\documentstyle[amssymb,12pt]{article}
\def\baselinestretch{1.2}
%\textwidth=18cm
%\oddsidemargin=-1cm
\begin{document}

\begin{center}
{\bf ASSIGNED EXERCISES OF MATH544: \\Integral Equations set  9\\} 
{\it April, 2001}
\end{center}



\vspace{0.4cm}

\noindent
{\bf  THE INTEGRAL EQUATIONS, GREEN'S FUNCTION  AND THE NEUMANN SERIES}\\
1. Ronald B Guenther and John W Lee, 2. Hildebrandt \\
See also Ch.4 of the lecture notes.

\vspace{0.3cm}

\noindent
{\bf 1}.\, An integral equation of the form
$$y(x)=f(x)+\lambda \int_{a}^{x}\, g(x,s) \, y(s) ds$$
is called a {\it Volterra equation}.

\vspace{0.2cm}

\noindent
{\bf a}.\, Show that , if $y(x)$ satisfies the differential equation
$$y^{\prime \prime}+x y=1$$
and the conditions $y(0)=y^{\prime}(0)=0$ , then $y$ satisfies the Volterra
equation
$$y(x)=\int_{0}^{x}\, (s-x)\,s \, y(s) ds+{x^2 \over 2}$$
{\bf b}.\, Prove that the converse of the preceding statement is also true.

\vspace{0.3cm}

\noindent
{\bf 2}. Transform the problem
$$y^{\prime \prime}+x y=1, ~~~ y(0)=y(1)=0$$
to the integral equation
$$y(x)=-{1 \over 2}s (1-x) +\int_{0}^{1}\, G(x,s) s \, y(s) ds $$
where $G(x,s)=x(1-s)$ when $x<s$ and $G(x,s)=s(1-x)$ when $x>s$.


\noindent
{\bf 3}.\, Consider the integral equation
$$y(x)=f(x)+\lambda\, \int_{a}^{b}\, e^{x-s}\, y(s) ds.$$
(i) Find the iterated kernels  $g_{n}(x,s)$ defined in the lecture
(ii) the resolvent kernel $\gamma(x,s)$  (iii) Find the solution
(iv) find the cases (restrictions on $\lambda$
where there is no solution and (v) the solution is not unique
(vi) find the solution when $f(x)=x^2$.


\vspace{0.3cm}

\noindent
{\bf 4}.\, {\bf a}. \,Consider the integral equation
$$y(x)=f(x)+\lambda \int_{a}^{b}\, q(x)\,q(s)\, y(s)$$
where $q(x)$ with $x \in [a,b]$ is a continuous function.
Answer all questions (i)-(v) as in Pr.3. Consider also
{\bf b}. \, let $q(x)=x$ discuss the cases (i)-(v) and
{\bf c}.\, let $f(x)=\sin x$ the consider part {\bf b}.


\vspace{0.3cm}

\noindent
{\bf 5}.\, Consider the integral equation
$$y(x)=f(x)+\lambda \int_{0}^{1}\, (1-k\,x \,s ) y(s) ds$$
where $f(x)$ is a continuous function in $[0,1]$ , $k$ is a constant
and $\lambda$ is the parameter of the integral equation. Follow the cases
(i)-(v) in Problem 3 and answer them. Discuss all possible cases.


\vspace{0.3cm}

\noindent
{\bf 6}.\, Solve the following equation
$$y(x)=\lambda \int_{0}^{2\pi}\, \sin(x+s)\, y(s) ds.$$
Discuss all possibilities

\vspace{0.3cm}

\noindent
{\bf 7}.\, Obtain the most general solution of the integral equation
$$y(x)=f(x)+\lambda \int_{0}^{2\pi}\, \sin(x+s)\, y(s) ds.$$
when $f(x)=x$ and $f(x)=1$. Discuss all cases.

\vspace{0.3cm}

\noindent
{\bf 8}.\, Prove that the equation
$$y(x)=f(x)+{1 \over \pi}\, \int_{0}^{2\pi}\, \sin(x+s) \,y(s)\,ds$$
has no solution when $f(x)=x$, but that it possesses infinitely many
solutions when $f(x)=1$. Determine all such solutions


\end{document} 

 





