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\begin{document}

\begin{center}
{\Large \bf b5 Differential and Integral Equations - ht99 Vacation
  Work}
\end{center}

\noindent
{\it Plane autonomous systems of ordinary differential equations:
  phase-plane and critical points, linear systems, elementary
  non-linear systems and linearisation. Bendixson-Dulac criterion.}

\begin{enumerate}

  \item Consider the system
    \begin{eqnarray}
      \dt{x} & = & f(x,y)  \label{eq:1} \\
      \dt{y} & = & g(x,y)  \label{eq:2} \\
    \end{eqnarray}
    
    Explain the significance of the {\em trajectories} (the solutions
    of $\dx{y} = \frac{f(x,y)}{g(x,y)}$). Explain the significance of
    the singular points\footnote{Also known as critical points or
      steady states.}  $(x_{s},y_{s})$ where $f=g=0$.
    
    Suppose (\ref{eq:1})--(\ref{eq:2}) has a singular point at
    $(0,0)$, and $f$ and $g$ are analytic\footnote{Please state what
      analytic means!} near $(0,0)$. Then we can expand $f$ and $g$
    in Taylor series about the origin. Do so, then linearise the
    resulting equations and write in matrix form:

    \begin{equation}
      \dtbig \left( \begin{array}{c}
          x \\
          y
        \end{array} \right) = A \left( \begin{array}{c}
          x \\
          y
        \end{array} \right)  \label{eq:3}
    \end{equation}
    
    Since this is a linear equation, try solutions $\left(
      \begin{array}{c} x \\ y \end{array} \right)
    = \underline{\bf w}e^{\lambda t}$ \hspace{2mm}[{\it cf} \hspace{2mm} ODEs for which we try
    $y(x)~=~ce^{mx}$]. Find the general solution of (\ref{eq:3}) and
    explain the significance of the eigenvalues and eigenvectors of
    $A$.
    
    Prove that $(0,0)$ is stable $\iff$ Tr $A < 0$, Det $A > 0$.
    
    Explain how these results generalize to {\em any} singular point.

\vspace{8mm}

\item Consider the Lotka-Volterra equations for the population of an
  island of rabbits ($r \geq 0$) and sheep ($s \geq 0$). After the
  population numbers are scaled suitably, the sheep population
  satisfies $\dt{s} = s(2-r-s)$.
  
  In each of the following cases for the rabbit equation, find the
  critical points in the $(r,s)$ phase plane, classify them (as one of
  saddle, stable/unstable node/focus) and sketch a plausible phase
  portrait compatible with what you find. Make sure that you work out
  the eigenvectors (and that you know why you need to do this, and
  that you {\em use} them).

  In each case, state briefly the ecological implications of your
  findings. Do you see anything about the steady states on the axes
  that might at first seem odd? Explain why the trajectories here do
  not contradict common sense after all.

  \begin{enumerate}
    \item $\dt{r}=r(3-r-s)$
    \item $\dt{r}=r(3-2r-s)$
    \item $\dt{r}=r(3-r-2s)$
    \item $\dt{r}=r(3-2r-2s)$
  \end{enumerate}

\end{enumerate}

\newpage

\noindent
{\it Fredholm alternative (proof for finite rank kernels only).}

\begin{enumerate}
\setcounter{enumi}{2}

\item Consider the integral equation

  \begin{equation}
    y(x) = f(x) + \underbrace{\lambda \int_{a}^{b}K(x,t)y(t)dt}_{(*)}
    \label{eq:nhf}
  \end{equation}
  
  Suppose that the kernel $K$ is degenerate, {\it ie,} it can be
  written as a finite linear combination of products of a function of
  $x$ and a function of $t$. Write the preceding statement in
  mathematical notation.
  
  Use this to show that term $(*)$ in (\ref{eq:nhf}) can be written as
  a linear combination of known functions of $x$, with coefficients
  which are unknown constants $X_{j}$. Thence show that
  (\ref{eq:nhf}) may be written as a system of linear algebraic
  equations

  \begin{equation}
    (\mu I - A) \underline{\bf X} = \underline{\bf b}  \label{eq:nhf-alg}
  \end{equation}

  where $\underline{\bf X} = \left( \begin{array}{c} X_{1} \\ \vdots \\ X_{n}
  \end{array} \right)$, $\mu = \lambda^{-1}$ and $A$ and
$\underline{\bf b}$ should be given. State conditions under which
(\ref{eq:nhf-alg}) has

\begin{enumerate}
  \item a unique solution
  \item no solutions
  \item many solutions
\end{enumerate}

and explain why the solutions of (\ref{eq:nhf-alg}) for
$\underline{\bf X}$ are relevant to the solutions of (\ref{eq:nhf})
for $y(x)$.

Explain what we mean by the {\it eigenvalues} and {\it eigenfunctions}
of (\ref{eq:nhf}). Explain how these affect the possible solutions of
(\ref{eq:nhf})\footnote{{\it ie,} state the Fredholm alternative.}.

\vspace{8mm}

\item In the previous question, we worked through a method for
  investigating solutions of the general non-homogeneous Fredholm
  equation (\ref{eq:nhf}). In this question, we will apply these
  methods to the particular problem

  \begin{displaymath}
    y(x) = f(x) + \lambda \int_{-\frac{\pi}{2}}^{\frac{pi}{2}}
    K(x,t)y(t)dt
  \end{displaymath} 
  
  where $K(x,y) = \sin x \sin y + \sin x \cos y + \cos x \cos y$ and
  $f$ is continuous. 
  
  Show that the only characteristic value\footnote{Another term for
    eigenvalue.} of $K$ is 2/$\pi$, and find the corresponding space
  of characteristic functions\footnote{`Characteristic functions' is
    another term for `eigenfunctions'. You're being asked to give the
    general form of an eigenfunction.}. If $\lambda = 2/\pi$, what
  condition(s) does $f$ have to satisfy for a solution to exist?

  Solve the equation when $\lambda \neq 2/\pi$.

\end{enumerate}

\newpage

\noindent
{\it Hilbert-Schmidt theory for equations with symmetric kernels:
  orthonormal families, Bessel's inequality and eigenvalues, the
  expansion theorem (proof of the existence of an eigenvalue
  excluded).}

\begin{enumerate}
\setcounter{enumi}{4}

\item Consider the homogeneous Fredholm equation
    \begin{equation}
    y(x) = \lambda \int_{a}^{b}K(x,t)y(t)dt
    \label{eq:hf}
  \end{equation}

where $K$ is real-valued, continuous and symmetric.

\begin{enumerate}
  \item Explain what we mean by the eigenvalues and eigenfunctions of
    $K$. It can be shown that (\ref{eq:hf}) has at least one
    eigenvalue.
  \item Show that the eigenvalues are real, and that eigenfunctions
    corresponding to distinct eigenvalues are orthogonal.
  \item Show that every eigenvalue has at least one {\em real}
    eigenfunction.
  \item Show that each eigenvalue has {\em finite multiplicity}, {\em
      ie,} there can be only a {\em finite} linearly independent set
      of eigenfunctions corresponding to each eigenvalue.
  \item Show that (\ref{eq:hf}) has a countable number\footnote{Which
      may be finite.} of distinct eigenvalues.
  \item Use the above results to explain how we can enumerate a set of
    real orthonormal eigenfunctions $y_{1}, y_{2}, \ldots$
    corresponding to a set of real (not necessarily distinct)
    eigenvalues $\lambda_{1}, \lambda_{2}, \ldots$.
\end{enumerate}

Clearly state the Hilbert-Schmidt Expansion Theorem for a uniformly
convergent series expansion of the continuous function $f(x)$, where
\begin{displaymath}
f(x) = \int_{a}^{b} K(x,t) g(t) dt
\end{displaymath}
stating clearly the conditions required on $K$ and $g$, and making
clear how the coefficients of the series expansion are found.

\vspace{8mm}

\item Let $f(x)$ be a continuous real-valued function of period
  $2\pi$, and let $a_{n}$ be its Fourier cosine coefficients for $n
  \geq 0$:

  \begin{displaymath}
    a_{n} = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos nx dx
  \end{displaymath} 

Let $K(x,t) = f(x-t)$ for $x,t \in [0,2\pi]$. Show that if $a_{n} \neq
0$ then $\cos nx$ and $\sin nx$ are eigenfunctions of $K$ and state
the corresponding eigenvalues. You may assume that these eigenvalues
are the {\em only} eigenvalues, and that the corresponding $\cos nx$,
$\sin nx$ span the space of all eigenfunctions.

Assuming the Hilbert-Schmidt expansion theorem\footnote{Which in an
  isolated question you should state, but you've done this in the
  previous question; if you use a different form here, however, please
  state it.}, show that for any continuous $u(x)$ on $[0, 2\pi]$,

\begin{displaymath}
\int_{0}^{2\pi} f(x-t)u(t) dt = \frac{1}{2}A_{0} +
\sum_{n=1}^{\infty}A_{n} \cos nx + \sum_{n=1}^{\infty}B_{n}\sin nx
\end{displaymath}

where the series converge uniformly in $x$. Express $A_{n}$ and
$B_{n}$ in terms of $a_{n}$ and $u$.

\end{enumerate}

\newpage

\noindent
{\it Iterative methods and Neumann series applied to Fredholm integral
  equations.}

\begin{enumerate}
\setcounter{enumi}{6}

\item Consider the non-homogeneous Fredholm equation (\ref{eq:nhf}).
  Define the {\em iterated kernels} and explain their significance.
  State the solution of (\ref{eq:nhf}) in terms of the {\em Neumann
    series}, and sketch a proof that this is in fact the solution.
  Give a necessary and sufficient condition for uniform convergence of
  the Neumann series (the proof of this is not on your
    syllabus).

\vspace{8mm}

\item Consider (\ref{eq:nhf}) with $K(x,t)$ as in Question 4. Show
  that the iterated kernels are given by
\begin{displaymath}
K_{n}(x,y) = \left( \frac{\pi}{2} \right)^{n-1} (\sin x \sin y + n
\sin x \cos y + \cos x \cos y) 
\end{displaymath} 

Show directly ({\em ie,} not by quoting the `neccessary and
sufficient' condition above) that the Neumann series for $y$ converges
when $|\lambda| < 2/\pi$, and verify that the sum of the series agrees
with the solution you have already found in Question 4.

\end{enumerate}

\vspace{5mm}

\noindent
{\it Calculus of variations: Euler's equation as a necessary
  condition. Extension to a finite number of dependent functions and
  their derivatives, a finite number of independent variables,
  integral constraints and variations of boundary values. Applications
  to geometry, mechanics and quantum mechanics.}

\begin{enumerate}
\setcounter{enumi}{8}

\item Use the Fundamental Lemma of the Calculus of Variations (which
  you should state) to show that a function $y(x)$ which minimises
  \begin{displaymath}
    I:= \int_{a}^{b} F(x, y, y^{\prime}) \; dx
   \end{displaymath} 
   
   over all twice-continuously differentiable functions satisfying
   given boundary conditions, must also satisfy Euler's equation
   \begin{displaymath} 
     \dxbig \left( F_{y^{\prime}} \right) = F_{y}
   \end{displaymath} 

\vspace{5mm}

\item In geometrical optics, the light ray between two points $A$ and
  $B$ in three-dimensional space satisfies Fermat's principle; {\em
    ie,} it minimises
  \begin{displaymath} 
    I := \int_{A}^{B} \mu \; ds
  \end{displaymath} 
where $\mu$ is the refractive index (a positive function of position)
and $ds$ is the element of distance along the ray.

Suppose that $\mu$ depends only on the height $z$. Explain why the ray
between $A$ and $B$ will lie in the vertical plane containing them.

Now suppose that $\mu = \sqrt{z}$, and suppose that $A$, $B$ and the
ray between them lie in the region $z>0$. Use the calculus of
variations to show that the ray satisfies
  $z^{2} = (z - 2K)^{2} + (x - \lambda)^{2}$,
where $\lambda$ and $K \geq 0$ are constants determined by $A$ and
$B$.

If $A=(-a, 0, c)$ and $B=(a, 0, c)$, explain why we must have
$\lambda=0$, and write down an equation that $K$ must satisfy.

\begin{enumerate}
  \item If $a < c$ then the equation for $K$ has two solutions. Which
    of them gives the correct minimum value for $I$, and why? Find
    this minimum value.
  \item If $a > c$ then the equation for $K$ has no solution. What
    does this imply physically?
\end{enumerate}

\item
\begin{enumerate}
  \item Consider the problem of minimising
    \begin{displaymath} 
      I := \int_{a}^{b} F(x, y, z, y^{\prime}, z^{\prime}) \; dx
    \end{displaymath} 
    
    Sketch a proof that minimisers $y(x)$ and $z(x)$ must satisfy the
    Euler equations, which you should state.

  \item Consider minimising
    \begin{displaymath} 
      I := \int_{A} \int F(x, y, u, u_{x}, u_{y}) \;dx \;dy
    \end{displaymath} 
    over all twice-continuously-differentiable functions $u(x,y)$
    satisfying $u(x,y) = f(x,y)$ on $\partial A$. Sketch a proof that
    a minimiser $u$ must satisfy
    \begin{displaymath}
      \pdiffxbig \left[F_{u_{x}} \right] + \pdiffybig \left[ F_{u_{y}}
      \right] = F_{u}
    \end{displaymath} 
    stating conditions on $A$, $\partial A$, $F$ and $f$ for your
    proof to work, and pointing out where in the proof you use each
    condition.

    \item Consider minimising
      \begin{displaymath}
        I := \int_{t_{0}}^{t_{1}} F(x,y,z,\dot{x},\dot{y},\dot{z}) \;dt
      \end{displaymath}
      over curves $\underline{\bf r}(t) = (x(t),y(t),z(t))$ passing
      through $\underline{\bf r}(t_{0})$ and $\underline{\bf
      r}(t_{1})$, and lying completely in the surface $z=g(x,y)$.
    
    Show that the problem reduces to minimising
    \begin{displaymath}
      J := \int_{t_{0}}^{t_{1}} H(x,y,\dot{x},\dot{y}) \;dt
    \end{displaymath} 
    over curves $x(t)$ and $y(t)$ passing through $(x(t_{0}),
    y(t_{0}))$ and $(x(t_{1}), y(t_{1}))$, where $H$ is a function
    which you should state in terms of $F$.

\end{enumerate}

\vspace{8mm}

\item A point moves in the plane from $(x_{0}, y_{0})$ at time $t_{0}$
  to $(x_{1}, y_{1})$ at time $t_{1}$ in such a way that
  \begin{displaymath}
    I:=\int_{t_{0}}^{t_{1}} \left( \frac{a\dot{x}^{2} +
    2b\dot{x}\dot{y} + c\dot{y}^{2}}{2} - V \right) \;dt
  \end{displaymath}
  is minimised, where $a$, $b$, $c$ and $V$ are prescribed functions
  of $x$, $y$ and $t$. Write down the Euler equations for the problem.

  If $a$, $b$, $c$ and $V$ are independent of $t$, show that
  \begin{displaymath} 
  \frac{a\dot{x}^{2} + 2b\dot{x}\dot{y} + c\dot{y}^{2}}{2} + V
  \end{displaymath} 
  is constant during the motion.

  If the end-point conditions are changed so that $y(t_{1})$ is no
  longer prescribed to be $y_{1}$, what is the new boundary condition
  that must be applied to the Euler equations in order to find the
  minimising path?

\end{enumerate} 

\newpage

\noindent
{\it Sturm-Liouville theorem, with application to eigenfunction
  expansions. Bessel functions and Legendre, Hermite and Laguerre
  polynomials as examples of systems of eigenfunctions.
  Sturm-Liouville equation derived from a variational principle.}

\begin{enumerate}
\setcounter{enumi}{12}

\item Describe the part played in the theory of the Sturm-Liouville
  equation
  \begin{equation}
    Ly(x) + \lambda \rho(x)y(x)=0       \label{eq:sl}
  \end{equation}
  and its non-homogeneous counterpart
  \begin{equation}
    Ly(x) + \lambda \rho(x)y(x)=f(x)       \label{eq:nsl}
  \end{equation}
  by the theory of Green's functions, the Fredholm Alternative
  Theorem, the Hilbert-Schmidt Expansion Theorem and the Calculus of
  Variations.

\end{enumerate}

\vspace{2cm}

\begin{flushright}
\begin{Large}
Kate L Pugh, Hilary 1999
\end{Large}
\end{flushright}

\end{document} 
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