Abstract
The general non-linear first-order partial differential equation requires a deeper analysis than its linear and quasi-linear counterparts. Instead of a field of characteristic directions, the non-linear equation delivers a one-parameter family of directions at each point of the space of dependent and independent variables. These directions subtend a local cone-like surface known as the Monge cone. Augmenting the underlying space by adding coordinates representing the first partial derivatives of the unknown field, however, it is possible to recover most of the features of the quasi-linear case so that, ultimately, even the solution of the general non-linear equation can be reduced to the integration of a system of ordinary differential equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In honour of Gaspard Monge (1764–1818), the great French mathematician and engineer, who made seminal contributions to many fields (descriptive geometry, differential geometry, partial differential equations).
- 2.
There are some mathematical subtleties. For example, we are tacitly assuming that the partial derivatives of the function F with respect to the arguments \(u_x\) and \(u_y\) do not vanish simultaneously. Also, we are considering a small range of tangent planes, where one of the slopes is a single-valued function of the other.
- 3.
See previous footnote.
- 4.
The Monge cone is a particular case of an envelope of surfaces. In Box 5.1 we present a more general derivation.
- 5.
We follow the terminology of [3].
- 6.
See [3].
- 7.
This example is suggested as an exercise in [4], p. 66.
- 8.
References
Duff GFD (1956) Partial differential equations. Toronto University Press, Toronto
Garabedian PR (1964) Partial differential equations. Wiley, London
John F (1982) Partial differential equations. Springer, Berlin
Sneddon IN (1957) Elements of partial differential equations. McGraw-Hill, New York (Republished by Dover (2006))
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Epstein, M. (2017). The Genuinely Nonlinear First-Order Equation. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-55212-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-55211-8
Online ISBN: 978-3-319-55212-5
eBook Packages: EngineeringEngineering (R0)