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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 713798, 14 pages
http://dx.doi.org/10.1155/2012/713798
Research Article

Matrix Exponentiation and the Frank-Kamenetskii Equation

Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

Received 29 August 2012; Revised 4 November 2012; Accepted 5 November 2012

Academic Editor: Oluwole Daniel Makinde

Copyright © 2012 E. Momoniat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Long time solutions to the Frank-Kamenetskii partial differential equation modelling a thermal explosion in a vessel are obtained using matrix exponentiation. Spatial derivatives are approximated by high-order finite difference approximations. A forward difference approximation to the time derivative leads to a Lawson-Euler scheme. Computations performed with a BDF approximation to the time derivative and a fourth-order Runge-Kutta approximation to the time derivative are compared to results obtained with the Lawson-Euler scheme. Variation in the central temperature of the vessel corresponding to changes in the shape parameter and Frank-Kamenetskii parameter are computed and discussed.