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{\bf ASSIGNED EXERCISES OF MATH544: \\IntegraL Equations set  8\\} 
{\it April, 2001}
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{\bf  THE INTEGRAL EQUATIONS AND THE GREEN'S FUNCTION}\\
Ronald B Guenther and John W Lee, Partial Differential Equations of
Mathematical Physics and Integral Equations \\
See also Ch.4 of the lecture notes.

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{\bf 1}.\, Show that the Green's function is unique, if it exists.
{\it Hint}. If both $G(x,y)$ and $H(x,y)$ provides the same $u(x)$
given by

\begin{eqnarray}
u(x)=\int_{a}^{b}\, G(x,y) f(y) dy,\\
u(x)=\int_{a}^{b}\, H(x,y) f(y) dy,
\end{eqnarray}

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then
$$\int_{a}^{b}[G(x,y)-H(x,y)] f(y) dy=0$$.

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{\bf 2}.\, Find Green's function of the following problem
$$-k\,u^{\prime \prime}+l\,u=0,~~u(0)-u^{\prime}(0)=0,~~u(1)=0.$$

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{\bf 3}.\, Discuss the existence and and the construction
 of the Green's function of the following problem

\begin{eqnarray}
-[p(x) u^{\prime}]^{\prime}+q(x) u=f(x), ~~ 0 < x <1, \\
u(0)=u(1),~~~u{\prime}(0)=u^{\prime}(1)
\end{eqnarray}

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{\bf 4}.\,Find Green's function for the initial value problem
$$L u=f(x),~~~u(0)=0,~~u^{\prime}(0)=0.$$
Show that the solution has the form
$$u(x)=\int_{0}^{x}\,G(x,y) f(y) dy.$$

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{\bf 5}.\, Let $u^{\prime \prime}-u=f(x)$. Find  Green's function for
the boundary conditions $u(x) \rightarrow 0$ ax $|x| \rightarrow \infty$.



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