Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2012, Article ID 162034, 15 pages doi:10.1155/2012/162034 Research Article On the Computation of Blow-up Solutions for Semilinear ODEs and Parabolic PDEs P. G. Dlamini and M. Khumalo Department of Mathematics, University of Johannesburg, Cnr Siemert & Beit Streets, Doornfontein 2028, South Africa Correspondence should be addressed to M. Khumalo, mkhumalo@uj.ac.za Received 31 August 2011; Revised 8 November 2011; Accepted 8 November 2011 Academic Editor: Robertt A. Fontes Valente Copyright q 2012 P. G. Dlamini and M. Khumalo. 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. We introduce an adaptive numerical method for computing blow-up solutions for ODEs and well- known reaction-diffusion equations. The method is based on the implicit midpoint method and the implicit Euler method. We demonstrate that the method produces superior results to the adaptive PECE-implicit method and the MATLAB solver of comparable order. 1. Introduction Reaction-diffusion equations model a wide range of problems in physics, biology, and chemistry. They explain how the concentration of one or more substances distributed in space changes under the influence of two processes: chemical reactions and diffusion. These sub- stances can be basic particles in physics, bacteria, molecules, cells, and so forth. The substances reside in a region Ω ⊂ R d ,d ≥ 1. The reaction-diffusion equation is a semilinear parabolic partial differential equation of the form u t t, x - Δut, x f t, x, u, t> 0,x ∈ Ω ⊂ R d , u0,x u 0 x ≥ 0, x ∈ Ω, ut, x 0, t> 0,x ∈ ∂Ω 1.1 Equation 1.1 can be viewed as a heat conduction problem, where ux, t is the temperature of a substance in a bounded domain Ω ⊂ R d and f t, x, u represents a heat