Many important physical processes may be modeled in terms of a
system of partial differential equations. Solutions to such a system
are then sought satisfying initial and boundary data according to the dynamics
and geometry of the process under consideration. Unlike the general theory
describing systems of ordinary differential equations there exists no
comparable theory that can accommodate arbitrary systems of partial
differential equations and analytic methods traditionally resort to ad hoc
methods that are tailored to specific situations. This lack of a general theory
is particularly acute for overdetermined systems that often arise when modeling
natural processes, such as problems in continuum mechanics and fluid flow.
The essential non-linearities inherent in the physics can lead to the growth of instabilities and the formation of shocks and special methods are necessary to control the development of solutions in such circumstances. While the use of numerical methods is natural for well posed problems considerable mathematical investigation is a necessary precursor for these to be efficient.
We have been studying partial differential in terms of the theory of exterior differential systems. In particular we have developed a new algorithm for solving the classical Cauchy problem, where consistent data is prescribed at an initial instant and its evolution in time is sought. The approach relies on reducing the problem to a system of ordinary differential equations which are much easier to handle in general. One may regard the manifold of initial data as forming a "surface" in a large space of variables on which we determine a vector field. Each data point on the initial "surface" is then dragged into the space of variables along the flow lines of the vector field to yield a solution satisfying the initial conditions. The main virtue of our approach over many traditional ones is that the determination of the evolution vector is algebraic. Thus the method can be coded into a symbolic algebra language and effectively automated on a machine.
For physical systems this is essential since the determination of the characteristic vector may require the manipulation of huge numbers of overdetermined algebraic equations. Once the algebraic part of the problem is completed the integration of the ordinary differential system follows in a straightforward manner. It is at this stage that deterministic numerical procedures may be contemplated rather that the traditional discretisation of the original problem.
We have exploited this approach to study a number of inherently non-linear problems. These have included the vibrations of stiff elastic fibers and membranes, the relativistic motion of charged gyroscopes and the thermal properties of fast flowing viscous fluids.
The basic idea of treating partial differential equations in this geometrical fashion provides conceptual clarifications and offers access to the powerful methods of differential geometry. From a practical viewpoint it enables one to examine any proposed system of partial differential equations prior to time consuming numerical analysis. Such equations may have inherent inconsistencies or they may be incomplete. Our methods, which are valid for both linear and non-linear systems, enable one to check for consistency and find any missing equations necessary to bring the system intoinvolution. Furthermore they provide a precise specification of the number of arbitrary functions that exist in the general solution. These techniques also yield adapted coordinates that often dramatically accelerate the process of analysis and in some cases even transform the system to a linear one. Such simplifications offer economic advantages when coupled with the automated exterior methods built into our algorithms.