Abstract
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems of PDEs, including schemes based on differential algebra and geometric approaches including the theory of exterior differential systems [5] and the so-called “formal theory” [4] built on the jet bundle formalism. This paper is a brief introduction to jet bundles, focusing on the completion of systems to equivalent involutive systems for which power series solutions may be constructed order by order. We will not consider the mathematical underpinnings of involution (which lie in the theory of combinatorial decompositions of polynomial modules [2,3]) nor other applications of the theory of jet bundles such as the theory of symmetries of systems of PDEs [6] or discretisation schemes based on discrete approximations to jet bundles [1].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
J. Marsden, G. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys. 199 (1998), 351–395.
J. Calmet, M. Hausdorf and W. Seiler, A Constructive Introduction to Involution, Proc. ISACA 2000, World Scientific, Singapore, 2000
W. Seiler, A Combinatorial Approach to Involution and δ-Regularity, Preprint, Universität Mannheim, 2000
Seiler, Analysis and Application of the Formal Theory of Partial Differential Equations, School of Physics and Materials, Lancaster University, 1994.
Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick, Analysis, Manifolds and Physics Part 1, North-Holland, 1982
P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 2000
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media New York
About this chapter
Cite this chapter
Baker, R., Doran, C.J.L. (2002). Jet Bundles and the Formal Theory of Partial Differential Equations. In: Dorst, L., Doran, C., Lasenby, J. (eds) Applications of Geometric Algebra in Computer Science and Engineering. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0089-5_12
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0089-5_12
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6606-8
Online ISBN: 978-1-4612-0089-5
eBook Packages: Springer Book Archive