\begin{align*}
 &\overline{u}_{0}(t,r)=0\\
 &\overline{u}_{1}(t,r)=\frac{1}{10}(11-r+11t-rt-53t^{2}+20et^{2}+3rt^{2}+32t^{3}-10et^{3}-2rt^{3})\\
 &\overline{u}_{2}(t,r)=\frac{1}{100}e^{-1-t}\left( 4er^{2}t^{6}+40e^{2}rt^{6}-128ert^{6}+...\right)\\
 &\text{and so on.}
 \end{align*} 
Thus the HAM solution 
\begin{equation*}
\overline{U}_{m}(t,r)=\sum_{i=0}^{m}\overline{u}_{i}(t,r)
\end{equation*}

\noindent
The numerical results obtained by Bernstein polynomial collocation method (BPCM) for $n_{1}=n_{2}=4$ and Legendre wavelet method (LWM) for $M_{1}=M_{2}=4,k_{1}=k_{2}=1$ have been compared with the results obtained by $2^{nd}$ order homotopy analysis method (HAM) of $u(t,r)=(\underline{u}(t,r),\overline{u}(t,r))$ for $r=0.25,~0.5,~0.75$. These results have been shown in Table 1-2. From Table 2, it may be easily observed that $L_{\infty}$ and $L_{2}$ errors for Bernstein polynomial collocation method are better than other method solutions.\\


\noindent
\textbf{Example 2}\\ 
Consider the following nonlinear Hammerstein fuzzy integral equation
\begin{equation}\label{6.4}
u(t)=g(t)+\int_{0}^{1}\frac{ts}{3}(u(s))^{2}ds,\quad t\in [0,1],
\end{equation}
where
\begin{equation*}
g(t,r)=\left[ \frac{11}{12}t-\frac{1}{6}(1-r),~~~\frac{11}{12}t+\frac{1}{6}(1-r)\right] ,\quad t,r\in [0,1]
\end{equation*}
Eq. \eqref{6.4}, in crisp sense, converted into a system as
\begin{align}\label{6.5}
\underline{u}(t,r)=\frac{11}{12}t-\frac{1}{6}(1-r)+\int_{0}^{1}\frac{ts}{3}\left(\underline{u}(s,r)\right)^{2}ds
\end{align}
\begin{align}\label{6.6}
\overline{u}(t,r)=\frac{11}{12}t+\frac{1}{6}(1-r)+\int_{0}^{1}\frac{ts}{3}\left(\overline{u}(s,r)\right)^{2}ds
\end{align}

The exact solution of this problem is not known. This problem has been solved by Bernstein polynomial collocation method and Legendre wavelet method, and compare with the approximate-analytical method like homotopy analysis method \cite{Saharay-2011,Abbasbandy-2012}. Since, exact solution of this problem is unknown, the solution in HAM has been considered as standard solution. \\

\begin{itemize}
\item \textbf{Comparison with HAM solution}
\end{itemize}

\noindent
In homotopy analysis method \cite{Saharay-2011,Abbasbandy-2012}, the $m^{th}$ order deformation equation approximating $\underline{u}(t,r)$ is given by
\begin{align*}
L\left[ \underline{u}_{m}(t,r)-\chi_{m}\underline{u}_{m-1}(t,r)\right] 
=\hbar \Re_{m}\left( \underline{u}_{0},\underline{u}_{1},...,\underline{u}_{m-1}\right) 
=\hbar\frac{1}{(m-1)!}\frac{\partial ^{m-1}}{\partial q^{m-1}}N(\varphi (t,r;q))\vert _{q=0}
\end{align*}
where 
\begin{align*}
N(\varphi (t,r;q))=\varphi (t,r;q)-\frac{11}{12}t+\frac{1}{6}(1-r)-\int_{0}^{1}\frac{ts}{3}(\varphi (s,r;q))^{2}ds
\end{align*}
and $\chi _{m}=\begin{cases}0,& m\leq 1\\ 1,&m>1 \end{cases}$.\\
\noindent

Here auxiliary parameter \cite{Abbasbandy-2012} $\hbar =-1$  belongs to the convergence region of HAM series solution.\\

Now using $m^{th}$ order deformation equation of $\underline{u}(t,r)$, we recursively obtain
\begin{align*}
 &\underline{u}_{0}(t,r)=0\\
 &\underline{u}_{1}(t,r)=-\frac{1}{6}+\frac{r}{6}+\frac{11t}{12}\\
 &\underline{u}_{2}(t,r)=\frac{211t}{5184}+\frac{2rt}{81}+\frac{r^{2}t}{216}\\
 &\text{and so on.}
 \end{align*} 
Thus the HAM solution 
\begin{equation*}
\underline{U}_{m}(t,r)=\sum_{i=0}^{m}\underline{u}_{i}(t,r)
\end{equation*}

Similarly, using $m^{th}$ order deformation equation of $\overline{u}(t,r)$, we recursively obtain
\begin{align*}
 &\overline{u}_{0}(t,r)=0\\
 &\overline{u}_{1}(t,r)=\frac{1}{6}-\frac{r}{6}+\frac{11t}{12}\\
 &\overline{u}_{2}(t,r)=\frac{563t}{5184}-\frac{7rt}{162}+\frac{r^{2}t}{216}\\
 &\text{and so on.}
 \end{align*} 
Thus the HAM solution 
\begin{equation*}
\overline{U}_{m}(t,r)=\sum_{i=0}^{m}\overline{u}_{i}(t,r)
\end{equation*}

\noindent
The numerical results obtained by Bernstein polynomial collocation method (BPCM) for $n_{1}=n_{2}=2$ and Legendre wavelet method (LWM) for $M_{1}=M_{2}=4,k_{1}=k_{2}=1$ have been compared with the results obtained by $3^{rd}$ order homotopy analysis method (HAM) of $u(t,r)=(\underline{u}(t,r),\overline{u}(t,r))$ for $r=0.25,~0.5,~0.75$. These results have been shown in Table 3-4. From Table 4, it may be easily observed that $L_{\infty}$ and $L_{2}$ errors for Bernstein polynomial collocation method are better than other method solutions.\\


\noindent
\textbf{Example 3}\\ 
Consider the following nonlinear Hammerstein fuzzy integral equation
\begin{equation}\label{6.7}
u(t)=g(t)+\int_{0}^{1}ts(u(s))^{3}ds,\quad t\in [0,1],
\end{equation}
where
$$g(t,r)=g_{1}(t,r)+g_{2}(t,r),\quad t,r\in [0,1],$$
where 
\begin{align*}
&g_{1}(t,r)=\left[ 2+r,3-r\right]\\
&g_{2}(t,r)=\left[ -t\left( \frac{87}{10}+\frac{47}{4}r+4r^{2}+\frac{1}{2}r^{3}\right), -t\left( \frac{49}{10}-\frac{9}{2}r+\frac{5}{2}r^{2}-\frac{1}{2}r^{3}\right) \right]
\end{align*}
The exact solution of this problem is $u(t,r)=[2+t+r,~~3-2t-r]$. Eq. \eqref{6.7}, in crisp sense, converted into a system as
\begin{align}\label{6.8}
\underline{u}(t,r)=\underline{g}(t,r)+\int_{0}^{1}ts\left(\underline{u}(s,r)\right)^{3}ds
\end{align}
\begin{align}\label{6.9}
\overline{u}(t,r)=\overline{g}(t,r)+\int_{0}^{1}ts\left(\overline{u}(s,r)\right)^{3}ds
\end{align}
where $$g(t,r)=(\underline{g}(t,r),\overline{g}(t,r))=(\underline{g_{1}}(t,r)+\underline{g_{2}}(t,r),\overline{g_{1}}(t,r)+\overline{g_{2}}(t,r))$$

Solving eq. \eqref{6.8} by Bernstein polynomial collocation method for $n_{1}=n_{2}=1$, we obtain the unknowns as
$$c_{0,0}=2,~~~~c_{0,1}=3,~~~~c_{1,0}=3,~~~~c_{1,1}=4$$
and the solution can be obtained as $\underline{u}(t,r)=\sum_{i=0}^{1}\sum_{j=0}^{1}c_{i,j}B_{i,1}(t)B_{j,1}(r)=2+t+r$.\\
\noindent
Similarly, solving eq. \eqref{6.9} by $1^{st}$ order Bernstein polynomial collocation method, we obtain the unknowns as
$$c_{0,0}=3,~~~~c_{0,1}=2,~~~~c_{1,0}=1,~~~~c_{1,1}=0$$
and the solution can be obtained as $\overline{u}(t,r)=\sum_{i=0}^{1}\sum_{j=0}^{1}c_{i,j}B_{i,1}(t)B_{j,1}(r)=3-2t-r$.

Solving eq. \eqref{6.8} by Legendre wavelet method for $M_{1}=M_{2}=2,k_{1}=k_{2}=1$, we obtain the unknowns as
$$C^{T}=[3.0003,0.28862,0.28885,-0.0000332746]$$
and the solution can be obtained as $\underline{u}(t,r)= C^{T}\Psi (t,r)=2+r+t-0.000399295rt$.\\
\noindent
Similarly, solving eq. \eqref{6.9} by Legendre wavelet method for $M_{1}=M_{2}=2,k_{1}=k_{2}=1$, we obtain the unknowns as
$$C^{T}=[1.52366,-0.216966,-0.563689,0.0414012]$$
and the solution can be obtained as $\overline{u}(t,r)= C^{T}\Psi (t,r)=3-r-2.20108t+0.496815rt$.

The numerical results obtained by Bernstein polynomial collocation method (BPCM) for $n_{1}=n_{2}=1$ and Legendre wavelet method (LWM) for $M_{1}=M_{2}=2,k_{1}=k_{2}=1$ have been compared with the results obtained by $4^{th}$ order homotopy analysis method (HAM) of $u(t,r)=(\underline{u}(t,r),\overline{u}(t,r))$ for $r=0.25,~0.5,~0.75$. These results have been shown in Table 5. It is justified that there is a good agreement between HAM results and exact results, and also, with other numerical results. 

\section{Conclusion}
\label{sec:7}
In this work, Bernstein polynomial collocation method and Legendre wavelet method have been applied to the nonlinear Hammerstein fuzzy Fredholm integral equations and the obtained results then compared with the results obtained by homotopy analysis method. The presented methods reduce the Hammerstein fuzzy Fredholm integral equation to system of nonlinear algebraic equations and this system has been solved by Newton's method. Since homotopy analysis method is an analytical method, then from the tables, it is justified that the  results obtained by presented methods are very accurate with regard to HAM results and also it is cleared that the Bernstein polynomial collocation method gives more accuracy than Legendre wavelet method. The illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. These examples also exhibit the accuracy and efficiency of the presented methods. However, Bernstein polynomial collocation method provides more accurate solutions in comparison to other discussed methods.

\section*{Acknowledgment}
The author would like to convey their sincere thanks to learned anonymous referees for their helpful
suggestions for the improvement and betterment of the manuscript.

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\end{document}


