A simple method for obtaining good bounds for solutions of reaction-diffusion equations with nonlinear kinetics

✍ Scribed by A.A. Parshotam; S.M. Rao Bhamidimarri; G.C. Wake

• Publisher: Elsevier Science
• Year: 1991
• Tongue: English
• Weight: 574 KB
• Volume: 46
• Category: Article
• ISSN: 0009-2509
• DOI: 10.1016/0009-2509(91)80115-f

✦ Synopsis

kinetic model is proposed for a surface-supported biological film (bioparticle) employed in many biological processes. The equations that govern kinetic and diffusion-controlled substrate uptake by the attached organisms are invariably nonlinear and analytical solutions if any are impossible to find. It is therefore desirable to determine approximate analytical solutions or, failing this, bounds for the exact numerical solution. This article represents an attempt to provide increasingly better bounds to such equations by a linearization technique. An iterative scheme relying on the maximum principle is presented for obtaining upper and lower bounds to solutions to resulting nonlinear reaction-dilXrsion equations. In particular, a spherical bioparticle with Michaelis-Menten-type reaction kinetics is considered. Its application to more general equations is also discussed. These bounds are in good agreement with the numerical solutions obtained by shooting and finite-difference procedures. The method, which can easily be generalized to other geometries, is relatively simple to use and converges rapidly to a very good upper and lower bound.