Many technological applications need of differential models either for their microscopic or macroscopic description and, frequently, they involve non-linear equations. There is a class of differential models, i.e. the diffusive-reaction type models that have a very large number of applications in different bio-engineering problems. These include transport and reaction in biological processes, heat and mass transfer in more than one phase, and electrostatic potentials in many current molecular processes, just to name a few. Solutions to these models are either by linear approximations of the non-linear source, by other approximations of such non-linearity, or by numerical methods that frequently are heavily dependent of the mesh size for a successful convergence. In this contribution, an analytical-logarithmic approach, proposed by Oyanader and Arce, is revisited to extend its application to more complex modelling of bio-systems for different geometries. Due to the lack of general methods to derive analytical solutions for second order differential models with non-linear sources and constant, this study has focused on developing an efficient and economical procedure to obtain a formal analytical solution for such models that is used in a simple predictor-corrector approach to predict the correct solution. The proposed extended method involves the use of a recursive function, ÆAO
, of the nonlinear dependable variable that work as a corrector function. The procedure converts the nonlinear ordinary differential equation in a simpler pseudo-linear ordinary differential equation whose analytical solution is later modified by means of the ÆAO
correction function to obtain the correct solution. Several numerical examples will be presented to illustrate the method.
(*) Oyanader, M., Arce, P., âA New and Simpler Approach for the Solution of the Electrostatic Potential Differential Equation. Enhanced Solution for Planar, Cylindrical and Annular Geometries,â Journal of Colloid and Interface Science, 2005, 284, 315.