Partial Differential Equations and Group Theory

The formal theory of systems of partial differential equations (PDEs) was developed by D.C. Spencer in the USA between 1960 and 1975. It studies the solution spaces of systems of PDEs without especially integrating them. It also allows the study of Lie pseudogroups, that is groups of transformation solutions of systems of PDEs. Although this work supersedes the classical approaches of M. Janet and E. Cartan, it is still largely unknown by mathematicians and has never been used by physicists. This book provides a self-contained introduction to these methods, with illustrations and specific examples coming from many branches of physics, the engineering sciences and applied mathematics. The algorithms involved are presented in a way that allows the use of computer algebra for the intrinsic study of nonlinear PDEs. The book also presents the group-theoretical unification of the finite element methods for elasticity, heat and electromagnetism. It contains the material of an intensive course and can be used for a one-year course at graduate level. It is also suitable for researchers in mathematics, mathematical physics, computer algebra, control theory and theoretical mechanics.