Donald Estep

An Essay on the Role of Numerical Analysis in Traditional Mathematics Education

 

This essay is excerpted from Computational Differential Equations, K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Cambridge University Press, 1996, ISBN 91-44-49311-8.

 

Mathematical modeling using differential and integral equations has formed the basis of science and engineering since the creation of calculus by Leibniz and Newton. Mathematical modeling has two basic dual aspects: one symbolic and the other constructive-numerical, which reflect the duality between the infinite and the finite, or the continuum and the discrete. The two aspects have been closely intertwined throughout the development of modern science from the development of calculus in the work of Euler, Lagrange, Laplace and Gauss into the work of von Neumann in our time. For example, Laplace's monumental {Mécanique Céleste} in five volumes presents a symbolic calculus for a mathematical model of gravitation taking the form of Laplace's equation, together with massive numerical computations giving concrete information concerning the motion of the planets in our solar system.

 

However, beginning with the search for rigor in the foundations of calculus in the 19th century, a split between the symbolic and constructive aspects gradually developed. The split accelerated with the invention of the electronic computer in the 1940s, after which the constructive aspects were pursued in the new fields of numerical analysis and computing sciences, primarily developed outside departments of mathematics. The unfortunate result today is that symbolic mathematics and constructive-numerical mathematics by and large are separate disciplines and are rarely taught together. Typically, a student first meets calculus restricted to its symbolic form and then much later, in a different context, is confronted with the computational side. This state of affairs lacks a sound scientific motivation and causes severe difficulties in courses in physics, mechanics and applied sciences building on mathematical modeling. The difficulties are related to the following two basic questions: (i) How to get applications into mathematics education? (ii) How to use mathematics in applications? Since differential equations are so fundamental in mathematical modeling, these questions may be turned around as follows: (i) How can we teach differential equations in mathematics education? (ii) How can we use differential equations in applications?

 

Traditionally, the topic of differential equations in basic mathematics education is restricted to separable scalar first order ordinary differential equations and constant coefficient linear scalar nth order equations for which explicit solution formulas are presented, together with some applications of separation of variables techniques for partial differential equations like the Poisson equation on a square. Even slightly more general problems have to be avoided because the symbolic solution methods quickly become so complex. Unfortunately, the presented tools are not sufficient for applications and as a result the student must be left with the impression that mathematical modeling based on symbolic mathematics is difficult and only seldom really useful. Furthermore, the numerical solution of differential equations, considered with disdain by many pure mathematicians, is often avoided altogether or left until later classes, where it is often taught in a "cookbook" style and not as an integral part of a mathematics education aimed at increasing understanding. The net result is that there seems to be no good answer to the first question in the traditional mathematics education.

 

The second question is related to the apparent principle of organization of a technical university with departments formed around particular differential equations: mechanics around Lagrange's equation, physics around Schrödinger's equation, electromagnetics around Maxwell's equations, fluid and gas dynamics around the Navier-Stokes equations, solid mechanics around Navier's elasticity equations, nuclear engineering around the transport equation, and so on. Each discipline has largely developed its own set of analytic and numerical tools for attacking its special differential equation independently and this set of tools forms the basic theoretical core of the discipline and its courses. The organization principle reflects both the importance of mathematical modeling using differential equations and the traditional difficulty of obtaining solutions.

 

Both of these questions would have completely different answers if it were possible to compute solutions of differential equations using a unified mathematical methodology simple enough to be introduced in the basic mathematics education and powerful enough to apply to real applications. In a natural way, mathematics education would then be opened to a wealth of applications and applied sciences could start from a more practical mathematical foundation. Moreover, establishing a common methodology opens the possibility of exploring ``multi-physics'' problems including the interaction of phenomena from solids, fluids, electromagnetics and chemical reactions, for example.

 

In our textbooks on computational differential equations, we seek to present such a unified mathematical methodology for solving differential equations numerically based on a principle of a fusion of mathematics and computation. The backbone of our approach is a new unified presentation of numerical solution techniques for differential equations based on Galerkin methods. But the point is not to write a textbook that is easily accessible for students in a specialized course on the numerical solution of differential equations. The material in this book takes time to digest, as much as the underlying mathematics itself. In our opinion, the optimal course involves the gradual integration of the material covered in our textbooks into the traditional mathematics curriculum right from the beginning.

 

We emphasize that we are not advocating the study of computational algorithms over the mathematics of calculus and linear algebra; it is always a fusion of analysis and numerical computation that appears to be the most fruitful. The material that we would like to see included in the mathematics curriculum offers a concrete motivation for the development of analytic techniques and mathematical abstraction. Computation does not make analysis obsolete, but gives the analytical mind a focus. Furthermore, the role of symbolic methods changes. Instead of being the workhorse of analytical computations requiring a high degree of technical complexity, symbolic analysis may focus on analytical aspects of model problems in order to increase understanding and develop intuition.