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November  2003, 3(4): 495-504. doi: 10.3934/dcdsb.2003.3.495

Using numerical experiments to discover theorems in differential equations

Joseph D. Fehribach 1,

1. 

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609-2247, United States

Received  December 2002 Revised  March 2003 Published  August 2003

This work explores the use of numerical experiments in two specific cases: (1) the discovery of two families of exact solutions to the elastic string equations, and approximately periodic solutions that appear to exist near pseudo-solutions formed from these families; (2) the study of the diffusion-reaction-conduction process in an electrolyte wedge (meniscus corner) of a current-producing porous electrode. This latter work establishes the well-posedness of the electrolyte wedge problem and provides asymptotic expansions for the current density and total current produced by such a wedge. The theme of this paper is the use of computing to discover a result that is difficult or impossible to find without a computer, but which once observed, can then be proven mathematically.
Keywords: Numerical experiments, maximum principle, electrolyte wedge problem, matched asymptotics., exact solutions, elastic string, periodic pseudosolutions, energy.
Mathematics Subject Classification: 35L65, 74J30, 35J2.
Citation: Joseph D. Fehribach. Using numerical experiments to discover theorems in differential equations. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 495-504. doi: 10.3934/dcdsb.2003.3.495
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