J Math Chem (2010) 48:145–158 DOI 10.1007/s10910-009-9646-x ORIGINAL PAPER Numerical modeling of oxygen diffusion in cells with Michaelis-Menten uptake kinetics Pedro M. Lima · Luisa Morgado Received: 7 October 2009 / Accepted: 25 November 2009 / Published online: 16 December 2009 © Springer Science+Business Media, LLC 2009 Abstract A class of singular boundary value problems is studied, which models the oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics. Suitable singular Cauchy problems are considered in order to determine one-parameter families of solutions in the neighborhood of the singularities. These families are then used to construct stable shooting algorithms for the solution of the considered problems and also to propose a variable substitution in order to improve the convergence order of the finite difference methods. Numerical results are presented and discussed. Keywords m-Laplacian · Singular Cauchy problem · Singular boundary value problem · Shooting method · Finite difference method 1 Introduction It is known that boundary value problems of the form y ′′ (x ) + 2 x y ′ (x ) = f ( y ), 0 < x < 1 (1) P. M. Lima (B ) Cemat and Dep. Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal e-mail: plima@math.ist.utl.pt L. Morgado Cemat, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal L. Morgado Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, Quinta de Prados, 5001-801 Vila Real, Portugal 123