Abstract
In this report, a first order nonlinear partial differential equation with a parameter dependent initial condition is examined. Even though an analytic solution of the equation is determined, a surprising bifurcation phenomenon is discovered via computer graphics. This “computer-discovered” bifurcation, in turn, leads to further mathematical analysis and deeper geometric understanding of the solution. Indeed, this is a simple example of an elementary catastrophe (in the sense of Thorn) and demonstrates the usefulness of numerical computations in providing qualitative information even in the presence of exact solutions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
G. F. Carey, B. N. Jiang, and R. Showalter.A Regularization-Stabilization Technique for Nonlinear Conservation Equation Computations ,Num. Methods for Par. Diff. Eqns., Vol. 4, No. 3, pp. 165–171, Fall 1988.
P. J. Costa.An Explicit Solution of a First Order Nonlinear Partial Differential Equation. MTT Lincoln Laboratory Technical Memorandum No. 34L-0016, 9 June 1986.
I. Ekeland.Mathematics and the Unexpected ,University of Chicago Press, 1988.
F. John.Partial Differential Equations ,Fourth Edition, Springer-Verlag, 1982.
G.Strang.Introduction to Applied Mathematics, Wellesley-Cambridgc Press ,1986.
G. Strang. Private communication, 1988.
E. C. Zachmanoglou and D. Thoe.Introduction to Partial Differential Equations ,Williams & Wilkins Company, 1976.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag New York Inc.
About this paper
Cite this paper
Costa, P.J., Westlake, R.H. (1989). An Example of Computer Enhanced Analysis. In: Kaltofen, E., Watt, S.M. (eds) Computers and Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9647-5_13
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9647-5_13
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97019-6
Online ISBN: 978-1-4613-9647-5
eBook Packages: Springer Book Archive