A partial difference equation and a minimal surface

J. H. Clarke and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 3, 1997, Number 3, Pages 170–172
Full paper (PDF, 216 Kb)

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Authors and affiliations

J. H. Clarke
University of Technology, Sydney, 2007, Australia

A. G. Shannon
University of Technology, Sydney, 2007, Australia

Abstract

Difference equations are often convenient in the mathematical modelling of medical problems because many physiological properties, such as the measurement of plasma glucose levels, tend to be assessed at discrete time intervals. Ollerton and Shannon [4] describe one such linear difference equation in detail in the context of the development of an artificial beta-cell. A partial non-linear difference equation which arose in a similar context is given by

(1 + u_i,j+i² − 2u_i,ju_i,j+1 + u_i,j²)(u_i+2,j − 2u_i+1,j + u_i,j) − 2((u_i+1,j − u_i,j) u_i,j+1 − (u_i+l,j − u_i,j) u_i,j)
+ (l + u_i+1,j² − 2u_i+l,ju_i,j + u_i,j²)(u_i,j+2 − 2u_i,j+1 + u_i,j) = 0.

The purpose of this note it to convert this difference equation into the corresponding partial differential equation and to examine the resulting minimal surface. This is done formally without digressing into the issues associated with mapping from the integers to the reals.

References

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3. L P Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York, 1966, p.251.
4. R L Ollerton and A G Shannon, Difference equations and an artificial pancreatic beta-cell, International Journal of Mathematical Education in Science and Technology, 22(4), 1991: 545-554.
5. H Skala, Contour maps – a visual experience, The College Mathematics Journal, 22(3), 1991: 241-244.