Finite-difference solutions are considered for the heat conduction equation in one space dimension subject to general boundary conditions involving linear combinations of the function and its space derivative. It is shown that under such conditions, instability can often arise even although "stable" formulae of the Crank-Nicolson type are used. In particular, the persistent error discussed by Parker and Crank (1964) is shown to be a weak case of this more serious instability.

Finite difference methods (including the Peaceman-Rachford method) are considered for the solution of the third boundary value problem for parabolic and elliptic equations. Conditions on the coefficients involved in the boundary conditions are obtained from the stability requirements of the difference methods and shown to coincide with those necessary for asymptotic stability of the differential system.

A self-adjoint parabolic equation in one space variable is considered, under boundary conditions which involve the function and its space derivative. A type of numerical instability can arise, which is traceable to the boundary conditions, and which is caused by the existence of unbounded solutions of the original differential equation.

I learned to play squash from Pat. At the end of one of Pat's 20 mile cycle runs I would join him on the squash court and he could still run energetically. I was determined to beat him within 20 sessions and I just about managed it. As well as being a colleague at St Andrews we even shared an office close to the main building (47 North Street) which is long since demolished.

In [Finite difference solution of the third boundary problem in elliptic and parabolic equations], the author discussed the numerical solution of the heat conduction equation in an open rectangular region, under boundary conditions involving a linear combination of the function and its normal derivative. It was shown that the instability which was observed in the difference methods examined, could be traced to the existence of solutions of the differential equation which grew, asymptotically, with time. This numerical instability was important in the solution of the heat equation only for large values of the time, and so did not affect calculations which were carried out over a few time steps. But, in the numerical solution of elliptic equations by iteration, such asymptotic instability will prevent convergence of the iterative process to the original system of equations. It is the purpose of this note to demonstrate this fact, and also to discuss the solution of Laplace's and Poisson's equations, when the Laplacian operator is singular, in a sense to be defined. It will be shown that, for certain boundary conditions, the numerical solution of Laplace's equation is best obtained by direct methods, rather than by iterative methods. In addition, a condition for the existence of a solution of the singular problem is obtained.

This has been an exciting year for the Department of Mathematics and Statistics, as the contents of this year's Chase Report show. The year seems to have raced by, at a breathtaking pace and, although it feels as if I have just begun my term as Chair, I am now one-third of the way through. ... During this spring and summer we will be hosting no less than 4 national conferences, with several hundred participants from many countries.

This is my second year as Chair of the Department of Mathematics and Statistics, a busy year with many changes taking place, and many new faces about to arrive. ...

It is said that time passes quickly when you are having fun. Whether that means that time drags when you aren't, I do not know, but I will say that the last three years have gone by incredibly quickly. My term as chair ends on June 30, when Dr Karl Dilcher takes over. ... Our Honours programmes continue to grow. Especially gratifying is the increase in enrolment in the Statistics Honours programme, which is at the highest it has ever been. The Graduate programmes are also growing, and we are attracting a number of excellent students in both Mathematics and Statistics. Our Outreach Programmes report describes a very good year of activities by the Outreach team, involving high school students and teachers. All in all, I think that the Department has had a wonderfully successful year, and I have had an enjoyable term as chair. I shall miss being in the centre of all the activity. I have had a good three years, made so by the hard work of the members of the Department, efforts that it is easy to take for granted because nobody complains! I would like to thank everyone for their help, advice and understanding, and for the willingness to take on tasks uncomplainingly, and for carrying them out efficiently. I think few chairs are so lucky as I have been.

Pat Keast's 3-year tenure as chair marked a period of rapid change, with numerous retirements and some resignations, and with just as many excellent new appointments. Such times of change often bring with them the danger of turbulences and uncertainties, but under Pat's capable leadership this beginning transition in the department's demographics went very smoothly, and this is still the friendly place it has always been.

I was a mature student at Dalhousie, and knew Professor Keast. When he discovered I was soon graduating he was overjoyed and pleased for me. He had a passion for teaching and opened up the minds of those who were eager to learn. It came naturally to him. He was a very popular professor amongst students and colleagues and received much respect from everyone.

Dr Keast. this is what I called him. When I graduated he told me I could call him Pat. I could not do it. He'll always be Dr Keast to me. This belies the tenderness I feel for this man who believed in me and was a feminist ahead of his time. He was strong. He was dedicated. He was brilliant and funny. He was absolutely relentless in his demand for commitment and curiosity. I have many good memories. Life shaping experiences. Thank you Dr Keast. Thank you Pat. You were a light, a warrior of the most gentle kind.

Pat Keast of Broxburn, now living in Halifax, Nova Scotia, was reading a local newspaper report about a driver accused by the police of illegally having alcohol in his car. He was found guilty, but Pat reckons he knew why the driver felt he had to have a go and have his day in court. The driver was called David Eisnor, and the constable was Walter Goliath. Sometimes a sling is not enough.