Differential Equations and Computation

J. C. Butcher

Differential equations are seen in all of nature, since they represent the behaviour of quantities which change with time at a rate determined by their current value. The growth of bacteria, the variation of nitrous oxide concentration in the atmosphere and the motion of planets are all expressed in terms of differential equations. To understand and predict physical phenomena requires "solving" differential equations. Because this is hardly ever possible in the sense of finding a formula for the solution, computational mathematics comes into play.

The part of mathematics I have enjoyed working on for the last 40 or more years has centred on finding conditions for various types of numerical methods to have properties that make them suitable for finding approximate solutions for differential equations as accurately and efficiently as possible. Even when I started on this subject, I was delighted by the connections that came to light between what I was trying to do and other parts of mathematics. As I have progressed further, such interesting relationships with other parts of mathematics and with other sciences have kept suggesting themselves. Within the last few years an algebraic system that I devised, more than 30 years ago, to study numerical accuracy has been rediscovered because of its application to the calculation of what are known as "Feynman integrals"; these are used to analyse certain aspects of nuclear reactions.

Over the years, I have also become interested in a lighter side of mathematics, concerned with solving problems in elementary mathematics, or explaining their solution to other people. I write two series of "recreational mathematics" articles which appear regularly in the New Zealand Mathematics Magazine and in the Newsletter of the New Zealand Mathematical Society. Some of the articles in these collections can be found in www.math.auckland.ac.nz/~butcher/miniature/index.html

It is difficult to say much about numerical methods for solving differential equations without making use of ideas from graph theory, combinatorics, abstract algebra, real and complex analysis and linear algebra. Hence, I will conclude this article by recalling an anecdote about the famous Indian mathematician S. K. Ramanujan. During an illness in England, he was visited by his friend and colleague G. H. Hardy. Ramanujan asked Hardy the number of his taxi in the hope that this would be an interesting number. Hardy apologised and told him that the number was 1729 - a number he regarded as boring. Ramanujan immediately said that 1729 can be written as the sum of two cubes in two different ways: 1729 = 10^3+9^3 = 12^3+1^3. Moreover, it is the lowest positive integer that can be written in this way. I thought I would try to find all solutions to this problem. This sort of equation, where the only allowed solutions are made up from integers, is known as a "Diophantine equation". For example, another solution is 39312 = 33^3+15^3 = 34^3+2^3 but if negative integers are permitted there is a very simple solution 91= 4^3+3^3 = 6^3 - 5^3. To learn more about my attempt to find all solutions to the ``Hardy's Taxi" problem, see Mathematical Miniature 9 on my website.

I don't know why Ramanujan became a mathematican, even though this required overcoming many obstacles to achieve this goal. Even less do I know why those of us with a fraction of his ability, want to follow in this same path as best we can. All I can suppose is that mathematics is a basic human need like music or literature or friendship and everyone feels the need for mathematics to some extent.