Using numerical experiments to discover theorems in differential equations

Joseph D. Fehribach

doi:10.3934/dcdsb.2003.3.495

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2003, Volume 3, Issue 4: 495-504. Doi: 10.3934/dcdsb.2003.3.495

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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

Received: November 30, 2002

Revised: February 28, 2003

Published: October 2003

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Abstract

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.

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Mathematics Subject Classification: 35L65, 74J30, 35J25.

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